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% The Project Gutenberg EBook of Space--Time--Matter, by Hermann Weyl %
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% This eBook is for the use of anyone anywhere at no cost and with %
% almost no restrictions whatsoever. You may copy it, give it away or %
% re-use it under the terms of the Project Gutenberg License included %
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% Title: Space--Time--Matter %
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% Author: Hermann Weyl %
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% Translator: Henry L. Brose %
% %
% Release Date: June 21, 2013 [EBook #43006] %
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% Language: English %
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The Project Gutenberg EBook of Space--Time--Matter, by Hermann Weyl
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org
Title: Space--Time--Matter
Author: Hermann Weyl
Translator: Henry L. Brose
Release Date: June 21, 2013 [EBook #43006]
Language: English
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*** START OF THIS PROJECT GUTENBERG EBOOK SPACE--TIME--MATTER ***
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\textbf{\Huge SPACE---TIME---MATTER} \\
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\footnotesize
BY \\[12pt]
\textbf{\LARGE HERMANN WEYL}
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\footnotesize
TRANSLATED FROM THE GERMAN BY \\
\normalsize
HENRY L. BROSE
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\footnotesize
WITH FIFTEEN DIAGRAMS
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METHUEN \& CO. LTD. \\
36 ESSEX STREET W.C. \\
LONDON
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\textit{First Published in 1922}
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\PageSep{v}
\SectTitle{FROM THE AUTHOR'S PREFACE TO
THE FIRST EDITION}
\First{Einstein's} Theory of Relativity has advanced our
ideas of the structure of the cosmos a step further. It
is as if a wall which separated us from Truth has
collapsed. Wider expanses and greater depths are now exposed
to the searching eye of knowledge, regions of which we
had not even a presentiment. It has brought us much nearer
to grasping the plan that underlies all physical happening.
Although very recently a whole series of more or less
popular introductions into the general theory of relativity has
appeared, nevertheless a systematic presentation was lacking.
I therefore considered it appropriate to publish the following
lectures which I gave in the Summer Term of~1917 at the
\textit{Eidgen.\ Technische Hochschule} in Zürich. At the same time
it was my wish to present this great subject as an illustration
of the intermingling of philosophical, mathematical, and
physical thought, a study which is dear to my heart. This
could be done only by building up the theory systematically
from the foundations, and by restricting attention throughout
to the principles. But I have not been able to satisfy these
self-imposed requirements: the mathematician predominates
at the expense of the philosopher.
The theoretical equipment demanded of the reader at the
outset is a minimum. Not only is the special theory of relativity
dealt with exhaustively, but even Maxwell's theory and
analytical geometry are developed in their main essentials.
This was a part of the whole scheme. The setting up of the
Tensor Calculus---by means of which, alone, it is possible to
\PageSep{vi}
express adequately the physical knowledge under discussion---occupies
a relatively large amount of space. It is therefore
hoped that the book will be found suit able for making physicists
better acquainted with this mathematical instrument, and
also that it will serve as a text-book for students and win
their sympathy for the new ideas.
\Signature{HERMANN WEYL}
{Ribbitz in Mecklenburg}
{\textit{Easter}, 1918}
\SectTitle{PREFACE TO THE THIRD EDITION}
\First{Although} this book offers fruits of knowledge in a
refractory shell, yet communications that have reached
me have shown that to some it has been a source of
comfort in troublous times. To gaze up from the ruins of
the oppressive present towards the stars is to recognise the
indestructible world of laws, to strengthen faith in reason, to
realise the ``harmonia mundi'' that transfuses all phenomena,
and that never has been, nor will be, disturbed.
My endeavour in this third edition has been to attune this
harmony more perfectly. Whereas the second edition was
a reprint of the first, I have now undertaken a thorough
revision which affects Chapters II~and IV above all. The
discovery by Levi-Civita, in~1917, of the conception of infinitesimal
parallel displacements suggested a renewed examination
of the mathematical foundation of Riemann's geometry.
The development of pure infinitesimal geometry in Chapter~II,
in which every step follows quite naturally, clearly, and
necessarily, from the preceding one, is, I believe, the final
result of this investigation as far as the essentials are concerned.
Several shortcomings that were present in my first
account in the \Title{Mathematische Zeitschrift} (Bd.~2, 1918) have
now been eliminated. Chapter~IV, which is in the main
devoted to Einstein's Theory of Gravitation has, in consideration
of the various important works that have appeared in the
meanwhile, in particular those that refer to the Principle of
Energy-Momentum, been subjected to a very considerable
\PageSep{vii}
revision. Furthermore, a new theory by the author has been
added, which draws the physical inferences consequent on the
extension of the foundations of geometry beyond Riemann,
as shown in Chapter~II, and represents an attempt to derive
from world-geometry not only gravitational but also electromagnetic
phenomena. Even if this theory is still only in its
infant stage, I feel convinced that it contains no less truth
than Einstein's Theory of Gravitation---whether this amount
of truth is unlimited or, what is more probable, is bounded by
the Quantum Theory.
I wish to thank Mr.~Weinstein for his help in correcting
the proof-sheets.
\Signature{HERMANN WEYL}
{Acla Pozzoli, near Samaden}
{\textit{August}, 1919}
\SectTitle{PREFACE TO THE FOURTH EDITION}
\First{In} this edition the book has on the whole preserved its
general form, but there are a number of small changes and
additions, the most important of which are: (1)~A paragraph
added to Chapter~II in which the problem of space is
formulated in conformity with the view of the Theory of
Groups; we endeavour to arrive at an understanding of the
inner necessity and uniqueness of Pythagorean space metrics
based on a quadratic differential form. (2)~We show that the
reason that Einstein arrives necessarily at uniquely determined
gravitational equations is that the scalar of curvature is the
only invariant having a certain character in Riemann's space.
(3)~In Chapter~IV the more recent experimental researches
dealing with the general theory of relativity are taken into consideration,
particularly the deflection of rays of light by the
gravitational field of the sun, as was shown during the solar
eclipse of 29th~May, 1919, the results of which aroused great
interest in the theory on all sides. (4)~With Mie's view of
matter there is contrasted another (\textit{vide} particularly §\,32 and
§\,36), according to which matter is a limiting singularity of
\PageSep{viii}
the field, but charges and masses are force-fluxes in the field.
This entails a new and more cautious attitude towards the
whole problem of matter.
Thanks are due to various known and unknown readers for
pointing out desirable modifications, and to Professor Nielsen
(at Breslau) for kindly reading the proof-sheets.
\Signature{HERMANN WEYL}
{Zürich, \textit{November}, 1920}{}
\PageSep{ix}
\newpage
\SectTitle{TRANSLATOR'S NOTE}
\First{In} this rendering of Professor Weyl's book into English,
pains have been taken to adhere as closely as possible to
the original, not only as regards the general text, but also
in the choice of English equivalents for technical expressions.
For example, the word \emph{affine} has been retained. It is used
by Möbius in his \Title{Der Barycentrische Calcul}, in which he
quotes a Latin definition of the term as given by Euler.
Veblen and Young have used the word in their \Title{Projective
Geometry}, so that it is not quite unfamiliar to English
mathematicians. \textit{Abbildung}, which signifies representation, is
generally rendered equally well by transformation, inasmuch
as it denotes a copy of certain elements of one space mapped
out on, or expressed in terms of, another space. In some
cases the German word is added in parenthesis for the sake
of those who wish to pursue the subject further in original
papers. It is hoped that the appearance of this English
edition will lead to further efforts towards extending Einstein's
ideas so as to embrace \emph{all} physical knowledge. Much has
been achieved, yet much remains to be done. The brilliant
speculations of the latter chapters of this book show how vast
is the field that has been opened up by Einstein's genius.
The work of translation has been a great pleasure, and I wish
to acknowledge here the courtesy with which suggestions
concerning the type and the symbols have been received and
followed by Messrs.\ Methuen \&~Co.\ Ltd. Acting on the
advice of interested mathematicians and physicists I have
used Clarendon type for the vector notation. My warm
thanks are due to Professor G.~H. Hardy of New College and
Mr.\ T.~W. Chaundy,~M.A., of Christ Church, for valuable suggestions
and help in looking through the proofs. Great care
has been taken to render the mathematical text as perfect as
possible.
\Signature{HENRY L. BROSE}
{Christ Church, Oxford}
{\textit{December}, 1921}
\PageSep{x}
\TableofContents
\iffalse
CONTENTS
PAGE
Introduction 1
CHAPTER I
Euclidean Space. Its Mathematical Form and its Rôle in Physics.
1. Derivation of the Elementary Conceptions of Space from that of
Equality
2. Foundations of Affine Geometry
3. Conception of $n$-dimensional Geometry, Linear Algebra, Quadratic
Forms
4. Foundations of Metrical Geometry
5. Tensors
6. Tensor Algebra. Examples
7. Symmetrical Properties of Tensors
8. Tensor Analysis. Stresses
9. The Stationary Electromagnetic Field
CHAPTER II
The Metrical Continuum
10. Note on Non-Euclidean Geometry
11. Riemann's Geometry
12. Riemann's Geometry (\textit{continued}). Dynamical View of Metrics
13. Tensors and Tensor-densities in an Arbitrary Manifold 102
14. Affinely Connected Manifolds 112
15. Curvature 117
16. Metrical Space 121
17. Remarks on the Special Case of Riemann's Space 129
18. Space Metrics from the Point of View of the Theory of Groups 138
CHAPTER III
Relativity of Space and Time
19. Galilei's and Newton's Principle of Relativity 149
20. Electrodynamics of Varying Fields. Lorentz's Theorem of Relativity 160
21. Einstein's Principle of Relativity 169
22. Relativistic Geometry, Kinematics, and Optics 179
23. Electrodynamics of Moving Bodies 188
24. Mechanics of the Principle of Relatirity 196
25. Mass and Energy 200
26. Mie's Theory 206
Concluding Remarks 217
\PageSep{xi}
CHAPTER IV
General Theory of Relativity
PAGE
27. Relativity of Motion, Metrical Field, and Gravitation 218
28. Einstein's Fundamental Law of Gravitation 229
29. Stationary Gravitational Field. Relationship with Experience 240
30. Gravitational Waves 248
31. Rigorous Solution of the Problem of One Body 252
32. Further Rigorous Solutions of the Statical Problem of Gravitation 259
33. Energy of Gravitation. Laws of Conservation 268
34. Concerning the Inter-connection of the World as a Whole 273
35. World Metrics as the Origin of Electromagnetic Phenomena 282
36. Application of the Simplest Principle of Action. Fundamental
Equations of Mechanics 295
Appendix I 313
Appendix II 315
Bibliographical References 319
Index 325
\fi
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The formulæ are numbered anew for each chapter. Unless otherwise stated,
references to formulæ are to those in the current chapter.
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\MainMatter
\Introduction{SPACE---TIME---MATTER}
\index{Space!form of@{(as form of phenomena)}}%
\index{Space!Euclidean|(}%
\First{\Emph{Space}} and \Emph{time} are commonly regarded as the \Emph{forms} of
existence of the real world, \Emph{matter} as its \Emph{substance}. A
definite portion of matter occupies a definite part of space
at a definite moment of time. It is in the composite idea of
\Emph{motion} that these three fundamental conceptions enter into intimate
relationship. Descartes defined the objective of the exact
sciences as consisting in the description of all happening in terms
of these three fundamental conceptions, thus referring them to
motion. Since the human mind first wakened from slumber, and
was allowed to give itself free rein, it has never ceased to feel the
profoundly mysterious nature of time-consciousness, of the progression
of the world in time,---of Becoming. It is one of those
ultimate metaphysical problems which philosophy has striven to
elucidate and unravel at every stage of its history. The Greeks
made Space the subject-matter of a science of supreme simplicity
and certainty. Out of it grew, in the mind of classical antiquity,
the idea of pure science. Geometry became one of the most powerful
expressions of that sovereignty of the intellect that inspired the
thought of those times. At a later epoch, when the intellectual
despotism of the Church, which had been maintained through the
Middle Ages, had crumbled, and a wave of scepticism threatened to
sweep away all that had seemed most fixed, those who believed
in Truth clung to Geometry as to a rock, and it was the highest
ideal of every scientist to carry on his science ``\emph{\Emph{more geometrico}}''.
Matter was imagined to be a substance involved in
every change, and it was thought that every piece of matter could
be measured as a quantity, and that its characteristic expression as a
``substance'' was the Law of Conservation of Matter which asserts
that matter remains constant in amount throughout every change.
This, which has hitherto represented our knowledge of space and
matter, and which was in many quarters claimed by philosophers
\PageSep{2}
as \textit{a~priori} knowledge, absolutely general and necessary, stands
to-day a tottering structure. First, the physicists in the persons of
Faraday and Maxwell, proposed the ``electromagnetic \Emph{field}'' in
\Chg{contradistinction}{contra-distinction} to \Emph{matter}, as a reality of a different category.
Then, during the last century, the mathematician, following a different
line of thought, secretly undermined belief in the evidence of
Euclidean Geometry. And now, in our time, there has been unloosed
a cataclysm which has swept away space, time, and matter
hitherto regarded as the firmest pillars of natural science, but only
to make place for a view of things of wider scope, and entailing a
deeper vision.
This revolution was promoted essentially by the thought of one
man, Albert Einstein. The working-out of the fundamental ideas
seems, at the present time, to have reached a certain conclusion;
yet, whether or not we are already faced with a new state of affairs,
we feel ourselves compelled to subject these new ideas to a close
analysis. Nor is any retreat possible. The development of scientific
thought may once again take us beyond the present achievement,
but a return to the old narrow and restricted scheme is out
of the question.
Philosophy, mathematics, and physics have each a share in the
problems presented here. We shall, however, be concerned above
all with the mathematical and physical aspect of these questions.
I shall only touch lightly on the philosophical implications for the
simple reason that in this direction nothing final has yet been
reached, and that for my own part I am not in a position to give
such answers to the epistemological questions involved as my conscience
would allow me to uphold. The ideas to be worked out in
this book are not the result of some speculative inquiry into the
foundations of physical knowledge, but have been developed in
the ordinary course of the handling of concrete physical problems---problems
arising in the rapid development of science which has, as
it were, burst its old shell, now become too narrow. This revision
of fundamental principles was only undertaken later, and then
only to the extent necessitated by the newly formulated ideas.
As things are to-day, there is left no alternative but that the
separate sciences should each proceed along these lines dogmatically,
that is to say, should follow in good faith the paths along
which they are led by reasonable motives proper to their own
peculiar methods and special limitations. The task of shedding
philosophic light on to these questions is none the less an important
one, because it is radically different from that which falls to
the lot of individual sciences. This is the point at which the
\PageSep{3}
philosopher must exercise his discretion. If he keep in view the
boundary lines determined by the difficulties inherent in these problems,
he may direct, but must not impede, the advance of sciences
whose field of inquiry is confined to the domain of concrete
objects.
Nevertheless I shall begin with a few reflections of a philosophical
character. As human beings engaged in the ordinary
activities of our daily lives, we find ourselves confronted in our
acts of perception by material things. We ascribe a ``real'' existence
to them, and we accept them in general as constituted,
shaped, and coloured in such and such a way, and so forth, as they
% [** TN: Commas inside quotes, periods outside]
appear to us in our perception in ``general,'' that is ruling out
possible illusions, mirages, dreams, and hallucinations.
These material things are immersed in, and transfused by, a
manifold, indefinite in outline, of analogous realities which unite
to form a single ever-present world of space to which I, with my
own body, belong. Let us here consider only these bodily objects,
and not all the other things of a different category, with which we
as ordinary beings are confronted; living creatures, persons, objects
of daily use, values, such entities as state, right, language, etc.
Philosophical reflection probably begins in every one of us who is
endowed with an abstract turn of mind when he first becomes
sceptical about the world-view of naïve realism to which I have
briefly alluded.
It is easily seen that such a quality as ``\Emph{green}'' has an existence
only as the correlate of the sensation ``green'' associated
with an object given by perception, but that it is meaningless to
attach it as a thing in itself to material things existing \Emph{in themselves}.
This recognition of the \Emph{subjectivity of the qualities
of sense} is found in Galilei (and also in Descartes and Hobbes) in
a form closely related to the principle underlying the \Emph{constructive
mathematical method of our modern physics which repudiates
``qualities''}. According to this principle, colours are
``really'' vibrations of the æther, i.e.\ motions. In the field of
philosophy Kant was the first to take the next decisive step towards
the point of view that not only the qualities revealed by the
senses, but also space and spatial characteristics have no objective
significance in the absolute sense; in other words, that \Emph{space, too,
is only a form of our perception}. In the realm of physics it is
perhaps only the theory of relativity which has made it quite
clear that the two essences, space and time, entering into our intuition
have no place in the world constructed by mathematical
physics. Colours are thus ``really'' not even æther-vibrations,
\PageSep{4}
but merely a series of values of mathematical functions in which
occur four independent parameters corresponding to the three
dimensions of space, and the one of time.
\index{Space!Euclidean|)}%
Expressed as a general principle, this means that the real
world, and every one of its constituents with their accompanying
characteristics, are, and can only be given as, intentional objects of
acts of consciousness. The immediate data which I receive are the
experiences of consciousness in just the form in which I receive
them. They are not composed of the mere stuff of perception,
as many Positivists assert, but we may say that in a sensation
an object, for example, is actually physically present for me---to
whom that sensation relates---in a manner known to every one,
yet, since it is characteristic, it cannot be described more fully.
Following Brentano, I shall call it the ``\Emph{intentional object}''.
In experiencing perceptions I see this chair, for example. My
attention is fully directed towards it. I ``have'' the perception,
but it is only when I make this perception in turn the intentional
object of a new inner perception (a free act of reflection enables
me to do this) that I ``know'' something regarding it (and not
the chair alone), and ascertain precisely what I remarked just
above. In this second act the intentional object is immanent,
i.e.\ like the act itself, it is a real component of my stream of
experiences, whereas in the primary act of perception the object
is transcendental, i.e.\ it is given in an experience of consciousness,
but is not a real component of it. What is immanent is \Emph{absolute},
i.e.\ it is exactly what it is in the form in which I have it, and I
can reduce this, its essence, to the axiomatic by acts of reflection.
On the other hand, transcendental objects have only a \Emph{phenomenal}
existence; they are appearances presenting themselves in manifold
ways and in manifold ``gradations''. One and the same leaf seems
to have such and such a size, or to be coloured in such and such
a way, according to my position and the conditions of illumination.
Neither of these modes of appearance can claim to present
the leaf just as it is ``in itself''. Furthermore, in every perception
there is, without doubt, involved the \Emph{thesis of reality} of the
object appearing in it; the latter is, indeed, a fixed and lasting
element of the general thesis of reality of the world. When,
however, we pass from the natural view to the philosophical attitude,
meditating upon perception, we no longer subscribe to this
thesis. We simply affirm that something real is ``supposed'' in
it. The meaning of such a supposition now becomes the problem
which must be solved from the data of consciousness. In addition
a justifiable ground for making it must be found. I do not by this
\PageSep{5}
in any way wish to imply that the view that the events of the
world are a mere play of the consciousness produced by the ego,
contains a higher degree of truth than naïve realism; on the contrary,
we are only concerned in seeing clearly that the datum of
consciousness is the starting-point at which we must place ourselves
if we are to understand the absolute meaning as well as the
right to the supposition of reality. In the field of logic we have an
analogous case. A judgment, which I pronounce, affirms a certain
set of circumstances; it takes them as true. Here, again, the philosophical
question of the meaning of, and the justification for, this
thesis of truth arises; here, again, the idea of objective truth is
not denied, but becomes a problem which has to be grasped from
what is given absolutely. ``Pure consciousness'' is the seat of
that which is philosophically \textit{a~priori}. On the other hand, a philosophic
examination of the thesis of truth must and will lead to
the conclusion that none of these acts of perception, memory, etc.,
which present experiences from which I seize reality, gives us a
conclusive right to ascribe to the perceived object an existence and
a constitution as perceived. This right can always in its turn be
over-ridden by rights founded on other perceptions, etc.
It is the nature of a real thing to be inexhaustible in content;
we can get an ever deeper insight into this content by the continual
addition of new experiences, partly in apparent contradiction,
by bringing them into harmony with one another. In this interpretation,
things of the real world are approximate ideas. From
this arises the empirical character of all our knowledge of reality.\footnote
{\Chg{Note 1.}{\textit{Vide} \FNote{1}.}}
Time is the primitive form of the stream of consciousness. It
\index{Later@{\emph{Later}}}%
is a fact, however obscure and perplexing to our minds, that the
contents of consciousness do not present themselves simply as
being (such as conceptions, numbers, etc.), but as \Emph{being now} filling
the form of the enduring present with a varying content. So that
one does not say this \Emph{is} but this is \Emph{now}, yet now no more. If we
project ourselves outside the stream of consciousness and represent
its content as an object, it becomes an event happening in
time, the separate stages of which stand to one another in the
relations of \Emph{earlier} and \Emph{later}.
Just as time is the form of the stream of consciousness, so one
may justifiably assert that space is the form of external material
reality. All characteristics of material things as they are presented
to us in the acts of external perception (\Chg{\emph{e.g.}}{e.g.}\ colour) are endowed
with the separateness of spatial extension, but it is only when
we build up a single connected real world out of all our experiences
that the spatial extension, which is a constituent of every
\PageSep{6}
perception, becomes a part of one and the same all-inclusive space.
Thus space is the \Emph{form} of the external world. That is to say,
every material thing can, without changing content, equally well
occupy a position in Space different from its present one. This immediately
gives us the property of the homogeneity of space which
is the root of the conception, Congruence.
Now, if the worlds of consciousness and of transcendental
reality were totally different from one another, or, rather, if only
the passive act of perception bridged the gulf between them, the
state of affairs would remain as I have just represented it, namely,
on the one hand a consciousness rolling on in the form of a lasting
present, yet spaceless; on the other, a reality spatially extended,
yet timeless, of which the former contains but a varying appearance.
Antecedent to all perception there is in us the experience of effort
and of opposition, of being active and being passive. For a person
leading a natural life of activity, perception serves above all to
place clearly before his consciousness the definite point of attack
of the action he wills, and the source of the opposition to it. As
the doer and endurer of actions I become a single individual with
a psychical reality attached to a body which has its place in space
among the material things of the external world, and by which I
am in communication with other similar individuals. Consciousness,
without surrendering its immanence, becomes a piece of
reality, becomes this particular person, namely myself, who was
born and will die. Moreover, as a result of this, consciousness
spreads out its web, in the form of time, over reality. Change,
motion, elapse of time, becoming and ceasing to be, exist in time
itself; just as my will acts on the external world through and
beyond my body as a motive power, so the external world is in its
turn \Emph{active} (as the German word ``Wirklichkeit,'' reality, derived
from ``wirken'' $=$ to act, indicates). Its phenomena are related
throughout by a \Emph{causal connection}. In fact physics shows that
cosmic time and physical form cannot be dissociated from one
another. The new solution of the problem of amalgamating space
and time offered by the theory of relativity brings with it a deeper
insight into the harmony of action in the world.
The course of our future line of argument is thus clearly outlined.
What remains to be said of time, treated separately, and
of grasping it mathematically and conceptually may be included in
this introduction. We shall have to deal with space at much
greater length. Chapter~I will be devoted to a discussion of
\Emph{Euclidean space} and its mathematical structure. In Chapter~II
will be developed those ideas which compel us to pass beyond the
\PageSep{7}
Euclidean scheme; this reaches its climax in the general space-conception
of the metrical continuum (Riemann's conception of
space). Following upon this Chapter~III will discuss the problem
mentioned just above of the \Emph{amalgamation} of Space and Time in
the world. From this point on the results of mechanics and
physics will play an important part, inasmuch as this problem by
its very nature, as has already been remarked, comes into our view
of the world as an active entity. The edifice constructed out of
the ideas contained in Chapters II and III will then in the final
Chapter~IV lead us to Einstein's \emph{General Theory of Relativity},
which, physically, entails a new Theory of \Emph{Gravitation}, and also
to an extension of the latter which embraces electromagnetic
phenomena in addition to gravitation. The revolutions which are
brought about in our notions of Space and Time will of necessity
affect the conception of matter too. Accordingly, all that has to
be said about matter will be dealt with appropriately in Chapters
III and~IV\@.
To be able to apply mathematical conceptions to questions of
\index{Earlier@{\emph{Earlier} and \emph{later}}}%
Time we must postulate that it is theoretically possible to fix
in Time, to any order of accuracy, an absolutely rigorous \Emph{now}
(present) as a \Emph{point of Time}---i.e.\ to be able to indicate points of
time, one of which will always be the earlier and the other the
later. The following principle will hold for this ``order-relation''.
If $A$~is earlier than~$B$ and $B$~is earlier than~$C$, then $A$~is earlier
than~$C$. Each two points of Time, $A$~and~$B$, of which $A$~is the
earlier, mark off a \Emph{length of time}; this includes every point
which is later than~$A$ and earlier than~$B$. The fact that Time is
a form of our stream of experience is expressed in the idea of
\Emph{equality}: the empirical content which fills the length of Time~$AB$
\index{Equality!of time-lengths}%
can in itself be put into any other time without being in any
way different from what it is. The length of time which it would
then occupy is equal to the distance~$AB$. This, with the help of
the principle of causality, gives us the following objective criterion
in physics for equal lengths of time. If an absolutely isolated
physical system (i.e.\ one not subject to external influences) reverts
once again to exactly the same state as that in which it was at
some earlier instant, then the same succession of states will be
repeated in time and the whole series of events will constitute a
cycle. In general such a system is called a \Emph{clock}. Each period
\index{Clocks}%
of the cycle lasts \Emph{equally} long.
The mathematical fixing of time by \Emph{measuring} it is based upon
these two relations, ``earlier (or later) times'' and ``equal times''.
The nature of measurement may be indicated briefly as follows:
\PageSep{8}
Time is homogeneous, i.e.\ a single point of time can only be given
by being specified individually. There is no inherent property
arising from the general nature of time which may be ascribed to
any one point but not to any other; or, every property logically
derivable from these two fundamental relations belongs either to
all points or to none. The same holds for time-lengths and
point-pairs. A property which is based on these two relations and
which holds for \Emph{one} point-pair must hold for every point-pair~$AB$
(in which $A$~is earlier than~$B$). A difference arises, however, in the
case of three point-pairs. If any two time-points $O$~and~$E$ are
given such that $O$~is earlier than~$E$, it is possible to fix conceptually
further time-points~$P$ by referring them to the unit-distance~$OE$.
This is done by constructing logically a relation~$t$ between three
points such that for every two points $O$~and~$E$, of which $O$~is the
earlier, there is one and only one point~$P$ which satisfies the
relation~$t$ between $O$,~$E$ and~$P$, i.e.\ symbolically,
\[
OP = t · OE
\]
(e.g.\ $OP = 2 · OE$ denotes the relation $OE = EP$). \Emph{Numbers} are
\index{Number}%
merely concise symbols for such relations as~$t$, defined logically
from the primary relations. $P$~is the ``time-point with the
\Emph{abscissa~$t$ in the co-ordinate system} (taking $OE$ as unit length)''.
Two different numbers $t$~and~$t^{*}$ in the same co-ordinate system
necessarily lead to two different points; for, otherwise, in consequence
of the homogeneity of the continuum of time-lengths,
the property expressed by
\[
t · AB = t^{*} · AB,
\]
since it belongs to the time-length $AB = OE$, must belong to \Emph{every}
time-length, and hence the equations $AC = t · AB$, $AC = t^{*} · AB$
would both express the same relation, i.e.\ $t$~would be equal to~$t^{*}$.
Numbers enable us to single out separate time-points relatively to
a unit-distance~$OE$ out of the time-continuum by a conceptual,
and hence objective and precise, process. But the objectivity of
things conferred by the exclusion of the ego and its data derived
directly from intuition, is not entirely satisfactory; the co-ordinate
system which can only be specified by an individual act (and then
only approximately) remains as an inevitable residuum of this
elimination of the percipient.
It seems to me that by formulating the principle of measurement
in the above terms we see clearly how mathematics has come to
play its rôle in exact natural science. \emph{An essential feature of
measurement is the difference between the ``determination'' of an
object by individual specification and the determination of the same
\PageSep{9}
object by some conceptual means.} The latter is only possible
relatively to objects which must be defined directly. That is why
a \Emph{theory of relativity} is perforce always involved in measurement.
The general problem which it proposes for an arbitrary
\index{Co-ordinates, curvilinear!generally@{(generally)}}%
domain of objects takes the form: (1)~What must be given such that
relatively to it (and to any desired order of precision) one can single
out conceptually a single arbitrary object~$P$ from the continuously
extended domain of objects under consideration? That which has
to be given is called the \Emph{co-ordinate system}, the conceptual
definition is called the \Emph{co-ordinate} (or abscissa) of~$P$ in the co-ordinate
\index{Abscissa}%
system. Two different co-ordinate systems are completely
\index{Co-ordinate systems}%
equivalent for an objective standpoint. There is no property, that
can be fixed conceptually, which applies to one co-ordinate system
but not to the other; for in that case too much would have been given
directly. (2)~What relationship exists between the co-ordinates
of one and the same arbitrary object~$P$ in two different co-ordinate
systems?
In the realm of time-points, with which we are at present concerned,
the answer to the first question is that the co-ordinate
system consists of a time-length~$OE$ (giving the origin and the
unit of measure). The answer to the second question is that the
required relationship is expressed by the formula of transformation
\[
t = at' + b\qquad (a > \Typo{o}{0})
\]
in which $a$~and $b$ are constants, whilst $t$~and~$t'$ are the co-ordinates
of the same arbitrary point~$P$ in an ``unaccented'' and ``accented''
system respectively. For all possible pairs of co-ordinate systems
the characteristic numbers, $a$~and~$b$, of the transformation may be
any real numbers with the limitation that $a$~must always be positive.
The aggregate of transformations constitutes a \Emph{group}, as\Pagelabel{9}
\index{Groups}%
their nature would imply, i.e.,
1. ``identity'' $t = t'$ is contained in it.
2. Every transformation is accompanied by its reciprocal in
the group, i.e.\ by the transformation which exactly cancels its
effect. Thus, the inverse of the transformation $(a, b)$, viz.\ $t = at' + b$,
is $\left(\dfrac{1}{a}, -\dfrac{b}{a}\right)$, viz.\ $t' = \dfrac{1}{a}t - \dfrac{b}{a}$.
3. If two transformations of a group are given, then the one
which is produced by applying these two successively also belongs to
the group. It is at once evident that, by applying the two transformations
\[
t = at' + b\qquad
t' = a't'' + b'
\]
\PageSep{10}
\index{Translation of a point!(in the geometrical sense)}%
in succession, we get
\[
t = a_{1} t'' + b_{1}
\]
where $a_{1} = a · a'$ and $b_{1} = (ab') + b$; and if $a$~and~$a'$ are positive,
so is their product.
The theory of relativity discussed in Chapters III~and~IV proposes
the problem of relativity, not only for time-points, but for
the physical world in its entirety. We find, however, that this
problem is solved once a solution has been found for it in the case
of the two forms of this world, space and time. By choosing a
co-ordinate system for space and time, we may also fix the physically
real content of the world conceptually in all its parts by
means of numbers.
All beginnings are obscure. Inasmuch as the mathematician
operates with his conceptions along strict and formal lines, he,
above all, must be reminded from time to time that the origins of
things lie in greater depths than those to which his methods enable
him to descend. Beyond the knowledge gained from the individual
sciences, there remains the task of \Emph{comprehending}. In
spite of the fact that the views of philosophy sway from one
system to another, we cannot dispense with it unless we are to
convert knowledge into a meaningless chaos.
\PageSep{11}
\Chapter[Euclidean Space]{I}
{Euclidean Space. Its Mathematical Formulation and
its Rôle in Physics}
\index{Euclidean!geometry|(}%
\Section{1.}{Deduction of the Elementary Conceptions of Space from
that of Equality}
\First{Just} as we fixed the present moment (``now'') as a geometrical
point in time, so we fix an exact ``here,'' a point in space,
as the first element of continuous spatial extension, which,
like time, is infinitely divisible. Space is not a one-dimensional
continuum like time. The principle by which it is continuously
extended cannot be reduced to the simple relation of ``earlier'' or
``later''. We shall refrain from inquiring what relations enable
us to grasp this continuity conceptually. On the other hand, space,
like time, is a \Emph{form} of phenomena. Precisely the same content,
identically the same thing, still remaining what it is, can equally
well be at some place in space other than that at which it is actually.
The new portion of Space~$\vS'$ then occupied by it is equal to that
portion~$\vS$ which it actually occupied. $\vS$~and~$\vS'$ are said to be
\Emph{congruent}. To every point~$P$ of~$\vS$ there corresponds one definite
\index{Congruent}%
\index{Congruent!transformations}%
\index{Homologous points}%
\index{Transformation or representation!congruent}%
\Emph{homologous} point~$P'$ of~$\vS'$ which, after the above displacement to a
new position, would be surrounded by exactly the same part of the
given content as that which surrounded $P$ originally. We shall call
this ``transformation'' (in virtue of which the point~$P'$ corresponds
to the point~$P$) a \Emph{congruent transformation}. Provided that the
appropriate subjective conditions are satisfied the given material
thing would seem to us after the displacement exactly the same as
before. There is reasonable justification for believing that a rigid
body, when placed in two positions successively, realises this idea
of the equality of two portions of space; by a \Emph{rigid} body we mean
one which, however it be moved or treated, can always be made to
appear the same to us as before, if we take up the appropriate
position with respect to it. I shall evolve the scheme of geometry
\index{Geometry!Euclidean|(}%
from the conception of equality combined with that of continuous
connection---of which the latter offers great difficulties to analysis---and
\PageSep{12}
shall show in a superficial sketch how all fundamental conceptions
of geometry may be traced back to them. My real object
in doing so will be to single out \Emph{translations} among possible congruent
transformations. Starting from the conception of translation
I shall then develop Euclidean geometry along strictly axiomatic
lines.
First of all the \Emph{straight line}. Its distinguishing feature is that
\index{Line, straight!Euclidean@{(in Euclidean geometry)}}%
it is determined by two of its points. Any \emph{other} line can, even
when two of its points are kept fixed, be brought into another
position by a congruent transformation (the test of straightness).
Thus, if $A$~and~$B$ are two different points, the straight line
$g = AB$ includes every point which becomes transformed into itself
by all those congruent transformations which transform $AB$ into
themselves. (In familiar language, the straight line lies evenly
between its points.) Expressed kinematically, this is tantamount
to saying that we regard the straight line as an axis of rotation.
It is homogeneous and a linear continuum just like time. Any
arbitrary point on it divides it into two parts, two ``rays''. If $B$~lies
on one of these parts and $C$~on the other, then $A$~is said to
be between $B$~and~$C$ and the points of one part lie to the right of~$A$,
the points of the other part to the left. (The choice as to
which is right or left is determined arbitrarily.) The simplest
fundamental facts which are implied by the conception ``between''
\index{Between@{\emph{Between}}}%
can be formulated as exactly and completely as a geometry which
is to be built up by deductive processes demands. For this reason
we endeavour to trace back all conceptions of continuity to the
conception ``between,'' i.e.\ to the relation ``$A$~is a point of the
straight line~$BC$ and lies between $B$ and~$C$'' (this is the reverse of
the real intuitional relation). Suppose $A'$~to be a point on~$g$ to
the right of~$A$, then $A'$~also divides the line~$g$ into two parts. We
call that to which $A$ belongs the left-hand side. If, however,
$A'$~lies to the left of~$A$ the position is reversed. With this convention,
analogous relations hold not only for $A$~and~$A'$ but also
for \Emph{any} two points of a straight line. The points of a straight
line are ordered by the terms left and right in precisely the same
way as points of time by the terms earlier and later.
Left and right are equivalent. There is one congruent transformation
which leaves $A$ fixed, but which interchanges the
two halves into which $A$~divides the straight line. Every finite
portion of straight line~$AB$ may be superposed upon itself in such
a way that it is reversed (i.e.\ so that $B$~falls on~$A$, and $A$~falls on~$B$).
On the other hand, a congruent transformation which transforms
$A$~into itself, and all points to the right of~$A$ into points to
\PageSep{13}
the right of~$A$, and all points to the left of~$A$ into points to the left
of~$A$, leaves every point of the straight line undisturbed. The
homogeneity of the straight line is expressed in the fact that the
straight line can be placed upon itself in such a way that any
point~$A$ of it can be transformed into any other point~$A'$ of it, and
that the half to the right of~$A$ can be transformed into the half to
the right of~$A'$, and likewise for the portions to the left of $A$ and
$A'$ respectively (this implies a mere translation of the straight
line). If we now introduce the equation $AB = A'B'$ for the points
of the straight line by interpreting it as meaning that $AB$~is transformed
into the straight line~$A'B'$ by a translation, then the same
things hold for this conception as for time. These same circumstances
enable us to introduce numbers, and to establish a reversible
and single correspondence between the points of a straight line
and real numbers by using a unit of length~$OE$.
Let us now consider the group of congruent transformations
which leaves the straight line~$g$ fixed, i.e.\ transforms every point
of~$g$ into a point of~$g$ again.
We have called particular attention to rotations among these
as having the property of leaving not only $g$~as a whole, but
also every single point of~$g$ unmoved in position. How can translations
in this group be distinguished from twists?
\index{Twists}%
I shall here outline a preliminary argument in which not only
the straight line, but also the plane is based on a property of
\index{Axis of rotation}%
\index{Plane!(in Euclidean space)}%
rotation.
\index{Rotation!geometrical@{(in geometrical sense)}}%
Two rays which start from a point~$O$ form an \Emph{angle}. Every
\index{Angles!measurement of}%
\index{Angles!right}%
angle can, when inverted, be superposed exactly upon itself, so
that one arm falls on the other, and \textit{vice versa}. Every \Emph{right} angle
is congruent with its complementary angle. Thus, if $h$~is a straight
line perpendicular to~$g$ at the point~$A$, then there is one rotation
about~$g$ (``inversion'') which interchanges the two halves into which
$h$~is divided by~$A$. All the straight lines which are perpendicular
to~$g$ at~$A$ together form the \Emph{plane}~$E$ through~$A$ perpendicular to~$g$.
Each pair of these perpendicular straight lines may be produced
from any other by a rotation about~$g$.
\Figure{1}
\PageSep{14}
If $g$~is inverted, and placed upon itself in some way, so that $A$~is
transformed into itself, but so that the two halves into which $A$
divides~$g$ are interchanged, then the plane~$E$ of necessity coincides
with itself. The plane may also be defined by taking this property
in conjunction with that of symmetry of rotation. Two
congruent tables of revolution (i.e.\ symmetrical with respect to
rotations) are plane if, by means of inverting one, so that its axis
is vertical in the opposite direction, and placing it on the other,
the two table-surfaces can be made to coincide. The plane is
homogeneous. The point~$A$ on~$E$ which appears as the centre in this
example is in no way unique among the points of~$E$. A straight
line~$g'$ passes through each one $A'$ of them in such a way that $E$~is
made up of all straight lines through~$A'$ perpendicular to~$g'$.
The straight lines~$g'$ which are perpendicular to~$E$ at its points~$A'$
respectively form a group of \Emph{parallel} straight lines. The straight
\index{Parallel}%
line~$g$ with which we started is in no wise unique among them.
The straight lines of this group occupy the whole of space in such
a way that only one straight line of the group passes through each
point of space. This in no way depends on the point~$A$ of the
straight line~$g$, at which the above construction was performed.
If $A^{*}$~is any point on~$g$, then the plane which is erected
normally to~$g$ at~$A^{*}$ cuts not only~$g$ perpendicularly, but also
\Emph{all} straight lines of the group of parallels. All such normal
planes~$E^{*}$ which are erected at all points~$A^{*}$ on~$g$ form a group
of parallel planes. These also fill space continuously and uniquely.
We need only take another small step to pass from the above
framework of space to the rectangular system of co-ordinates.
We shall use it here, however, to fix the conception of spatial
translation.
Translation is a congruent transformation which transforms
not only~$g$ but every straight line of the group of parallels into
itself. There is one and only one translation which transfers the
arbitrary point~$A$ on~$g$ to the arbitrary point~$A^{*}$ on the same
straight line.
I shall now give an alternate method of arriving at the conception
of translation. The chief characteristic of translation is
that all points are of equal importance in it, and that the behaviour
of a point during translation does not allow any objective assertion
to be made about it, which could not equally well be made of any
other point (this means that the points of space for a given translation
can only be distinguished by specifying each one singly
[``that one there''], whereas in the case of rotation, for example,
the points on the axis are distinguished by the property that they
\PageSep{15}
\index{Groups!of translations}%
preserve their positions). By using this as a basis we get the
following definition of translation, which is quite independent of
the conception of rotation. Let the arbitrary point~$P$ be transformed
into~$P'$ by a congruent transformation: we shall call $P$~and~$P'$
connected points. A second congruent transformation
which has the property of again transforming every pair of connected
points into connected points, is to be called \Emph{interchangeable}
with the first transformation. A congruent transformation
is then called a translation, if it gives rise to interchangeable congruent
transformations, which transform the arbitrary point~$A$
into the arbitrary point~$B$. The statement that two congruent
transformations I~and~II are interchangeable signifies (as is easily
proved from the above definition) that the congruent transformation
resulting from the successive application of I~and~II is identical
with that which results when these two transformations are
performed in the reverse order. It is a fact that one translation
(and, as we shall see, \Emph{only} one) exists, which transforms the
arbitrary point~$A$ into the arbitrary point~$B$. Moreover, not only
is it a fact that, if $\vT$~denote a translation and $A$~and~$B$ any two
points, there is, according to our definition, a congruent transformation,
interchangeable with~$\vT$, which transforms $A$ into~$B$,
but also that the particular \Emph{translation} which transforms $A$ into~$B$
has the required property. A translation is therefore interchangeable
with all other translations, and a congruent transformation
which is interchangeable with all translations is also
necessarily a translation. From this it follows that the congruent
transformation which results from successively performing two
translations, and also the ``inverse'' of a translation (i.e.\ that
transformation which exactly reverses or neutralises the original
translation) is itself a translation. Translations possess the
``group'' property.\footnote
{\Chg{Note 2.}{\textit{Vide} \FNote{2}.}}
There is no translation which transforms
$A$ into~$A$ except \Emph{identity}, in which every point remains undisturbed.
For if such a translation were to transform $P$ into~$P'$,
then, according to definition, there must be a congruent transformation,
which transforms $A$ into~$P$ and simultaneously $A$ into~$P'$;
$P$~and~$P'$ must therefore be identical points. Hence there
cannot be two different translations both of which transform~$A$
into another point~$B$.
As the conception of translation has thus been defined independently
of that of rotation, the translational view of the
straight line and plane may thus be formed in contrast with the
above view based on rotations. Let $\va$ be a translation which
transfers the point~$A_{0}$ to~$A$. This same translation will transfer~$A_{1}$
\PageSep{16}
to a point~$A_{2}$, $A_{2}$~to~$A_{3}$, etc. Moreover, through it $A_{0}$~will
be derived from a certain point~$A_{-1}$, $A_{-1}$~from~$A_{-2}$, etc. This
does not yet give us the whole straight line, but only a series of
\Chg{equi-distant}{equidistant} points on it. Now, if $n$~is a natural number (integer),
a translation~$\dfrac{\va}{n}$ exists which, when repeated $n$~times, gives~$\va$. If,
then, starting from the point~$A_{0}$ we use~$\dfrac{\va}{n}$ in the same way as we
just now used~$\va$ we shall obtain an array of points on the straight
line under construction, which will be $n$~times as dense.
If we take all possible whole numbers as values of~$n$ this array
will become denser in proportion as $n$~increases, and all the points
which we obtain finally fuse together into a linear continuum, in
which they become embedded, giving up their individual existences
(this description is founded on our intuition of continuity). We
may say that the straight line is derived from a point by an infinite
repetition of the same infinitesimal translation and its inverse. A
plane, however, is derived by translating one straight line,~$g$, along
another,~$h$. If $g$~and~$h$ are two different straight lines passing
through the point~$A_{0}$, then if we apply to~$g$ all the translations
which transform $h$ into itself, all straight lines which thus result
from~$g$ together form the \Emph{common} plane of $g$~and~$h$.
We succeed in introducing logical order into the structure of
geometry only if we first narrow down the general conception of
\index{Geometry!affine}%
congruent transformation to that of translation, and use this as an
axiomatic foundation (§§\,2 and~3). By doing this, however, we
arrive at a geometry of translation alone, viz.\ affine geometry
\index{Affine!geometry!(linear Euclidean)}%
within the limits of which the general conception of congruence
has later to be re-introduced~(§\,4). Since intuition has now
furnished us with the necessary basis we shall in the next
paragraph enter into the region of deductive mathematics.
\Section{2.}{The Foundations of Affine Geometry}
For the present we shall use the term vector to denote a
\index{Vector}%
translation or a displacement~$\va$ in the space. Later we shall have
occasion to attach a wider meaning to it. The statement that the
displacement~$\va$ transfers the point~$P$ to the point~$Q$ (``transforms''
$P$ into~$Q$) may also be expressed by saying that $Q$~is the end-point
of the vector~$\va$ whose starting-point is at~$P$. If $P$~and~$Q$ are any
two points then there is one and only one displacement~$\va$ which
transfers $P$ to~$Q$. We shall call it the vector defined by $P$~and~$Q$,
and indicate it by~$\Vector{PQ}$.
\PageSep{17}
The translation~$\vc$ which arises through two successive translations
\index{Co-ordinates, curvilinear!linear@{(in a linear manifold)}}%
$\va$~and~$\vb$ is called the sum of $\va$~and~$\vb$, i.e.\ $\vc = \va + \vb$. The
\index{Addition of tensors!of vectors}%
\index{Sum of!vectors}%
definition of summation gives us: (1)~the meaning of multiplication
\index{Multiplication!of a vector by a number}%
(repetition) and of the division of a vector by an integer; (2)~the
purport of the operation which transforms the vector~$\va$ into its
inverse~$-\va$; (3)~the meaning of the nil-vector~$\0$, viz.\ ``identity,''
which leaves all points fixed, i.e.\ $\va + \0 = \va$ and $\va + (-\va) = \0$.
It also tells us what is conveyed by the symbols $±\dfrac{m\va}{n} = \lambda\va$, in
which $m$~and $n$ are any two natural numbers (integers) and $\lambda$~denotes
the fraction~$±\dfrac{m}{n}$. By taking account of the postulate of
continuity this also gives us the significance of~$\lambda\va$, when $\lambda$~is \Emph{any}
real number. The following system of axioms may be set up for
\index{Axioms!of affine geometry}%
affine geometry:---
%[** TN: Headings changed to match the text, cf. Chapter II, pp. 141 ff.]
\Subsection{\Chg{1}{I}. Vectors}
Two vectors $\va$ and $\vb$ uniquely determine a vector $\va + \vb$ as their
sum. A number~$\lambda$ and a vector~$\va$ uniquely define a vector~$\lambda\va$,
which is ``$\lambda$~times~$\va$'' (multiplication). These operations are
subject to the following laws:---
($\alpha$) Addition---
(1) $\va + \vb = \vb + \va$ (Commutative Law).
\index{Commutative law}%
(2) $(\va + \vb) + \vc = \va + (\vb + \vc)$ (Associative Law).
\index{Associative law}%
(3) If $\va$ and $\vc$ are any two vectors, then there is one and only
one value of~$\vx$ for which the equation $\va + \vx = \vc$ holds. It is
called the difference between $\vc$~and~$\va$ and signifies $\vc - \va$ (Possibility
\index{Subtraction of vectors}%
of Subtraction).
($\beta$) Multiplication---
(1) $(\lambda + \mu) \va = (\lambda\va) + (\mu\va)$ (First Distributive Law).
\index{Distributive law}%
(2) $\lambda(\mu\va) = (\lambda\mu)\va$ (Associative Law).
(3) $1\Add{ · }\va = \va$.
(4) $\lambda(\va + \vb) = (\lambda\va) + (\lambda\vb)$ (Second Distributive Law).
For rational multipliers $\lambda$,~$\mu$, the laws~$(\beta)$ follow from the
axioms of addition if multiplication by such factors be \Emph{defined}
from addition. In accordance with the principle of continuity we
shall also make use of them for any arbitrary real numbers, but we
purposely formulate them as separate axioms because they cannot
be derived in the general form from the axioms of addition by
logical reasoning alone. By refraining from reducing multiplication
to addition we are enabled through these axioms to banish
continuity, which is so difficult to fix precisely, from the logical
\PageSep{18}
% [** TN: Idiosyncratic item number]
structure of geometry. The law~\Eq{(\beta)}~4 comprises the theorems of
similarity.
($\gamma$) The ``Axiom of Dimensionality,'' which occupies the next
place in the system, will be formulated later.
\Subsection{\Chg{2}{II}. Points and Vectors}
1. Every pair of points $A$ and $B$ determines a vector~$\va$; expressed
symbolically $\Vector{AB} = \va$. If $A$~is any point and $\va$~any vector,
there is one and only one point~$B$ for which $\Vector{AB} = \va$.
2. If $\Vector{AB} = \va$, $\Vector{BC} = \vb$, then $\Vector{AC} = \va + \vb$.
In these axioms two fundamental categories of objects occur,
viz.\ points and vectors; and there are three fundamental relations,
those expressed symbolically by---
\[
\va + \vb = \vc\qquad
\vb = \lambda\va\qquad
\Vector{AB} = \va\Add{.}
\Tag{(1)}
\]
All conceptions which may be defined from~\Inum{(1)} by logical reasoning
alone belong to affine geometry. The doctrine of affine geometry
is composed of all theorems which can be deduced logically from
the axioms~\Inum{(1)}, and it can thus be erected deductively on the
axiomatic basis \Inum{(1)}~and~\Inum{(2)}. The axioms are not all logically
independent of one another for the axioms of addition for vectors
\Inum{(\Chg{\textit{I}}{I}$\alpha$, 2 and~3)} follow from those~\Inum{(\Chg{\textit{II}}{II})} which govern the relations
between points and vectors. It was our aim, however, to make
the vector-axioms~\Inum{\Chg{\textit{I}}{I}} suffice in themselves, so that we should be
able to deduce from them all those facts which involve vectors
exclusively (and not the relations between vectors and points).
From the axioms of addition~\Chg{\textit{I}}{I}$\alpha$ we may conclude that a definite
vector~$\0$ exists which, for every vector~$\va$, satisfies the equation
$\va + \0 = \va$. From the axioms~\Chg{\textit{II}}{II} it further follows that $\Vector{AB}$~is
equal to this vector~$\0$ when, and only when, the points $A$ and~$B$
coincide.
If $O$~is a point and $\ve$~is a vector differing from~$\0$, the end-points
\index{Line, straight!generally@{(generally)}}%
of all vectors~$OP$ which have the form~$\xi\ve$ ($\xi$~being an arbitrary real
number) form a \Emph{straight line}. This explanation gives the translational
or affine view of straight lines the form of an exact definition
which rests solely upon the fundamental conceptions involved in
the system of affine axioms. Those points~$P$ for which the abscissa~$\xi$
is positive form one-half of the straight line through~$O$, those for
which $\xi$~is negative form the other half. If we write~$\ve_{1}$ in place of~$\ve$,
and if $\ve_{2}$~is another vector, which is not of the form~$\xi\ve_{1}$, then the
end-points~$P$ of all vectors~$\Vector{OP}$ which have the form $\xi_{1}\ve_{1} + \xi_{2}\ve_{2}$
form a \Emph{plane}~$\vE$ (in this way the plane is derived affinely by sliding
\index{Plane}%
\PageSep{19}
one straight line along another). If we now displace the plane~$\vE$
along a straight line passing through~$O$ but not lying on~$\vE$, the
plane passes through all space. Accordingly, if $\ve_{3}$~is a vector not
expressible in the form $\Typo{\xi_{1}\ve + \xi_{2}\ve}{\xi_{1}\ve_{1} + \xi_{2}\ve_{2}}$, then every vector can be represented
in one and only one way as a linear combination of $\ve_{1}$,~$\ve_{2}$,
and~$\ve_{3}$, viz.
\[
\xi_{1}\ve_{1} + \xi_{2}\ve_{2} + \xi_{3}\ve_{3}.
\]
We thus arrive at the following set of definitions:---
A finite number of vectors $\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$ is said to be \Emph{linearly
independent} if
\[
\xi_{1}\ve_{1} + \xi_{2}\ve_{2} + \dots + \xi_{h}\ve_{h}
\Tag{(2)}
\]
only vanishes when all the \Chg{coefficients}{co-efficients}~$\xi$ vanish simultaneously.
\index{Dimensions}%
\index{Linear equation!vector manifold}%
\index{Vector!manifold@{-manifold, linear}}%
With this assumption all vectors of the form~\Eq{(2)} together constitute
a so-called \Emph{$\Chg{\mathbf{h}}{h}$-dimensional linear vector-manifold} (or simply
% [** TN: Here "vector-field" clearly refers to an arbitrary expression (2)]
vector-field); in this case it is the one mapped out by the vectors
$\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$. An $h$-dimensional linear vector-manifold~$\vM$ can
be characterised without referring to its particular base~$\ve$, as
follows:---
(1) The two fundamental operations, viz.\ addition of two
vectors and multiplication of a vector by a number do not transcend
the manifold, i.e.\ the sum of two vectors belonging to~$\vM$ as also
the product of such a vector and any real number also lie in~$\vM$.
(2) There are $h$~linearly independent vectors in~$\vM$, but every
\index{Independent vectors}%
\index{Linearly independent}%
$h + 1$ are linearly dependent on one another.
From the property~(2) (which may be deduced from our original
definition with the help of elementary results of linear equations)
it follows that~$h$, the dimensional number, is as such characteristic
of the manifold, and is not dependent on the special vector base by
which we map it out. The dimensional axiom which was omitted
in the above table of axioms may now be formulated.
\Emph{There are $n$~linearly independent vectors, but every $n + 1$
are linearly dependent on one another,} \\
or: The vectors constitute an $n$-dimensional linear manifold.
If $n = 3$ we have affine geometry of space, if $n = 2$ plane
\index{Geometry!n-dimensional@{$n$-dimensional}}%
geometry, if $n = 1$ geometry of the straight line. In the deductive
treatment of geometry it will, however, be expedient to leave the
value of~$n$ undetermined, and to develop an ``$n$-dimensional geometry''
in which that of the straight line, of the plane, and of space
are included as special cases. For we see (at present for affine
geometry, later on for \Emph{all} geometry) that there is nothing in the
mathematical structure of space to prevent us from exceeding the
dimensional number~$3$. In the light of the mathematical uniformity
of space as expressed in our axioms, its special dimensional
\PageSep{20}
number~$3$ appears to be accidental, so that a systematic deductive
theory cannot be restricted by it. We shall revert to the idea of
an $n$-dimensional geometry, obtained in this way, in the next paragraph.\footnote
{\Chg{Note 3.}{\textit{Vide} \FNote{3}.}}
We must first complete the definitions outlined.
If $O$~is an arbitrary point, then the sum-total of all the end-points~$P$
\index{Configuration, linear point}%
\index{Linear equation!point-configuration}%
of vectors, the origin of which is at~$O$ and which belong
to an $h$-dimensional vector field~$\vM$ as represented by~(2), occupy
fully \Emph{an $\Chg{\mathbf{h}}{h}$-dimensional point-configuration}. We may, as before,
say that it is \Emph{mapped out} by the vectors $\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$, which
start from~$O$. The one-dimensional configuration of this type is
called a straight line, the two-dimensional a plane. The point~$O$
does not play a unique part in this linear configuration. If $O'$~is
any other point of it, then $\Vector{O'P}$~traverses the same vector manifold~$\vM$
if all possible points of the linear aggregate are substituted for~$P$
in turn.
If we measure off all vectors of the manifold~$\vM$ firstly from the
point~$O$ and then from any other arbitrary point~$O'$ the two resulting
linear point aggregates are said to be \Emph{parallel} to one another.
The definition of parallel planes and parallel straight lines
is contained in this. That part of the $h$-dimensional linear assemblage
which results when we measure off all the vectors~(2)
from~$O$, subject to the limitation
\[
0 \leq \xi_{1} \leq 1,\qquad
0 \leq \xi_{2} \Erratum{\geq}{\leq} 1,\quad \dots\Add{,}\qquad
0 \leq \xi_{h} \Erratum{\geq}{\leq} 1,
\]
will be called the $h$-dimensional \Emph{parallelepiped} which has its
\index{Parallelepiped}%
origin at~$O$ and is mapped out by the vectors $\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$. (The
\index{Distance (generally)!(in Euclidean geometry)}%
one-dimensional parallelepiped is called \emph{distance}, the two-dimensional
one is called \emph{parallelogram}. None of these conceptions
is limited to the case $n = 3$, which is presented in ordinary experience.)
A point~$O$ in conjunction with $n$~linear independent vectors
$\ve_{1}$, $\ve_{2}$,~\dots\Add{,} $\ve_{h}$ will be called a co-ordinate system~$(\Typo{\vc}{\vC})$. Every vector~$\vx$
can be presented in one and only one way in the form
\[
\vx = \xi_{1}\ve_{1} + \xi_{2}\ve_{2} + \dots + \xi_{n}\ve_{n}\Add{.}
\Tag{(3)}
\]
The numbers~$\xi_{i}$ will be called its \Emph{components} in the co-ordinate
\index{Components, co-variant, and contra-variant!vector@{of a vector}}%
system~$(\vC)$. If $P$~is any arbitrary point and if $\Vector{OP}$~is equal to the
vector~\Eq{(3)}, then the~$\xi_{i}$ are called the \Emph{co-ordinates} of~$P$. All co-ordinate
systems are equivalent in affine geometry. There is no
property of this geometry which distinguishes one from another. If
\[
O' \Chg{;}{\mid} \ \ve_{1}',\ \ve_{2}'\Add{,}\ \dots\Add{,} \ve_{n}'
\]
denote a second co-ordinate system, equations
\PageSep{21}
\index{Parallel}%
\[
\ve_{i}' = \sum_{k=1}^{n} \Chg{\alpha_{ki}}{\alpha_{k}^{i}} \Typo{\ve^{k}}{\ve_{k}}
\Tag{(4)}
\]
will hold in which the~$\Chg{\alpha_{ki}}{\alpha_{k}^{i}}$ form a number system which must have
a non-vanishing determinant (since the~$\Typo{\ve_{1}'}{\ve_{i}'}$ are linearly independent).
If $\xi_{i}$ are the components of a vector~$\vx$ in the first co-ordinate
\index{Vector!transformation, linear}%
system and $\xi_{i}'$~the components of the same vector in the second
co-ordinate system, then the relation
\[
\xi_{i} = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} \xi_{k}'
\Tag{(5)}
\]
holds; this is easily shown by substituting the expressions~\Eq{(4)} in
the equation
\[
\sum_{i} \xi_{i} \ve_{i} = \sum_{i} \xi_{i}' \ve_{i}'.
\]
Let $\alpha_{1}$, $\alpha_{2}$,~\dots\Add{,} $\alpha_{n}$ be the co-ordinates of~$O'$ in the first co-ordinate
system. If $x_{i}$~are the co-ordinates of any arbitrary point in the
first system and $x_{i}'$~its co-ordinates in the second, the equations
\[
x_{i} = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} x_{k}' + \alpha_{i}
\Tag{(6)}
\]
hold. For $x_{i} - \alpha_{i}$ are the components of
\[
\Vector{O'P} = \Vector{OP} - \Vector{OO'}
\]
in the first system; $x_{i}'$~are the components of~$\Vector{O'P}$ in the second.
Formulæ~\Eq{(6)} which give the transformation for the co-ordinates are
\index{Linear equation!vector manifold!transformation}%
\index{Transformation or representation!affine}%
\index{Transformation or representation!linear-vector}%
thus linear. Those (viz.~5) which transform the vector components
are easily derived from them by cancelling the terms~$\alpha_{i}$ which do
not involve the variables. An analytical treatment of affine geometry
\index{Affine!transformation}%
is possible, in which every vector is represented by its components
and every point by its co-ordinates. The geometrical
relations between points and vectors then express themselves as
relations between their components and co-ordinates respectively
of such a kind that they are not destroyed by linear arbitrary
transformations.
Formulæ \Eq{(5)}~and~\Eq{(6)} may also be interpreted in another way.
They may be regarded as a mode of representing an affine \Emph{transformation}
in a definite co-ordinate system. A transformation,
i.e.\ a rule which assigns a vector~$\vx'$ to every vector~$\vx$ and a point~$P'$
to every point~$P$, is called linear or affine if the fundamental
affine relations~\Eq{(1)} are not disturbed by the transformation: so
\PageSep{22}
that if the relations~\Eq{(1)} hold for the original points and vectors
they also hold for the transformed points and vectors:
\[
\va' + \vb' = \vc'\qquad
\vb' = \lambda\va'\qquad
\Vector{A'B'} = \va' - \vb'
\]
and if in addition no vector differing from~$\0$ transforms into the
vector~$\0$. Expressed in other words this means that two points
are transformed into one and the same point only if they are
themselves identical. Two figures which are formed from one
another by an affine transformation are said to be affine. From
the point of view of affine geometry they are identical. There can
be no affine property possessed by the one which is not possessed
by the other. The conception of linear transformation thus plays
the same part in affine geometry as congruence plays in general
geometry; hence its fundamental importance. In affine transformations
linearly independent vectors become transformed into
linearly independent vectors again; likewise an $h$-dimensional
linear configuration into a like configuration; parallels into parallels;
a co-ordinate system $O \mid \ve_{1},\ \ve_{2}\Add{,}\ \dots\Add{,} \ve_{n}$ into a new co-ordinate
system $O' \mid \ve_{1}',\ \ve_{2}'\Add{,}\ \dots\Add{,} \ve_{n}'$.
Let the numbers $\Chg{\alpha_{ki}}{\alpha_{k}^{i}}$, $\alpha_{i}$, have the same meaning as above. The
\index{Linear equation!vector manifold!transformation}%
\index{Transformation or representation!linear-vector}%
\index{Vector!transformation, linear}%
vector~\Eq{(3)} is changed by the affine transformation into
\[
\vx' = \xi_{1} \ve_{1}' + \xi_{2} \ve_{2}' + \dots + \xi_{n} \ve_{n}'.
\]
If we substitute in this the expressions for~$\ve_{i}'$ and use the original
co-ordinate system $O \mid \ve_{1},\ \ve_{2}\Add{,}\ \dots\Add{,} \ve_{n}$ to picture the affine transformation,
then, interpreting $\xi_{i}$ as the components of any vector
and $\xi_{i}'$ as the components of its transformed vector,
\[
\xi_{i}' = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} \xi_{k}\Add{.}
\Tag{(5')}
\]
If $P$ becomes~$P'$, the vector~$\Vector{OP}$ becomes~$\Vector{O'P'}$, and it follows from
this that if $x_{i}$~are the co-ordinates of~$P$ and $x_{i}'$~those of~$P'$, then
\[
x_{i}' = \sum_{k=1}^{n} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} x_{k} + \alpha_{i}.
\]
In analytical geometry it is usual to characterise linear configurations
by linear equations connecting the co-ordinates of the
``current'' point (variable). This will be discussed in detail in the
next paragraph. Here we shall just add the fundamental conception
of ``linear forms'' upon which this discussion is founded. A
function~$L(\vx)$, the argument~$\vx$ of which assumes the value of every
vector in turn, these values being real numbers only, is called a
\Emph{linear form}, if it has the functional properties
\index{Form!linear}%
\[
L(\va + \vb) = L(\va) + L(\vb);\qquad
L(\lambda \va) = \lambda · L(\va).
\]
\PageSep{23}
In a co-ordinate system $\ve_{1}$,~$\ve_{2}$,~\dots\Add{,} $\ve_{n}$ each of the $n$ vector-components~$\xi_{i}$
of~$\vx$ is such a linear form. If $\vx$~is defined by~\Eq{(3)}, then
any arbitrary linear form~$L$ satisfies
\[
L(\vx) = \xi_{1} L(\ve_{1}) + \xi_{2} L(\ve_{2}) + \dots + \xi_{n} L(\ve_{n}).
\]
Thus if we put $L(\ve_{i}) = a_{i}$, the linear form, expressed in terms of
components, appears in the form
\[
a_{1} \xi_{1} + a_{2} \xi_{2} + \dots + a_{n} \xi_{n}
\quad\text{(the $a_{i}$'s are its constant co-efficients).}
\]
Conversely, every expression of this type gives a linear form. A
number of linear forms $L_{1}$,~$L_{2}$, $L_{3}$,~\dots\Add{,} $L_{h}$ are linearly independent,
if no constants~$\lambda_{i}$ exist, for which the identity-equation holds:
\[
\lambda_{1} L_{1}(\vx) + \lambda_{2} L_{2}(\vx) + \dots \Add{+} \lambda_{h} L_{h}(\vx) = 0
\]
except $\lambda_{i} = 0$. $n + 1$~linear forms are \emph{always} linearly inter-dependent.
\Section{3.}{The Conception of $n$-dimensional Geometry. Linear
Algebra. Quadratic Forms}
To recognise the perfect mathematical harmony underlying the
laws of space, we must discard the particular dimensional number
$n = 3$. Not only in geometry, but to a still more astonishing
degree in physics, has it become more and more evident that as
soon as we have succeeded in unravelling fully the natural laws
which govern reality, we find them to be expressible by mathematical
relations of surpassing simplicity and architectonic
perfection. It seems to me to be one of the chief objects of
mathematical instruction to develop the faculty of perceiving this
simplicity and harmony, which we cannot fail to observe in the
theoretical physics of the present day. It gives us deep satisfaction
in our quest for knowledge. Analytical geometry, presented
in a compressed form such as that I have used above in exposing
its principles, conveys an idea, even if inadequate, of this perfection
of form. But not only for this purpose must we go beyond the
dimensional number $n = 3$, but also because we shall later require
four-dimensional geometry for concrete physical problems such as
are introduced by the theory of relativity, in which Time becomes
added to Space in a four-dimensional geometry.
We are by no means obliged to seek illumination from the
mystic doctrines of spiritists to obtain a clearer vision of multi-dimensional
geometry. Let us consider, for instance, a homogeneous
mixture of the four gases, hydrogen, oxygen, nitrogen, and
carbon dioxide. An arbitrary quantum of such a mixture is specified
if we know how many grams of each gas are contained
in it. If we call each such quantum a vector (we may bestow
names at will) and if we interpret addition as implying the
\PageSep{24}
\index{Mechanics!of the principle of relativity}%
union of two quanta of the gases in the ordinary sense, then
all the axioms~\Inum{\Chg{\textit{I}}{I}} of our system referring to vectors are fulfilled
for the dimensional number $n = 4$, provided we agree also to
talk of negative quanta of gas. One gram of pure hydrogen, one
gram of oxygen, one gram of nitrogen, and one gram of carbon dioxide
are four ``vectors,'' independent of one another from which
all other gas quanta may be built up linearly; they thus form a co-ordinate
system. Let us take another example. We have five
parallel horizontal bars upon each of which a small bead slides.
A definite condition of this primitive ``adding-machine'' is defined
if the position of each of the five beads upon its respective rod is
known. Let us call such a condition a ``point'' and every simultaneous
displacement of the five beads a ``vector,'' then all of our
\index{Vector}%
axioms are satisfied for the dimensional number $n = 5$. From
this it is evident that constructions of various types may be
evolved which, by an appropriate disposal of names, satisfy our
axioms. Infinitely more important than these somewhat frivolous
examples is the following one which shows that \Emph{our axioms
characterise the basis of our operations in the theory of
linear equations}. If $\alpha_{i}$~and~$\alpha$ are given numbers,
\[
\alpha_{1} x_{1} + \alpha_{2} x_{2} + \dots \Add{+} \alpha_{n} x_{n} = 0
\Tag{(7)}
\]
is usually called a \Emph{homogeneous} linear equation in the unknowns~$x_{i}$,
\index{Homogeneous linear equations}%
whereas
\[
\alpha_{1} x_{1} + \alpha_{2} x_{2} + \dots \Add{+} \alpha_{n} x_{n} = \alpha
\Tag{(8)}
\]
is called a \Emph{non-homogeneous} linear equation. In treating the theory
\index{Non-homogeneous linear equations}%
of linear homogeneous equations, it is found useful to have a short
name for the system of values of the variables~$x_{i}$; we shall call it
``vector''. In carrying out calculations with these vectors, we
shall define the sum of the two vectors
\[
(a_{1}, a_{2}, \dots\Add{,} a_{n})
\quad\text{and}\quad
(b_{1}, b_{2}, \dots\Add{,} b_{n})
\]
to be the vector
\[
(a_{1} + b_{1}, a_{2} + b_{2}, \dots\Add{,} a_{n} + b_{n})
\]
and $\lambda$~times the first vector to be
\[
(\lambda a_{1}, \lambda a_{2}, \dots\Add{,} \lambda a_{n}).
\]
The axioms~\Inum{\Chg{\textit{I}}{I}} for vectors are then fulfilled for the dimensional number~$n$.
\index{Space!n-dimensional@{$n$-dimensional}}%
\begin{align*}
\ve_{1} &= (1, 0, 0, \dots\Add{,} 0), \\
\ve_{2} &= (0, 1, 0, \dots\Add{,} 0), \\
\multispan{2}{\dotfill} \\
\ve_{n} &= (0, 0, 0, \dots\Add{,} 1)
\end{align*}
form a system of independent vectors. The components of any
arbitrary vector $(x_{1}, x_{2}, \dots\Add{,} x_{n})$ in this co-ordinate system are the
\PageSep{25}
numbers $x_{i}$ themselves. The fundamental theorem in the solution
\index{Geometry!n-dimensional@{$n$-dimensional}}%
of linear homogeneous equations may now be stated thus:---
\[
\text{if}\quad
L_{1}(\vx),\quad
L_{2}(\vx),\quad \dots\Add{,}\quad
L_{h}(\vx)
\]
are $h$~linearly independent linear forms, the solutions~$\vx$ of the
equations
\[
L_{1}(\vx) = 0,\quad
L_{2}(\vx) = 0,\quad \dots\Add{,}\quad
L_{h}(\vx) = 0
\]
form an $(n - h)$-dimensional linear vector manifold.
In the theory of non-homogeneous linear equations we shall
find it advantageous to denote a system of values of the variables~$x_{i}$
a ``point''. If $x_{i}$~and~$x_{i}'$ are two systems which are solutions
of equation~\Eq{(8)}, their difference
\[
x_{1}' - x_{1},\quad
x_{2}' - x_{2},\ \dots\Add{,}\quad
x_{n}' - x_{n}
\]
is a solution of the corresponding homogeneous equation~\Eq{(7)}. We
shall, therefore, call this difference of two systems of values of the
variables~$x_{i}$ a ``vector,'' viz.\ the ``vector'' defined by the two
``points'' $(x_{i})$ and~$(x_{i}')$; we make the above conventions for the
addition and multiplication of these vectors. \Emph{All the axioms then
hold.} In the particular co-ordinate system composed of the vectors~$\ve_{i}$
given above, and having the ``origin'' $O = (0, 0, \dots\Add{,} 0)$,
the co-ordinates of a point~$(x_{i})$ are the numbers $x_{i}$ themselves.
The fundamental theorem concerning linear equations is: those
points which satisfy $h$~independent linear equations, form a point-configuration
of $n - h$~dimensions.
In this way we should not only have arrived quite naturally at
our axioms without the help of geometry by using the theory of linear
equations, but we should also have reached the wider conceptions
which we have linked up with them. In some ways, indeed, it
would appear expedient (as is shown by the above formulation of
the theorem concerning homogeneous equations) to build up the
theory of linear equations upon an axiomatic basis by starting from
the axioms which have here been derived from geometry. A theory
developed along these lines would then hold for any domain of
operations, for which these axioms are fulfilled, and not only for a
``system of values in $n$~variables''. It is easy to pass from such
a theory which is more conceptual, to the usual one of a more
formal character which operates from the outset with numbers~$x_{i}$ by
taking a definite co-ordinate system as a basis, and then using in
place of vectors and points their components and co-ordinates
respectively.
It is evident from these arguments that the whole of affine
geometry merely teaches us that space is a \Emph{region of three dimensions
in linear quantities} (the meaning of this statement
\PageSep{26}
will be sufficiently clear without further explanation). All the
separate facts of intuition which were mentioned in~§\,1 are simply
disguised forms of this one truth. Now, if on the one hand it is very
satisfactory to be able to give a common ground in the theory of
knowledge for the many varieties of statements concerning space,
spatial configurations, and spatial relations which, taken together,
constitute geometry, it must on the other hand be emphasised that
this demonstrates very clearly with what little right mathematics
may claim to expose the intuitional nature of space. Geometry
contains no trace of that which makes the space of intuition what it
\Emph{is} in virtue of its own entirely distinctive qualities which are not
shared by ``states of addition-machines'' and ``gas-mixtures'' and
``systems of solutions of linear equations''. It is left to metaphysics
to make this ``comprehensible'' or indeed to show why
and in what sense it is incomprehensible. We as mathematicians
have reason to be proud of the wonderful insight into the knowledge
of space which we gain, but, at the same time, we must recognise
with humility that our conceptual theories enable us to grasp only
one aspect of the nature of space, that which, moreover, is most
formal and superficial.
To complete the transition from affine geometry to complete
metrical geometry we yet require several conceptions and facts
which occur in linear algebra and which refer to \Emph{bilinear and
quadratic forms}. A function $Q(\vx\Com \vy)$ of two arbitrary vectors $\vx$
and~$\vy$ is called a bilinear form if it is a linear form in~$\vx$ as well as
\index{Bilinear form}%
\index{Form!bilinear}%
in~$\vy$. If in a certain co-ordinate system $\xi_{i}$~are the components of~$\vx$,
$\eta_{i}$~those of~$\vy$, then an equation
\[
Q(\vx\Com \vy) = \sum_{i, k=1}^{n} \Typo{\alpha}{a}_{ik} \xi_{i} \eta_{k}
\]
with constant co-efficients~$\Typo{\alpha}{a}_{ik}$ holds. We shall call the form ``non-degenerate''
if it vanishes identically in~$\vy$ only when the vector
$\vx = \Typo{0}{\0}$. This happens when, and only when, the homogeneous
equations
\[
\sum_{i=1}^{n} \Typo{\alpha}{a}_{ik} \xi_{i} = 0
\]
have a single solution $\xi_{i} = 0$ or when the determinant $|\Typo{\alpha}{a}_{ik}| \neq 0$.
From the above explanation it follows that this condition, viz.\ the
non-vanishing of the determinant, persists for arbitrary linear transformations.
The bilinear form is called \Emph{symmetrical} if $Q(\vy\Com \vx) = Q(\vx\Com \vy)$.
\index{Symmetry}%
This manifests itself in the co-efficients by the symmetrical
\PageSep{27}
property $\Typo{\alpha}{a}_{ki} = \Typo{\alpha}{a}_{ik}$. Every bilinear form~$Q(\vx\Com \vy)$ gives rise to a
\index{Form!quadratic}%
\Emph{quadratic form} which depends on only one variable vector~$\vx$
\[
Q(\vx) = Q(\vx\Com \vx) = \sum_{i,k=1}^{n} \Typo{\alpha}{a}_{ik} \xi_{i} \xi_{k}.
\]
In this way every quadratic form is derived in general from one,
and only one, \Emph{symmetrical} bilinear form. The quadratic form~$Q(\vx)$
which we have just formed may also be produced from the
symmetrical form
\[
\tfrac{1}{2}\bigl\{Q(\vx\Com \vy) + Q(\vy\Com \vx)\bigr\}
\]
by identifying $\vx$ with~$\vy$.
To prove that one and the same quadratic form cannot arise
from two different symmetrical bilinear forms, one need merely
show that a symmetrical bilinear form~$Q(\vx\Com \vy)$ which satisfies the
equation~$Q(\vx\Com \vx)$ identically for~$\vx$, vanishes identically. This,
however, immediately results from the relation which holds for
every symmetrical bilinear form
\[
Q(\vx + \vy\Com \vx + \vy)
= Q(\vx\Com \vx) + 2Q(\vx\Com \vy) + Q(\vy\Com \vy)\Add{.}
\Tag{(9)}
\]
If $Q(\vx)$ denotes any arbitrary quadratic form then $Q(\vx\Com \vy)$~is always
%[** TN: Original entry points to page 17]
\index{Definite@{\emph{Definite, positive}}}%
\index{Non-degenerate bilinear and quadratic forms}%
to signify the symmetrical bilinear form from which $Q(\vx)$~is derived
(to avoid mentioning this in each particular case). When we say
that a quadratic form is non-degenerate we wish to convey that the
above symmetrical bilinear form is non-degenerate. A quadratic
form is \Emph{positive definite} if it satisfies the inequality $Q(\vx) > 0$ for
\index{Positive definite}%
every value of the vector $\vx \neq \Typo{0}{\0}$. Such a form is certainly non-degenerate,
for no value of the vector $\vx \neq \Typo{0}{\0}$ can make $Q(\vx\Com \vy)$~vanish
identically in~$\vy$, since it gives a positive result for $\vy = \vx$.
\Section{4.}{The Foundations of Metrical Geometry}
\index{Axioms!of metrical geometry!(Euclidean)}%
\index{Geometry!metrical}%
To bring about the transition from affine to metrical geometry
we must once more draw from the fountain of intuition. From it
we obtain for three-dimensional space the definition of the \Emph{scalar
\index{Scalar!product}%
product} of two vectors $\va$~and~$\vb$. After selecting a definite vector
\index{Product!scalar}%
as a unit we measure out the length of~$\va$ and the length (negative
or positive as the case may be) of the perpendicular projection of~$\vb$
upon~$\va$ and multiply these two numbers with one another. This
means that the lengths of not only parallel straight lines may be
compared with one another (as in affine geometry) but also such
as are arbitrarily inclined to one another. The following rules
hold for scalar products:---
\[
\lambda \va · \vb = \lambda(\va · \vb)\qquad
(\va + \va') · \vb = (\va · \vb) + (\va' · \vb)
\]
\PageSep{28}
and analogous expressions with reference to the second factor; in
addition, the commutative law $\va · \vb = \vb · \va$. The scalar product
of~$\va$ with $\va$~itself, viz.\ $\va · \va = \va^{2}$, is always positive except when
$\va = \Typo{0}{\0}$, and is equal to the square of the length of~$\va$. These laws
signify that the scalar product of two arbitrary vectors, i.e.\ $\vx · \vy$ is
a symmetrical bilinear form, and that the quadratic form which
arises from it is positive definite. We thus see that not the length,
but the square of the length of a vector depends in a simple rational
way on the vector itself; it is a quadratic form. This is the real
content of Pythagoras' Theorem. The scalar product is nothing
more than the symmetrical bilinear form from which this quadratic
form has been derived. We accordingly formulate the following:---
\begin{Axiom}[Metrical Axiom:]
If a unit vector~$\ve$, differing from zero, be
chosen, every two vectors $\vx$~and~$\vy$ uniquely determine a number
$(\vx · \vy) = Q(\vx\Com \vy)$; the latter, being dependent on the two vectors, is a
symmetrical bilinear form.
\end{Axiom}
The quadratic form $(\vx · \vx) = Q(\vx)$ which
arises from it is positive definite. $Q(\ve) = 1$.
We shall call~$Q$ the \Emph{metrical groundform}. We then have
\index{Co-ordinates, curvilinear!linear@{(in a linear manifold)}}%
\index{Groundform, metrical!linear@{(of a linear manifold)}}%
\index{Metrical groundform}%
that
\begin{Axiom}
an affine transformation which, in general, transforms the vector~$\vx$
into~$\vx'$ is a congruent one if it leaves the metrical groundform
\index{Congruent!transformations}%
\index{Transformation or representation!congruent}%
unchanged:---
\[
Q(\vx') = Q(\vx)\Add{.}
\Tag{(10)}
\]
Two geometrical figures which can be transformed into one another
by a congruent transformation are congruent.\footnotemark
\end{Axiom}
\footnotetext{We take no notice here of the difference between direct congruence and
mirror congruence (lateral inversion). It is present even in affine transformations,
in $n$-dimensional space as well as $3$-dimensional space.}%
The conception of
congruence is \Emph{defined} in our axiomatic scheme by these statements.
If we have a domain of operation in which the axioms
of~§\,2 are fulfilled, we can choose any arbitrary positive definite
quadratic form in it, ``promote'' it to the position of a fundamental
metrical form, and, using it as a basis, define the conception
of congruence as was just now done. This form then endows the
affine space with metrical properties and Euclidean geometry in
its entirety now holds for it. The formulation at which we have
arrived is not limited to any special dimensional number.
It follows from~\Eq{(10)}, in virtue of relation~\Eq{(9)} of~§\,3, that for a
congruent transformation the more general relation
\[
Q(\vx'\Com \vy') = Q(\vx\Com \vy)
\]
holds.
Since the conception of congruence is defined by the metrical
groundform it is not surprising that the latter enters into all
formulæ which concern the measure of geometrical quantities.
Two vectors $\va$~and~$\va'$ are congruent if, and only if,
\[
Q(\va) = Q(\va').
\]
\PageSep{29}
We could accordingly introduce~$Q(\va)$ as a measure of the vector~$\va$.
Instead of doing this, however, we shall use the positive square
root of~$Q(\va)$ for this purpose and call it the length of the vector~$\va$
(this we shall adopt as our definition) so that the further condition
is fulfilled that the length of the sum of two parallel vectors pointing
in the same direction is equal to the sum of the lengths of the
two single vectors. If $\va$,~$\vb$ as well as $\va'$,~$\vb'$ are two pairs of
vectors, all of length unity, then the figure formed by the first two
is congruent with that formed by the second pair, if, and only if,
$Q(\va, \vb) = Q(\va', \vb')$.
In this case again we do not introduce the number $Q(\va, \vb)$ itself
as a measure of the \Emph{angle}, but a number~$\theta$ which is related to it by
the transcendental function cosine thus\Add{:}---
\[
\cos \theta = Q(\va, \vb)
\]
so as to be in agreement with the theorem that the numerical
measure of an angle composed of two angles in the same plane is
\index{Angles!measurement of}%
\index{Angles!right}%
the sum of the numerical values of these angles. The angle which
is formed from any two arbitrary vectors $\va$~and~$\vb$ ($\neq \Typo{0}{\0}$) is then
calculated from
\[
\cos \theta = \frac{Q(\va, \vb)}{\sqrt{Q(\va\Com \va) · Q(\vb\Com \vb)}}\Add{.}
\Tag{(11)}
\]
In particular, two vectors $\va$,~$\vb$ are said to be \Emph{perpendicular} to one
\index{Perpendicularity!(in general)}%
\index{Right angle}%
another if $Q(\va\Com \vb) = 0$. This reminder of the simplest metrical
formulæ of analytical geometry will suffice.
The angle defined by~\Eq{(11)} which has been formed by two vectors
is shown always to be real by the inequality
\[
Q^{2}(\va\Com \vb) \leq Q(\va) · Q(\vb)
\Tag{(12)}
\]
which holds for every quadratic form~$Q$ which is $\geq 0$ for all values
of the argument. It is most simply deduced by forming
\[
Q(\lambda\va + \mu\vb)
= \lambda^{2} Q(\va) + 2\lambda\mu Q(\va\Com \vb) + \mu^{2} Q(\vb) \geq 0.
\]
Since this quadratic form in $\lambda$~and~$\mu$ cannot assume both positive
and negative values its ``discriminant'' $Q^{2}(\va\Com \vb) - \Typo{(Q)}{Q}(\va) · \Typo{(Q)}{Q}(\vb)$
cannot be positive.
A number,~$n$, of independent vectors form a \Emph{Cartesian co-ordinate
system} if for every vector
\index{Cartesian co-ordinate systems}%
\index{Co-ordinate systems!Cartesian}%
\begin{gather*}
\vx = x_{1}\ve_{1} + x_{2}\ve_{2} + \dots \Add{+} x_{n}\ve_{n} \\
Q(\vx) = x_{1}^{2} + x_{2}^{2} + \dots \Add{+} x_{n}^{2}
\Tag{(13)}
\end{gather*}
holds, i.e.\ if
\[
Q(\ve_{i}, \ve_{j})
= \begin{cases}
1 & (i = k)\Add{,} \\
0 & (i \neq k).
\end{cases}
\]
\PageSep{30}
From the standpoint of metrical geometry all co-ordinate
systems are of equal value. A proof (appealing directly to our
geometrical sense) of the theorem that such systems exist will
now be given not only for a ``definite'' but also for any arbitrary
non-degenerate quadratic form, inasmuch as we shall find later in
the theory of relativity that it is just the ``indefinite'' case that
plays the decisive rôle. We enunciate as follows:---
\emph{Corresponding to every non-degenerate quadratic form~$Q$ a co-ordinate
system~$\ve_{i}$ can be introduced such that}
\[
Q(\vx) = \epsilon_{1} x_{1}^{2}
+ \epsilon_{2} x_{2}^{2} + \dots
+ \epsilon_{n} x_{n}^{2}\quad (\epsilon_{i} = ± 1)\Add{.}
\Tag{(14)}
\]
\Proof.---Let us choose any arbitrary vector~$\ve_{1}$ for which $Q(\ve_{1}) \Typo{=}{}\neq 0$.
By multiplying it by an appropriate positive constant we
can arrange so that $Q(\ve_{1}) = ±1$. We shall call a vector~$\vx$ for which
$Q(\ve_{1}\Com \vx) = 0$ \Emph{orthogonal} to~$\ve_{1}$. If $\vx^{*}$~is a vector which is orthogonal
to~$\ve_{1}$, and if $x_{1}$~is any arbitrary number, then
\[
\vx = x_{1}\ve_{1} + x^{*}
\Tag{(15)}
\]
satisfies Pythagoras' Theorem:---
\[
Q(\vx) = x_{1}^{2} Q(\ve_{1}) + 2x_{1}Q(\ve_{1}\Com \vx^{*}) + Q(\vx^{*})
= ± x_{1}^{2} + Q(\vx^{*}).
\]
{\Loosen The vectors orthogonal to~$\ve_{1}$ constitute an $(n - 1)$-dimensional
linear manifold, in which $Q(\vx)$~is a non-degenerate quadratic form.
Since our theorem is self-evident for the dimensional number $n = 1$,
%[** TN: "n - 1" grouped with a viniculum in the original]
we may assume that it holds for $(n - 1)$~dimensions (proof by
successive induction from the case $(n - 1)$ to that of~$n$). According
to this, $n - 1$~vectors $\ve_{\Typo{3}{2}}$,~\dots\Add{,} $\ve_{n}$, orthogonal to~$\ve_{1}$ exist, such that
for}
\[
\vx^{*} = x_{2}\ve_{2} + \dots + x_{n}\ve_{n}
\]
the relation
\[
Q(\vx^{*}) = ± x_{2}^{2} ± \dots ± x_{n}^{2}
\]
holds.
This enables $Q(\vx)$~to be expressed in the required form.
Then
\[
Q(\ve_{i}) = \epsilon_{i}\qquad
Q(\ve_{i}, \ve_{k}) = 0\quad (i \neq k).
\]
These relations result in all the~$\ve_{i}$'s being independent of one
another and in each vector~$\vx$ being representable in the form~\Eq{(13)}.
They give
\[
x_{i} = \epsilon_{i} · Q(\ve_{i}, \vx)\Add{.}
\Tag{(16)}
\]
An important corollary is to be made in the ``indefinite'' case.
\index{Inertial force!index}%
\index{Inertial force!law of quadratic forms}%
The numbers $r$~and~$s$ attached to the~$\epsilon_{i}$'s, and having positive and
negative signs respectively, are uniquely determined by the quadratic
form: it may be said to have $r$~positive and $s$~negative
dimensions. ($s$~may be called the inertial index of the quadratic
form, and the theorem just enunciated is known by the name
``Law of Inertia''. The classification of surfaces of the second
\PageSep{31}
\index{Quadratic forms}%
order depends on it.) The numbers $r$~and~$s$ may be characterised
invariantly thus:---
There are $r$~mutually orthogonal vectors~$\ve$, for which $Q(\ve) > 0$;
but for a vector~$\vx$ which is orthogonal to these and not equal to~$\Typo{0}{\0}$,
it necessarily follows that $Q(\vx) < 0$. Consequently there cannot
be more than~$r$ such vectors. A corresponding theorem holds
for~$s$.
$r$~vectors of the required type are given by \Emph{those} $r$ fundamental
vectors~$\ve_{i}$ of the co-ordinate system upon which the
expression~\Eq{(14)} is founded, to \Emph{which} the positive signs~$\epsilon_{i}$ correspond.
The corresponding components~$x_{i}$ ($i = 1, 2, 3,~\dots\Add{,} r$) are
definite linear forms of~$\vx$ [cf.~\Eq{(16)}]: $x_{i} = L_{i}(\vx)$. If, now, $\ve_{i}$
($i = 1, 2, \dots\Add{,} r$) is any system of vectors which are mutually
orthogonal to one another, and satisfy the condition $Q(\ve_{i}) > 0$, and
if $\vx$~is a vector orthogonal to these~$\ve_{i}$, we can set up a linear combination
\[
\vy = \lambda_{1}\ve_{1} + \dots \Add{+} \lambda_{r}\ve_{r} + \mu \vx
\]
in which not all the co-efficients vanish and which satisfies the $r$~homogeneous
equations
\[
L_{1}(\vy) = 0,\quad \dots\Add{,\quad}
L_{r}(\vy) = 0.
\]
It is then evident from the form of the expression that $Q(\vy)$~must
be negative unless $\vy = \Typo{0}{\0}$. In virtue of the formula
\[
Q(\vy) - \bigl\{\lambda_{1}^{2} Q(\ve_{1}) + \dots + \lambda_{r}^{2} Q(\ve_{r})\bigr\}
= \mu^{2} Q(\vx)
\]
it then follows that $Q(\vx) < 0$ except in the case in which if $\vy = \Typo{0}{\0}$,
$\lambda_{1} = \dots = \lambda_{r}$ also $= 0$. But then, by hypothesis, $\mu$~must $\neq 0$,
i.e.\ $\vx = \Typo{0}{\0}$.
\begin{Remark}
In the theory of relativity the case of a quadratic form with one negative
and $n - 1$~positive dimensions becomes important. In three-dimensional
\index{Dimensions!(positive and negative, of a quadratic form)}%
space, if we use affine co-ordinates,
\[
-x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 0
\]
is the equation of a cone having its vertex at the origin and consisting of
two sheets, as expressed by the negative sign of~$x_{1}^{2}$, which are only connected
with one another at the origin of co-ordinates. This division into
two sheets allows us to draw a distinction between past and future in the
theory of relativity. We shall endeavour to describe this by an elementary
analytical method here instead of using characteristics of continuity.
Let $Q$~be a non-degenerate quadratic form having only one negative
dimension. We choose a vector, for which $Q(\ve) = -1$. We shall call
these vectors~$\vx$, which are not zero and for which $Q(\vx) \leq 0$ ``negative
vectors''. According to the proof just given for the Theorem of Inertia,
no negative vector can satisfy the equation $Q(\ve\Com \vx) = 0$. Negative vectors
thus belong to one of two classes or ``cones'' according as $Q(\ve\Com \vx) < 0$ or~$> 0$;
\PageSep{32}
$\ve$~itself belongs to the former class, $-\ve$~to the latter. A negative
vector~$\vx$ lies ``inside'' or ``on the sheet'' of its cone according as $Q(\vx) < 0$
or~$= 0$. To show that the two cones are independent of the choice of
the vector~$\ve$, one must prove that, from $Q(\ve) = Q(\ve') = -1$, and $Q(\vx) \leq 0$,
it follows that the sign of~$\dfrac{Q(\ve'\Com \vx)}{Q(\ve\Com \vx)}$ is the same as that of~$-Q(\ve\Com \ve')$.
Every vector~$\vx$ can be resolved into two summands
\[
\vx = x\ve + \vx^{*}
\]
such that the first is proportional and the second~($\vx^{*}$) is orthogonal to~$\ve$.
One need only take $\vx = - Q(\ve\Com \vx)$ and we then get
\[
Q(\vx) = -x^{2} + Q(\vx^{*})\Add{;}
\]
$Q(\vx^{*})$~is, as we know, necessarily $\geq 0$. Let us denote it by~$Q^{*}$.
The equation
\[
Q^{*} = x^{2} + Q(\vx) = Q^{2}(\ve\Com \vx) + Q(\vx)
\]
then shows that $Q^{*}$~is a quadratic form (degenerate), which satisfies the
identity or inequality, $Q^{*}(\vx) \geq 0$. We now have
\[
\begin{gathered}
Q(\vx) = -x^{2} + Q^{*}(\vx) \leq 0\Add{,} \\
\{x = -Q(\ve\Com \vx)\}\Add{;}
\end{gathered}
\qquad
\begin{gathered}
Q(\ve') = -e'^{2} + Q^{*}(\ve') < 0\Add{,} \\
\{e' = -Q(\ve\Com \ve)\}\Add{.}
\end{gathered}
\]
From the inequality~\Eq{(12)} which holds for~$Q^{*}$, it follows that
\[
\bigl\{Q^{*}(\ve'\Com \vx)\bigr\}^{2}
\leq Q^{*}(\ve') · Q^{*}(\vx)
< e'^{2} x^{2};
\]
consequently
\[
-Q(\ve'\Com \vx) = e'x - Q^{*}(\ve'\Com \vx)
\]
has the same sign as the first summand~$e'x$.
\end{Remark}
Let us now revert to the case of a definitely positive metrical
groundform with which we are at present concerned. If we use
a Cartesian co-ordinate system to represent a congruent transformation,
the co-efficients of transformation~$\Chg{\alpha_{ik}}{\alpha_{i}^{k}}$ in formula~\Eq{(5')}, §\,2,
will have to be such that the equation
\[
\xi_{1}'^{2} + \xi_{2}'^{2} + \dots + \xi_{n}'^{2}
= \xi_{1}^{2} + \xi_{2}^{2} + \dots + \xi_{n}^{2}
\]
is identically satisfied by the~$\xi$'s. This gives the ``conditions for
orthogonality''
\[
\sum_{r=1}^{n} \Chg{\alpha_{ri}}{\alpha_{r}^{i}}\Chg{\alpha_{rj}}{\alpha_{r}^{j}}
= \begin{cases}
1 & (i = j)\Add{,} \\
0 & (i \neq j)\Add{.}
\end{cases}
\Tag{(17)}
\]
They signify that the transition to the inverse transformation converts
the co-efficients~$\Chg{\alpha_{ik}}{\alpha_{i}^{k}}$ into~$\Chg{\alpha_{ki}}{\alpha_{k}^{i}}$:---
\[
\xi_{i} = \sum_{k=1}^{n} \Chg{\alpha_{ki}}{\alpha_{k}^{i}} \xi_{k}'.
\]
It furthermore follows that the determinant $\Delta = |\Chg{\alpha_{ik}}{\alpha_{i}^{k}}|$ of a congruent
transformation is identical with that of its inverse, and since
their product must equal~$1$, $\Delta = ±1$. The positive or the negative
\PageSep{33}
sign would occur according as the congruence is real or inverted as
in a mirror (``lateral inversion'').
Two possibilities present themselves for the analytical treatment
\index{Space!metrical}%
of metrical geometry. \Emph{Either} one imposes no limitation upon the
affine co-ordinate system to be used: the problem is then to develop
a theory of invariance with respect to arbitrary linear transformations,
in which, however, in contra-distinction to the case of
affine geometry, we have a definite invariant quadratic form, viz.\
the metrical groundform
\[
Q(\vx) = \sum_{i,k=1}^{n} g_{ik} \xi_{i} \xi_{k}
\]
once and for all as an absolute datum. \Emph{Or}, we may use Cartesian
co-ordinate systems from the outset: in this case, we are concerned
with a theory of invariance for orthogonal transformations, i.e.\
linear transformations, in which the co-efficients satisfy the secondary
conditions~\Eq{(17)}. We must here follow the first course so as to
be able to pass on later to generalisations which extend beyond the
limits of Euclidean geometry. This plan seems advisable from the
\index{Euclidean!geometry|)}%
\index{Geometry!Euclidean|)}%
algebraic point of view, too, since it is easier to gain a survey of
those expressions which remain unchanged for \Emph{all} linear transformations
than of those which are only invariant for orthogonal
transformations (a class of transformations which are subjected to
secondary limitations not easy to define).
We shall here develop the Theory of Invariance as a ``Tensor
\index{Tensor!linear@{(in linear space)}}%
Calculus'' along lines which will enable us to express in a convenient
mathematical form, not only geometrical laws, but also
all physical laws.
\Section{5.}{Tensors}
Two linear transformations,
\begin{alignat*}{2}
\xi^{i} &= \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k}, \qquad
&&\bigl(|\alpha_{k}^{i}| \neq 0\bigr)
\Tag{(18)} \\
%
\eta_{i} &= \sum_{k} \breve{\alpha}_{i}^{k} \bar{\eta}_{k}, \qquad
&&\bigl(|\breve{\alpha}_{i}^{k}| \neq 0\bigr)
\Tag{(18')}
\end{alignat*}
in the variables $\xi$~and~$\eta$ respectively, leading to the variables $\bar{\xi}$,~$\bar{\eta}$
are said to be \Emph{contra-gredient} to one another, if they make the
bilinear form $\sum_{i} \eta_{i} \xi^{i}$ transform into itself, i.e.\
\[
\sum_{i} \eta_{i} \xi^{i} = \sum_{i} \bar{\eta}_{i} \bar{\xi}^{i}\Add{.}
\Tag{(19)}
\]
\PageSep{34}
Contra-gredience is thus a reversible relationship. If the variables
$\xi$,~$\eta$ are transformed into $\bar{\xi}$,~$\bar{\eta}$ by one pair of contra-gredient transformations
$A$,~$\breve{A}$, and then $\bar{\xi}$,~$\bar{\eta}$ into $\bbar{\xi}$,~$\bbar{\eta}$ by a second pair $B$,~$\breve{B}$ it
follows from
\[
\sum_{i} \eta_{i} \xi^{i}
= \sum_{i} \bar{\eta}_{i} \bar{\xi}^{i}
= \sum_{i} \bbar{\eta}_{i} \bbar{\xi}^{i}\Typo{,}{}
\]
that the two transformations combined, which transform $\xi$ directly
into~$\bbar{\xi}$, and $\eta$~into $\bbar{\eta}$ are likewise contra-gredient. The co-efficients
of two contra-gredient substitutions satisfy the conditions
\[
\sum_{r} \alpha_{i}^{r} \breve{\alpha}_{r}^{k} = \delta_{i}^{k}
= \begin{cases}
1 & (i = k)\Add{,} \\
0 & (i \neq k)\Add{.}
\end{cases}
\Tag{(20)}
\]
If we substitute for the~$\xi$'s in the left-hand member of~\Eq{(19)} their
values in terms of~$\bar{\xi}$ obtained from~\Eq{(18)}, it becomes evident that
the equations~\Eq{(18')} are derived by reduction from
\[
\bar{\eta}_{i} = \sum_{k} \alpha_{i}^{k} \eta_{k}\Add{.}
\Tag{(21)}
\]
There is thus one and only one contra-gredient transformation
\index{Contra-gredient transformation}%
corresponding to every linear transformation. For the same reason
as~\Eq{(21)}
\[
\Typo{\bar{\xi}_{i}}{\bar{\xi}^{i}} = \sum_{k} \breve{\alpha}_{k}^{i} \xi^{k}
\]
holds. By substituting these expressions and~\Eq{(21)} in~\Eq{(19)}, we
find that the co-efficients, in addition to satisfying the conditions~\Eq{(20)},
satisfy
\[
\sum_{r} \alpha_{r}^{i} \breve{\alpha}_{k}^{r} = \delta_{k}^{i}.
\]
An orthogonal transformation is one which is contra-gredient to
\index{Orthogonal transformations}%
itself. If we subject a linear form in the variables~$\xi_{i}$ to any
arbitrary linear transformation the co-efficients become transformed
contra-grediently to the variables, or they assume a ``contra-variant''
relationship to these, as it is sometimes expressed.
In an affine co-ordinate system $O \Chg{;}{\mid} \ve_{1}, \ve_{2}, \dots\Add{,} \ve_{n}$ we have up
to the present characterised a displacement~$\vx$ by the uniquely defined
components~$\xi^{i}$ given by the equation
\[
\vx = \xi^{1} \ve_{1} + \xi^{2} \ve_{2} + \dots + \xi^{n} \ve_{n}.
\]
\PageSep{35}
% [** TN: Interrupt math mode below to allow line break]
If we pass over into another affine co-ordinate system $\bar{O} \mid
\bar{\ve}_{1}$, $\bar{\ve}_{2}, \dots\Add{,} \bar{\ve}_{n}$, whereby
\[
\bar{\ve}_{i} = \sum_{k} \alpha_{i}^{k} \ve_{k}\Add{,}
\]
the components of~$\vx$ undergo the transformation
\[
\xi^{i} = \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k}
\]
as is seen from the equation
\[
\vx = \sum_{i} \xi^{i} \ve_{i} = \sum_{i} \bar{\xi}^{i} \bar{\ve}_{i}.
\]
These components thus transform themselves contra-grediently
\index{Components, co-variant, and contra-variant!displacement@{of a displacement}}%
\index{Contra-variant tensors}%
to the fundamental vectors of the co-ordinate system, and are related
contra-variantly to them; they may thus be more precisely
termed the \Emph{contra-variant components} of the vector~$\vx$. In
\Emph{metrical} space, however, we may also characterise a displacement
in relation to the co-ordinate system by the values of its scalar
product with the fundamental vectors~$\ve_{i}$ of the co-ordinate system
\[
\xi_{i} = (\vx · \ve_{i}).
\]
In passing over into another co-ordinate system these quantities
transform themselves---as is immediately evident from their definition---``co-grediently''
to the fundamental vectors (just like the
latter themselves), i.e.\ in accordance with the equations
\[
\bar{\xi}_{i} = \sum_{k} \alpha_{i}^{k} \xi_{k};
\]
they behave ``co-variantly''. We shall call them the \Emph{co-variant
components} of the displacement. The connection between co-variant
and contra-variant components is given by the formulæ
\[
\xi_{i} = \sum_{k} (\ve_{i} · \ve_{k}) \xi^{k}
= \sum_{k} g_{ik} \xi^{k}
\Tag{(22)}
\]
or by their inverses (which are derived from them by simple resolution)
respectively
\[
\xi^{i} = \sum_{k} g^{ik} \xi_{k}\Add{.}
\Tag{(22')}
\]
In a Cartesian co-ordinate system the co-variant components coincide
with the contra-variant components. It must again be emphasised
that the contra-variant components alone are at our disposal
in affine space, and that, consequently, wherever in the following
\PageSep{36}
pages we speak of the components of a displacement without
specifying them more closely, the contra-variant ones are implied.
Linear forms of one or two arbitrary displacements have already
\index{Order of tensors}%
been discussed above. We can proceed from two arguments to
three or more. Let us take, for example, a trilinear form $A(\vx\Com \vy\Com \vz)$.
If in an arbitrary co-ordinate system we represent the two displacements
$\vx$,~$\vy$ by their contra-variant components, $\vz$~by its
co-variant components, i.e.\ $\xi^{i}$,~$\eta^{i}$, and~$\zeta_{i}$ respectively, then $A$~is
algebraically expressed as a trilinear form of these three series of
variables with definite number-\Chg{coefficients}{co-efficients}
\[
\sum_{i\Add{,} j\Add{,} k} \Typo{\alpha}{a}_{ik}^{l} \xi^{i} \eta^{k} \zeta_{l}\Add{.}
\Tag{(23)}
\]
Let the analogous expression in a different co-ordinate system,
indicated by bars, be
\[
\sum_{i\Add{,} j\Add{,} k} \bar{\Typo{\alpha}{a}}_{ik}^{l} \bar{\xi}^{i} \bar{\eta}^{k}
\Typo{\bar{\zeta}^{l}}{\bar{\zeta}_{l}}\Add{.}
\Tag{(23')}
\]
{\Loosen A connection between the two algebraic trilinear forms \Eq{(23)} and
\Eq{(23')} then exists, by which the one resolves into the other if the
two series of variables $\xi$,~$\eta$ are transformed contra-grediently to the
fundamental vectors, but the series~$\zeta$ co-grediently to the latter.
This relationship enables us to calculate the co-efficient~$\bar{\Typo{\alpha}{a}}_{ik}^{l}$ of~$A$
in the\Erratum{}{ second} co-ordinate system if the co-efficients~$\Typo{\alpha}{a}_{ik}^{l}$ and also the
transformation co-efficient~$\alpha_{i}^{k}$ leading from one co-ordinate system
to the other are known. We have thus arrived at the conception
of the ``$r$-fold co-variant, $s$-fold contra-variant tensor of the
% [** TN: Ordinal]
$(r + s)$th~degree'': it is not confined to metrical geometry but only
assumes the space to be affine. We shall now give an explanation
of this tensor \textit{in abstracto}. To simplify our expressions we shall
take special values for the numbers $r$~and~$s$ as in the example
quoted above: $r = 2$, $s = l$, $r + s = 3$. We then enunciate:---}
\emph{A trilinear form of three series of variables which is \Erratum{independent of}{dependent on}
the co-ordinate system is called a doubly co-variant, singly contra-variant
tensor of the third degree if the above relationship is as
follows. The expressions for the linear form in any two co-ordinate
systems, viz.:---
\[
\sum \Typo{\alpha}{a}_{ik}^{l} \xi^{i} \eta^{k} \zeta_{l},\qquad
\sum \bar{\Typo{\alpha}{a}}_{ik}^{l} \bar{\xi}^{i} \bar{\eta}^{k} \bar{\zeta}_{l}
\]
resolve into one another, if two of the series of variables (viz.\ the
first two $\xi$~and~$\eta$) are transformed contra-grediently to the fundamental
vectors of the co-ordinate system and the third co-grediently
\PageSep{37}
to the same.} The co-efficients of the linear form are called the
components of the tensor in the co-ordinate system in question.
Furthermore, they are called co-variant in the indices, $i$,~$k$, which
are associated with the variables to be transformed contra-grediently,
and contra-variant in the others (here only the one index~$l$).
The terminology is based upon the fact that the co-efficients of
a uni-linear form behave co-variantly if the variables are transformed
contra-grediently, but contra-variantly if they are transformed
co-grediently. Co-variant indices are always attached as suffixes
to the co-efficients, contra-variant ones written at the top of the
co-efficients. Variables with lowered indices are always to be
transformed co-grediently to the fundamental vectors of the co-ordinate
system, those with raised indices are to be transformed
contra-grediently to the same. A tensor is fully known if its components
in a co-ordinate system are given (assuming, of course,
that the co-ordinate system itself is given); these components may,
however, be prescribed arbitrarily. The tensor calculus is concerned
with setting out the properties and relations of tensors,
which are independent of the co-ordinate system. \emph{In an extended
sense a quantity in geometry and physics will be called a tensor if it
defines uniquely a Linear algebraic form depending on the co-ordinate
system in the manner described above; and conversely the tensor is
fully characterised if this form is given.} For example, a little
earlier we called a function of three displacements which depended
linearly and homogeneously on each of their arguments a tensor
of the third degree---one which is twofold co-variant and singly
contra-variant. This was possible in \Emph{metrical} space. In this
\index{Space!metrical}%
space, indeed, we are at liberty to represent this quantity by a
``none'' fold, single, twofold or threefold co-variant tensor. In
affine space, however, we should only have been able to express
it in the last form as a co-variant tensor of the third degree.
We shall illustrate this general explanation by some examples
\index{Components, co-variant, and contra-variant!tensor@{of a tensor}}%
in which we shall still adhere to the standpoint of affine geometry
alone.
1. If we represent a displacement~$\va$ in an arbitrary co-ordinate
system by its (contra-variant) components~$\Typo{\alpha}{a}^{i}$ and assign to it the
linear form
\[
\Typo{\alpha}{a}^{1} \xi_{1}
+ \Typo{\alpha}{a}^{2} \xi_{2} + \dots
+ \Typo{\alpha}{a}^{n} \xi_{n}
\]
having the variables~$\xi_{i}$ in this co-ordinate system, we get a contra-variant
tensor of the first order.
From now on we shall no longer use the term ``vector'' as
being synonymous with ``displacement'' but to signify a ``tensor
\PageSep{38}
of the first order,'' so that we shall say, \Emph{displacements are contra-variant
\index{Displacement current!space@{of space}}%
vectors}. The same applies to the \Emph{velocity} of a moving
point, for it is obtained by dividing the infinitely small displacement
which the moving point suffers during the time-element~$dt$
by~$dt$ (in the limiting case when $dt \to 0$). The present use of the
word vector agrees with its usual significance which includes not
only displacements but also every quantity which, after the choice
of an appropriate unit, can be represented uniquely by a displacement.
2. It is usually claimed that \Emph{force} has a geometrical character
\index{Force}%
on the ground that it may be represented in this way. In opposition,
however, to this representation there is another which, we
nowadays consider, does more justice to the physical nature of force,
inasmuch as it is based on the conception of \Emph{work}. In modern
physics the conception work is gradually usurping the conception
of force, and is claiming a more decisive and fundamental rôle. We
shall define the \Emph{components of a force} in a co-ordinate system~$\Typo{0}{O} \Chg{;}{\mid} \ve_{i}$
to be those numbers~$p_{i}$ which denote how much work it performs
during each of the virtual displacements~$\ve_{i}$ of its point of
application. These numbers completely characterise the force.
The work performed during the arbitrary displacement
\[
\vx = \xi^{1} \ve_{1} + \xi^{2} \ve_{2} + \dots + \xi^{n} \ve_{n}
\]
of its point of application is then $= \sum_{i} p_{i} \xi^{i}$. Hence it follows that
for two definite co-ordinate systems the relation
\[
\sum_{i} p_{i} \xi^{i} = \sum_{i} \bar{p}_{i} \bar{\xi}^{i}
\]
holds, if the variables~$\xi^{i}$, as signified by the upper indices, are
transformed contra-grediently with respect to the co-ordinate
system. According to this view, then, \Emph{forces are co-variant
vectors}. The connection between this representation of forces
and the usual one in which they are displacements will be discussed
when we pass from affine geometry, with which we are at present
dealing, to metrical geometry. The components of a co-variant
vector become transformed co-grediently to the fundamental vectors
in passing to a new co-ordinate system.
\Par{Additional Remarks.}---Since the transformations of the components~$a^{i}$
of a co-variant vector and of the components~$b^{i}$ of a
contra-variant vector are contra-gredient to one another, $\sum_{i} a_{i} b^{i}$~is
a definite number which is defined by these two vectors and is
independent of the co-ordinate system. This is our first example
\PageSep{39}
of an invariant tensor operation. Numbers or \Emph{scalars} are to be
classified as tensors of zero order in the system of tensors.
It has already been explained under what conditions a bilinear
form of two series of variables is called \Emph{symmetrical} and what
makes a symmetrical bilinear form non-degenerate. A bilinear
form~$F(\xi\Com \eta)$ is called \Emph{skew-symmetrical} if the interchange of
\index{Skew-symmetrical}%
the two sets of variables converts it into its negative, i.e.\ merely
changes its sign
\[
F(\eta\Com \xi) = -F(\xi\Com \eta).
\]
%[** TN: [sic] "a", not "\alpha" in the original]
This property is expressed in the \Typo{co-officients}{co-efficients}~$a_{ik}$ by the equations
$a_{ki} = -a_{ik}$. These properties persist if the two sets of variables are
subjected to the same linear transformations. The property of
being skew-symmetrical, symmetrical or (symmetrical and) non-degenerate,
possessed by co-variant or contra-variant tensors of the
second order is thus independent of the co-ordinate system.
Since the bilinear unit form resolves into itself after a contra-gredient
transformation of the two series of variables there is
among the \Emph{mixed} tensors of the second order (i.e.\ those which are
simply co-variant \Erratum{or}{and} simply contra-variant) one, called the unit
tensor, which has the components $\delta_{i}^{k} = \begin{gathered}1\ (i = k) \\ 0\ (i \neq k)\end{gathered}$ in every co-ordinate
system.
3. The metrical structure underlying Euclidean space assigns
to every two displacements
\[
\vx = \sum_{i} \xi^{i} \ve_{i}\qquad
\vy = \sum_{i} \eta^{i} \ve_{i}
\]
a number which is independent of the co-ordinate system and is
\index{Number}%
their scalar product
\[
\vx · \vy = \sum_{i\Com k} g_{ik} \xi^{i}\eta^{k}\qquad
g_{ik} = (\ve_{i} · \ve_{k}).
\]
Hence the bilinear form on the right depends on the co-ordinate
system in such a way that a co-variant tensor of the second order
is given by it, viz.\ the \Emph{fundamental metrical tensor}. The
metrical structure is fully characterised by it. It is symmetrical
and non-degenerate.
4. A \Emph{linear vector transformation} makes any displacement~$\vx$
\index{Matrix}%
correspond linearly to another displacement,~$\vx'$, i.e.\ so that the sum
$\vx' + \vy'$ corresponds to the sum $\vx + \vy$ and the product~$\lambda \vx'$ to the
product~$\lambda \vx$. In order to be able to refer conveniently to such
linear vector transformations, we shall call them \Emph{matrices}. If
the fundamental vectors~$\ve_{i}$ of a co-ordinate system become
\[
\ve_{i}' = \sum_{k} \alpha_{i}^{k} \ve_{k}
\]
\PageSep{40}
as a result of the transformation it will in general convert the
arbitrary displacement
\[
\vx = \sum_{i} \xi^{i} \ve_{i}\quad\text{into}\quad
\vx' = \sum_{i} \xi^{i} \ve_{i}' = \sum_{i\Com k} \alpha_{i}^{k} \xi^{i} \ve_{k}\Add{.}
\Tag{(24)}
\]
We may, therefore, characterise the matrix in the particular co-ordinate
system chosen by the bilinear form
\[
\sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \eta_{k}.
\]
It follows from~\Eq{(24)} that the relation
\[
\sum_{i\Com k} \bar{\Typo{\alpha}{a}}_{i}^{k} \bar{\xi}^{i} \ve_{k}
= \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \ve_{k}\ (= \vx')
\]
holds between two co-ordinate systems (we have used the same
terminology as above) if
\[
\sum_{i} \bar{\xi}^{i} \bar{\ve}_{i}
= \sum_{i} \xi^{i} \ve_{i}\ (\Add{=} \vx)\Add{;}
\]
thus
\[
\sum_{i\Com k} \bar{\Typo{\alpha}{a}}_{i}^{k} \bar{\xi}^{i} \bar{\eta}_{k}
= \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \eta_{k}
\]
if the~$\eta^{i}$ are transformed co-grediently to the fundamental vectors
and the~$\xi^{i}$ are transformed contra-grediently to them (the latter
remark about the transformations of the variables is self-evident
so that in future we shall simply omit it in similar cases). In
this way matrices are represented as tensors of the second order.
In particular, the unit tensor corresponds to ``identity'' which
assigns to every displacement~$\vx$ itself.
As was shown in the examples of force and metrical space it
often happens that the representation of geometrical or physical
quantities by a tensor becomes possible only after a unit measure
\index{Measure!unit of}%
has been chosen: this choice can only be made by specifying it in
each particular case. If the unit measure is altered the representative
tensors must be multiplied by a universal constant, viz.\ the
ratio of the two units of measure.
The following criterion is manifestly equivalent to this exposition
of the conception tensor. \emph{A linear form in several series
of variables, which is dependent on the co-ordinate system, is a tensor
if in every case it assumes a value independent of the co-ordinate
system \Inum{(\ia)}~whenever the components of an arbitrary contra-variant
vector are substituted for every contra-gredient series of variables, \Erratum{or}{and}
\PageSep{41}
\Inum{(\ib)}~whenever the components of an arbitrary co-variant vector are
substituted for a co-gredient series.}
If we now return from \Emph{affine} to \Emph{metrical} geometry, we see
\index{Co-gredient transformations}%
from the arguments at the beginning of the paragraph that the
difference between co-variants and contra-variants which affects
the tensors themselves in affine geometry shrinks to a mere
difference in the mode of representation.
Instead of talking of co-variant, mixed, and contra-variant
\emph{tensors} we shall hence find it more convenient here to talk only of
the co-variant, mixed, and contra-variant \emph{components} of a tensor.
After the above remarks it is evident that the transition from
one tensor to another which has a different character of co-variance
may be formulated simply as follows. If we interpret the contra-gredient
variables in a tensor as the contra-variant components
of an arbitrary displacement, and the co-gredient variables as
co-variant components of an arbitrary displacement, the tensor becomes
transformed into a linear form of several arbitrary displacements
which is independent of the co-ordinate system. By
representing the arguments in terms of their co-variant or contra-variant
components in any way which suggests itself as being
appropriate we pass on to other representations of the same
tensor. From the purely algebraic point of view the conversion
of a co-variant index into a contra-variant one is performed by
substituting new~$\xi_{i}$'s for the corresponding variables~$\xi^{i}$ in the linear
form in accordance with~\Eq{(22)}. The invariant nature of this process
depends on the circumstance that this substitution transforms
contra-gredient variables into co-gredient ones. The converse
process is carried out according to the inverse equations\Eq{(22')}.
The components themselves are changed (on account of the
symmetry of the~$g_{ik}$'s) from contra-variants to co-variants, i.e.\ the
indices are ``lowered'' according to the rule:
\[
\text{Substitute } \Typo{\alpha}{a}_{i}
= \sum_{j} g_{ij} \Typo{\alpha}{a}^{j} \text{ for } \Typo{\alpha}{a}^{i}
\]
irrespective of whether the numbers~$\Typo{\alpha}{a}^{i}$ carry any other indices or
not: the raising of the index is effected by the inverse equations.
If, in particular, we apply these remarks to the fundamental
metrical tensor, we get
\[
\sum_{i\Com k} g_{ik} \xi^{i} \eta^{k}
= \sum_{i} \xi^{i} \eta_{i}
= \sum_{k} \xi_{k} \eta^{k}
= \sum_{i\Com k} g^{ik} \xi_{i} \eta_{k}.
\]
Thus its mixed components are the numbers~$\delta_{k}^{i}$, its contra-variant
components are the co-efficients~$g^{ik}$ of the equations~\Eq{(22')}, which
\PageSep{42}
are the inverse of~\Eq{(22)}. It follows from the symmetry of the tensor
that these as well as the~$g_{ik}$'s satisfy the condition of symmetry
$g^{ki} = g^{ik}$.
With regard to notation we shall adopt the convention of denoting
the co-variant, mixed, and contra-variant components of
the same tensor by similar letters, and of indicating by the position
of the index at the top or bottom respectively whether the components
are contra-variant or co-variant with respect to the index,
as is shown in the following example of a tensor of the second
order:
\[
\sum_{i\Com k} \Typo{\alpha}{a}_{ik} \xi^{i} \eta^{k}
= \sum_{i\Com k} \Typo{\alpha}{a}_{k}^{i} \xi_{i} \eta^{k}
= \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \xi^{i} \eta_{k}
= \sum_{i\Com k} \Typo{\alpha}{a}^{ik} \xi_{i} \eta_{k}
\]
(in which the variables with lower and upper indices are connected
in pairs by~\Eq{(22)}).
In metrical space it is clear, from what has been said, that the
\index{Co-gredient transformations}%
difference between a co-variant and a contra-variant vector disappears:
in this case we can represent a force, which, according
to our view, is by nature a co-variant vector, as a contra-variant
vector, too, i.e.\ by a displacement. For, as we represented it
above by the linear form $\sum_{i} p_{i} \xi^{i}$ in the contra-gredient variables~$\xi^{i}$,
we can now transform the latter by means of~\Eq{(22')} into one having
co-gredient variables~$\xi_{i}$, viz.\ $\sum_{i} p^{i} \xi_{i}$. We then have
\[
\sum_{i} p^{i} \xi_{i}
= \sum_{i\Com k} g_{ik} p^{i} \xi^{k}
= \sum_{i\Com k} g_{ik} p^{k} \xi^{i}
= \sum_{i} p_{i} \xi^{i}\Add{;}
\]
the representative displacement~$\vp$ is thus defined by the fact that
the work which the force performs during an arbitrary displacement
is equal to the scalar product of the displacements $\vp$ and~$\vx$.
In a Cartesian co-ordinate system in which the fundamental
tensor has the components
\[
g_{ik} = \begin{cases}
1 & (i = k)\Add{,} \\
0 & (i \neq k)\Add{,}
\end{cases}
\]
the connecting equations~\Eq{(22)} are simply: $\xi_{i} = \xi^{i}$. If we confine
ourselves to the use of Cartesian co-ordinate systems, the difference
between co-variants and contra-variants ceases to exist, not only
for tensors but also for the tensor components. It must, however,
be mentioned that the conceptions which have so far been outlined
concerning the fundamental tensor~$g_{ik}$ assume only that it is
symmetrical and non-degenerate, whereas the introduction of a
\PageSep{43}
Cartesian co-ordinate system implies, in addition, that the corresponding
quadratic form is definitely positive. This entails a
difference. In the Theory of Relativity the time co-ordinate is
added as a fully equivalent term to the three-space co-ordinates,
and the measure-relation which holds in this four-dimensional
manifold is not based on a definite form but on an indefinite one
(Chapter~III). In this manifold, therefore, we shall not be able to
introduce a Cartesian co-ordinate system if we restrict ourselves to
real co-ordinates; but the conceptions here developed which are
to be worked out in detail for the dimensional number $n = 4$ may
be applied without alteration. Moreover, the algebraic simplicity
of this calculus advises us against making exclusive use of Cartesian
co-ordinate systems, as we have already mentioned at the end of
§\,4. Above all, finally, it is of great importance for later extensions
which take us beyond Euclidean geometry that the affine view
should even at this stage receive full recognition independently of
the metrical one.
Geometrical and physical quantities are scalars, vectors, and
tensors: this expresses the mathematical constitution of the space
in which these quantities exist. The mathematical symmetry
which this conditions is by no means restricted to geometry but,
on the contrary, attains its full validity in physics. As natural
phenomena take place in a metrical space this tensor calculus is
the natural mathematical instrument for expressing the uniformity
underlying them.
\Section{6.}{Tensor Algebra. Examples}
\Par{Addition of Tensors.}---The multiplication of a linear form,
\index{Addition of tensors}%
\index{Multiplication!of a tensor by a number}%
\index{Product!tensor@{of a tensor and a number}}%
\index{Sum of!tensors}%
bilinear form, trilinear form~\dots\ by a number, likewise the
addition of two linear forms, or of two bilinear forms~\dots\
always gives rise to a form of the same kind. Vectors and tensors
may thus be multiplied by a number (a scalar), and two or more
tensors of the same order may be added together. These operations
are carried out by multiplying the components by the number in
question or by addition, respectively. Every set of tensors of the
same order contains a unique tensor~$\Typo{0}{\0}$, of which all the components
vanish, and which, when added to any tensor of the same order,
leaves the latter unaltered. The state of a physical system is
described by specifying the values of certain scalars and tensors.
The fact that a tensor which has been derived from them by
mathematical operations and is an invariant (i.e.\ dependent upon
them alone and not upon the choice of the co-ordinate system) is
equal to zero is what, in general, the expression of a physical law
amounts to.
\PageSep{44}
\Par{Examples.}---The motion of a point is represented analytically
by giving the position of the moving-point or of its co-ordinates,
respectively, as functions of the time~$t$. The derivatives~$\dfrac{dx_{i}}{dt}$ are
the contra-variant components~$u^{i}$ of the vector ``velocity''. By
multiplying it by the mass~$m$ of the moving-point, $m$~being a scalar
which serves to express the inertia of matter, we get the ``impulse''
(or ``momentum''). By adding the impulses of several points
\index{Impulse (momentum)}%
\index{Moment!mechanical}%
\index{Momentum}%
of mass or of those, respectively, of which one imagines a rigid
body to be composed in the mechanics of point-masses, we get the
\index{Mechanics!fundamental law of!Newton@{of Newton's}}%
total impulse of the point-system or of the rigid body. In the case
of continuously extended matter we must supplant these sums by
integrals. The fundamental law of motion is
\[
\frac{dG^{i}}{dt} = p^{i};\quad
G^{i} = mu^{i}
\Tag{(25)}
\]
where $G^{i}$~denote the contra-variant components of the impulse of a
mass-point and $p^{i}$~denote those of the force.
Since, according to our view, force is primarily a co-variant
vector, this fundamental law is possible only in a metrical space,
but not in a purely affine one. The same law holds for the total
impulse of a rigid body and for the total force acting on it.
\Par{Multiplication of Tensors.}---By multiplying together two linear
\index{Multiplication!of tensors}%
forms $\sum_{i} a_{i} \xi^{i}$, $\sum_{i} b_{i} \eta^{i}$ in the variables $\xi$~and~$\eta$, we get a bilinear form
\[
\sum_{i\Com k} a_{i} b_{k} \xi^{i} \eta^{k}
\]
and hence from the two vectors $a$~and~$b$ we get a tensor~$c$ of the
second order, i.e.\
\[
a_{i} b_{k} = c_{ik}\Add{.}
\Tag{(26)}
\]
Equation~\Eq{(26)} represents an invariant relation between the vectors
$a$~and~$b$ and the tensor~$c$, i.e.\ if we pass over to a new co-ordinate
system precisely the same equations hold for the components
(distinguished by a bar) of these quantities in this new co-ordinate
system, i.e.\
\[
\bar{a}_{i} \bar{b}_{k} = \bar{c}_{ik}.
\]
In the same way we may multiply a tensor of the first order by
one of the second order (or generally, a tensor of any order by a
tensor of any order). By multiplying
\[
\sum_{i} a_{i} \xi^{i} \text{ by }
\sum_{i\Com k} b_{i}^{k} \eta^{i} \zeta_{k}
\]
\PageSep{45}
in which the Greek letters denote variables which are to be transformed
contra-grediently or co-grediently according as the indices
are raised or lowered, we derive the trilinear form
\[
\sum_{i\Com k\Com l} a_{i} b_{k}^{l} \xi^{i} \eta^{k} \zeta_{l}
\]
and, accordingly, by multiplying the two tensors of the first and
second order, a tensor~$c$ of the third order, i.e.\
\[
a_{i} · b_{k}^{l} = c_{ik}^{l}.
\]
This multiplication is performed on the components by merely
multiplying each component of one tensor by each component of
the other, as is evident above. It must be noted that the co-variant
components (with respect to the index~$l$, for example) of the resultant
tensor of the third order, i.e.\ $c_{ik}^{l} = a_{i} b_{k}^{l}$, are given by: $c_{ikl} = a_{i} b_{kl}$.
It is thus immediately permissible in such multiplication
formulæ to transfer an index on both sides of the equation from
below to above or \textit{vice versa}.
\Par{Examples of Skew-symmetrical and Symmetrical Tensors.}---If
two vectors with the contra-variant components $a^{i}$,~$b^{i}$, are multiplied
first in one order and then in the reverse order, and if we then
subtract the one result from the other, we get a skew-symmetrical
tensor~$c$ of the second order with the contra-variant components
\[
c^{ik} = a^{i} b^{k} - a^{k} b^{i}.
\]
This tensor occurs in ordinary vector analysis as the ``vectorial product''
\index{Product!vectorial}%
\index{Vector!product}%
of the two vectors $a$~and~$b$. By specifying a certain direction
of twist in three-dimensional space, it becomes possible to establish
a reversible one-to-one correspondence between these tensors and
the vectors. (This is impossible in four-dimensional space for the
obvious reason that, in it, a skew-symmetrical tensor of the second
order has six independent components, whereas a vector has only
four; similarly in the case of spaces of still higher dimensions.)
In three-dimensional space the above method of representation is
founded on the following. If we use only Cartesian co-ordinate
systems and introduce in addition to $a$~and~$b$ an arbitrary displacement~$\xi$,
the determinant
\[
\left\lvert
\begin{array}{@{}rrr@{}}
a^{1} & a^{2} & a^{3} \\
b^{1} & b^{2} & b^{3} \\
\xi^{1} & \xi^{2} & \Typo{c^{3}}{\xi^{3}} \\
\end{array}
\right\rvert = c^{23} \xi^{1} + c^{31} \xi^{2} + c^{12} \xi^{3}
\]
becomes multiplied by the determinant of the co-efficients of transformation,
when we pass from one co-ordinate system to another.
In the case of orthogonal transformations this determinant $= ±1$.
If we confine our attention to ``proper'' orthogonal transformations,
\PageSep{46}
i.e.\ such for which this determinant $= +1$ the above linear form in
the~$\xi$'s remains unchanged. Accordingly, the formulæ
\[
c^{23} = c_{1}^{*}\qquad
c^{31} = c_{2}^{*}\qquad
c^{12} = c_{3}^{*}
\]
express a relation between the skew-symmetrical tensor~$c$ and a
vector~$c^{*}$, this relation being invariant for proper orthogonal transformations.
The vector~$c^{*}$ is perpendicular to the two vectors
$a$~and~$b$, and its magnitude (according to elementary formulæ of
analytical geometry) is equal to the area of the parallelogram of
which the sides are $a$~and~$b$. It may be justifiable on the ground
of economy of expression to replace skew-symmetrical tensors by
vectors in ordinary vector analysis, but in some ways it hides the
essential feature; it gives rise to the well-known ``swimming-rules''
in \Chg{electro-dynamics}{electrodynamics}, which in no wise signify that there is a unique
direction of twist in the space in which \Chg{electro-dynamic}{electrodynamics} events
occur; they become necessary only because the magnetic intensity
of field is regarded as a vector, whereas it is, in reality, a skew-symmetrical
tensor (like the so-called vectorial product of two
vectors). If we had been given one more space-dimension, this
error could never have occurred.
In mechanics the skew-symmetrical tensor product of two
vectors occurs---
1. As moment of momentum (angular momentum) about a
\index{Angular!momentum}%
\index{Torque of a force}%
\index{Turning-moment of a force}%
point~$O$. If there is a point-mass at~$P$ and if $\xi^{1}$,~$\xi^{2}$,~$\xi^{3}$ are the
components of~$\Vector{OP}$ and $u^{i}$~are the (contra-variant) components of
the velocity of the points at the moment under consideration, and
$m$~its mass, the momentum of momentum is defined by
\[
L^{ik} = m(u^{i} \xi^{k} - u^{k} \xi^{i}).
\]
The moment of momentum of a rigid body about a point~$O$ is the
sum of the moments of momentum of each of the point-masses
of the body.
2. As the \Emph{turning-moment (torque) of a force}. If the
latter acts at the point~$P$ and if $p^{i}$~are its contra-variant components,
the torque is defined by
\[
q^{ik} = p^{i} \xi^{k} - p^{k} \xi^{i}.
\]
The turning-moment of a system of forces is obtained by simple
addition. In addition to~\Eq{(25)} the law
\[
\frac{dL^{ik}}{dt} = q^{ik}
\Tag{(27)}
\]
holds for a point-mass as well as for a rigid body free from constraint.
The turning-moment of a rigid body about a fixed point~$O$
is governed by the law~\Eq{(27)} alone.
\PageSep{47}
A further example of a skew-symmetrical tensor is the \Emph{rate of
rotation} (angular velocity) of a rigid body about the fixed point~$O$.
\index{Angular!velocity}%
\index{Rotation!kinematical@{(in kinematical sense)}}%
\index{Velocity!rotation@{of rotation}}%
If this rotation about~$O$ brings the point~$P$ in general to~$P'$, the
vector~$\Vector{OP'}$ is produced, and hence also~$PP'$, by a linear transformation
from~$\Vector{OP}$. If $\xi^{i}$~are the components of~$\Vector{OP}$, $\delta\xi^{i}$~those of~$PP'$,
$v_{k}^{i}$~the components of this linear transformation (matrix), we
have
\[
\delta\xi^{i} = \sum_{k} v_{k}^{i} \xi^{k}\Add{.}
\Tag{(28)}
\]
We shall concern ourselves here only with infinitely small rotations.
They are distinguished among infinitesimal matrices by the additional
property that, regarded as an identity in~$\xi$\Add{,}
\[
\delta\biggl(\sum_{i} \xi_{i} \xi^{i}\biggr)
= \delta\biggl(\sum_{i\Com k} g_{ik} \xi^{i} \xi^{k}\biggr)
= 0.
\]
This gives
\[
\sum_{i} \Typo{\xi^{i}}{\xi_{i}}\, \delta\xi^{i} = 0.
\]
By inserting the expressions~\Eq{(28)} we get\Pagelabel{47}
\[
\sum_{i\Com k} v_{k}^{i} \xi_{i} \xi^{k}
= \sum_{i\Com k} v_{ik} \xi^{i} \xi^{k}
= 0.
\]
This must be identically true in the variables~$\xi_{i}$, and hence
\[
v_{ki} + v_{ik} = 0
\]
i.e.\ the tensor which has $v_{ik}$~for its co-variant components is skew-symmetrical.
A rigid body in motion experiences an infinitely small rotation
during an infinitely small element of time~$\delta t$. We need only to
divide by~$\delta t$ the infinitesimal rotation-tensor~$v$ just formed to
derive (in the limit when $\delta t \to 0$) the skew-symmetrical tensor
``angular velocity,'' which we shall again denote by~$v$. If $u^{i}$~signify
the contra-variant components of the velocity of the point~$P$,
and $u_{i}$~its co-variant components in the formulæ~\Eq{(28)}, the latter
resolves into the fundamental formula of the kinematics of a rigid
body, viz.\
\[
u_{i} = \sum_{k} v_{ik} \xi^{k}\Add{.}
\Tag{(29)}
\]
\PageSep{48}
The existence of the ``instantaneous axis of rotation'' follows from
the circumstance that the linear equations
\[
\sum_{k} v_{ik} \xi^{k} = 0
\]
with the skew-symmetrical co-efficients~$v_{ik}$ always have solutions
\Emph{in the case $n = 3$}, which differ from the trivial one $\xi^{1} = \xi^{2} = \xi^{3} = 0$.
One usually finds angular velocity, too, represented as
a vector.
Finally the ``moment of inertia'' which presents itself in the
\index{Inertia!moment of}%
\index{Inertial force!moment}%
\index{Moment!of momentum}%
rotation of a body offers a simple example of a symmetrical tensor
of the second order.
If a point-mass of mass~$m$ is situated at the point~$P$ to which
the vector~$\Vector{OP}$ starting from the centre of rotation~$O$ and with the
components~$\xi^{i}$ leads, we call the symmetrical tensor of which the
contra-variant components are given by~$m \xi^{i} \xi^{k}$ (multiplication!), the
``inertia of rotation'' of the point-mass (with respect to the
centre of rotation~$O$). The inertia of rotation~$T^{ik}$ of a point-system
or body is defined as the sum of these tensors formed
separately for each of its points~$P$. This definition is different
from the usual one, but is the correct one if the intention of
regarding the velocity of rotation as a skew-symmetrical tensor and
not as a vector is to be carried out, as we shall presently see.
The tensor~$T^{ik}$ plays the same part with regard to a rotation about~$O$
as that of the scalar~$m$ in translational motion.
\Par{Contraction.}---If $a_{i}^{k}$~are the mixed components of a tensor of the
\index{Contraction-hypothesis of Lorentz and Fitzgerald!process of}%
second order, $\sum_{i} a_{i}^{i}$~is an invariant. Thus, if\Typo{,}{} $\bar{a}_{i}^{k}$~are the mixed components
of the same tensor after transformation to a new co-ordinate
system, then
\[
\sum_{i} a_{i}^{i} = \sum_{i} \bar{a}_{i}^{i}.
\]
\Proof.---The variables $\xi^{i}$,~$\eta_{i}$ of the bilinear form
\[
\sum_{i\Com k} a_{i}^{k} \xi^{i} \eta_{k}
\]
must be subjected to the contra-gredient transformations
\[
\xi^{i} = \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k},\qquad
\eta_{i} = \sum_{k} \breve{\alpha}_{i}^{k} \bar{\eta}_{k}
\]
\PageSep{49}
if we wish to bring them into the form
\[
\sum_{i\Com k} \bar{\Typo{\alpha}{a}}_{i}^{k} \bar{\xi}^{i} \bar{\eta}_{k}.
\]
From this it follows that
\begin{align*}
\bar{\Typo{\alpha}{a}}_{r}^{r}
&= \sum_{i\Com k} \Typo{\alpha}{a}_{i}^{k} \alpha_{r}^{i} \breve{\alpha}_{k}^{r},
\intertext{and}
\sum_{r} \bar{\Typo{\alpha}{a}}_{r}^{r}
&= \sum_{i\Com k} \biggl(\Typo{\alpha}{a}_{i}^{k} \sum_{r} \alpha_{r}^{i} \Typo{\alpha}{\breve{\alpha}}_{k}^{r}\biggr) \\
&= \sum_{i} a_{i}^{i}\quad\text{by~\Eq{(20')}.}
\end{align*}
The invariant $\sum_{i} \Typo{\alpha}{a}_{i}^{i}$ which has been formed from the components~$\Typo{\alpha}{a}_{i}^{k}$
\index{Trace of a matrix}%
of a matrix is called the \Emph{trace (spur) of the matrix}.
This theorem enables us immediately to carry out a general
operation on tensors, called ``contraction,'' which takes a second
place to multiplication. By making a definite upper index in the
mixed components of a tensor coincide with a definite lower one,
and summing over this index, we derive from the given tensor a
new one the order of which is two less than that of the original
one, e.g.\ we get from the components~$\Typo{\alpha}{a}_{hik}^{lm}$ of a tensor of the fifth
order a tensor of the third order, thus:---
\[
\sum_{r} \Typo{\alpha}{a}_{hir}^{lr} = c_{hi}^{l}\Add{.}
\Tag{(30)}
\]
The connection expressed by~\Eq{(30)} is an invariant one, i.e.\ it preserves
its form when we pass over to a new co-ordinate system, viz.\
\[
\sum_{r} \bar{\Typo{\alpha}{a}}_{hir}^{lr} = \bar{c}_{hi}^{l}\Add{.}
\Tag{(31)}
\]
To see this we only need the help of two arbitrary contra-variant
vectors $\xi^{i}$,~$\eta^{i}$ and a co-variant one~$\zeta_{i}$. By means of them we form
the components,
\[
\sum_{h\Com i\Com l} \Typo{\alpha}{a}_{hik}^{lm} \xi^{h} \eta^{i} \zeta_{l} = f_{k}^{m},
\]
of a mixed tensor of the second order: to this we apply the
theorem
\[
\sum_{r} f_{r}^{r} = \sum_{r} \bar{f}_{r}^{r}
\]
\PageSep{50}
which was just proved. We then get the formula
\[
\sum_{h\Com i\Com l} c_{hi}^{l} \xi^{h} \eta^{i} \zeta_{l}
= \sum_{h\Com i\Com l} \bar{c}_{hi}^{l} \bar{\xi}^{h} \bar{\eta}^{i} \bar{\zeta}_{l}
\]
in which the~$c$'s are defined by~\Eq{(30)}, the~$\bar{c}$'s by~\Eq{(31)}. The~$\bar{c}_{hi}^{l}$'s are
thus, in point of fact, the components of the same tensor of the
\index{Tensor!general@{(general)}}%
third order in the new system, of which the components in the old
one $= c_{ih}^{l}$.
\Par{Examples} of this process of contraction have been met with
in abundance in the above. Wherever summation took place with
respect to certain indices, the summation index appeared twice in
the general member of summation, once above and once below the
co-efficient: each such summation was an example of contraction.
For example, in formula~\Eq{(29)}: by multiplication of~$v_{ik}$ with~$\xi^{i}$ one
can form the tensor~$v_{ik} \xi^{l}$ of the third order; by making $k$ coincide
with~$l$ and summing for~$k$, we get the contracted tensor of the first
order~$u_{i}$. If a matrix~$A$ transforms the arbitrary displacement~$\vx$
into $\vx' = A(\vx)$, and if a second matrix~$B$ transforms this~$\vx'$ into
$\vx'' = -B(\vx')$, a combination~$BA$ results from the two matrices,
which transforms $\vx$ directly into $\vx'' = BA(\vx)$. If $A$~has the components~$\Typo{\alpha}{a}_{i}^{k}$
and $B$~components~$b_{i}^{k}$, the components of the combined
matrix~$BA$ are
\[
c_{i}^{k} = \sum_{r} b_{i}^{r} \Typo{\alpha}{a}_{r}^{k}\Add{.}
\]
Here, again, we have the case of multiplication followed by contraction.
The process of contraction may be applied simultaneously for
several pairs of indices. From the tensors of the 1st, 2nd, 3rd,~\dots\
order with the co-variant components $\Typo{\alpha}{a}_{i}$,~$\Typo{\alpha}{a}_{ik}$, $\Typo{\alpha}{a}_{ikl}$,~\dots, we thus
get, in particular, the invariants
\[
\sum_{i} \Typo{\alpha}{a}_{i} \Typo{\alpha}{a}^{i},\qquad
\sum_{i\Com k} \Typo{\alpha}{a}_{ik} \Typo{\alpha}{a}^{ik},\qquad
\sum_{i\Com k\Com l} \Typo{\alpha}{a}_{ikl} \Typo{\alpha}{a}^{ikl},\ \dots\Add{.}
\]
If, as is here assumed, the quadratic form corresponding to the
fundamental metrical tensor is definitely positive, these invariants
are all positive, for, in a Cartesian co-ordinate system they disclose
themselves directly as the sums of the squares of the components.
Just as in the simplest case of a vector, the square root of these
invariants may be termed the measure or the magnitude of the
tensor of the 1st, 2nd, 3rd,~\dots\ order.
We shall at this point make the convention, once and for all,
that if an index occurs twice (once above and once below) in a
\PageSep{51}
term of a formula to which indices are attached, this is always to
signify that summation is to be carried out with respect to the
index in question, and we shall not consider it necessary to put a
summation sign in front of it.
The operations of addition, multiplication, and contraction only
require affine geometry: they are not based upon a ``fundamental
metrical tensor''. The latter is only necessary for the process of
passing from co-variant to contra-variant components and the
reverse.
\Section{}{Euler's Equations for a Spinning Top}
\index{Euler's equations}%
\index{Top, spinning}%
As an exercise in tensor calculus, we shall deduce Euler's equations
for the motion of a rigid body under no forces about a fixed
\index{Motion!(under no forces)}%
point~$O$. We write the fundamental equations~\Eq{(27)} in the co-variant
form
\[
\frac{dL_{ik}}{dt} = 0
\]
and multiply them, for the sake of briefness, by the contra-variant
components~$w^{ik}$ of an arbitrary skew-symmetrical tensor which is
constant (independent of the time), and apply contraction with respect
to $i$~and~$k$. If we put $H_{ik}$ equal to the sum
\[
\sum_{m} mu_{i} \xi^{k}
\]
which is to be taken over all the points of mass, we get
\[
\tfrac{1}{2} L_{ik} w^{ik} = H_{ik} w^{ik} = H,
\]
an invariant, and we can compress our equation into
\[
\frac{dH}{dt} = 0\Add{.}
\Tag{(32)}
\]
If we introduce the expressions~\Eq{(29)} for~$u_{i}$, and the tensor of inertia~$T$,
then
\[
H_{ik} = v_{ir} T_{k}^{r}\Add{.}
\Tag{(33)}
\]
We have hitherto assumed that a co-ordinate system which is
fixed in \Emph{space} has been used. The components~$T$ of inertia then
change with the distribution of matter in the course of time. If,
however, in place of this we use a co-ordinate system which is fixed
in the \Emph{body}, and consider the symbols so far used as referring to
the components of the corresponding tensors with respect to this
co-ordinate system, whereas we distinguish the components of the
same tensors with respect to the co-ordinate system fixed in space
by a horizontal bar, the equation~\Eq{(32)} remains valid on account of
\PageSep{52}
the invariance of~$H$. The $T_{i}^{k}$'s are now constants; on the other
hand, however, the $w^{ik}$'s vary with the time. Our equation gives us
\[
\frac{dH_{ik}}{dt} · w^{ik} + H_{ik} · \frac{dw^{ik}}{dt} = 0\Add{.}
\Tag{(34)}
\]
To determine $\dfrac{dw^{ik}}{dt}$, we choose two arbitrary vectors fixed in the
body, of which the co-variant components in the co-ordinate system
attached to the body are $\xi_{i}$~and $\eta_{i}$ respectively. These quantities
are thus constants, but their components $\bar{\xi}_{i}$,~$\bar{\eta}_{i}$ in the space co-ordinate
system are functions of the time. Now,
\[
w^{ik} \xi_{i} \eta_{k} = \bar{w}^{ik} \bar{\xi}_{i} \bar{\eta}_{k},
\]
and hence, differentiating with respect to the time
\[
\frac{dw^{ik}}{dt} · \xi_{i} \eta_{k}
= \bar{w}^{ik} \left(
\frac{d\bar{\xi}_{i}}{dt} · \bar{\eta}_{k}
+ \bar{\xi}_{i} · \frac{d\bar{\eta}_{k}}{dt}\right)\Add{.}
\Tag{(35)}
\]
By formula~\Eq{(29)}
\[
\frac{d\bar{\xi}_{i}}{dt}
= \bar{v}_{ir} \bar{\xi}^{r}
= \bar{v}_{i}^{r} \Typo{\bar{\xi}^{r}}{\bar{\xi}_{r}}.
\]
We thus get for the right-hand side of~\Eq{(35)}
\[
\bar{w}^{ik} (\bar{v}_{i}^{r} \bar{\xi}_{r} \bar{\eta}_{k}
+ \bar{v}_{k}^{r} \bar{\xi}_{i} \bar{\eta}_{r}),
\]
and as this is an invariant, we may remove the bars, obtaining
\[
\xi_{i} \eta_{k} \frac{dw^{ik}}{dt}
= w^{ik} (\xi_{r} \eta_{k} v_{i}^{r} + \xi_{i} \eta_{r} v_{k}^{r}).
\]
This holds identically in $\xi$~and~$\eta$; thus if the~$H^{ik}$ are arbitrary
numbers,
\[
H_{ik} \frac{dw^{ik}}{dt}
= w^{ik} (v_{i}^{r} H_{rk} + v_{k}^{r} H_{ir}).
\]
If we take the $H_{ik}$'s to be the quantities which we denoted above
by this symbol, the second term of~\Eq{(34)} is determined, and our
equation becomes
\[
\left\{\frac{dH_{ik}}{dt} + (v_{i}^{r} H_{rk} + v_{k}^{r} H_{ir})\right\} w^{ik} = 0,
\]
which is an identity in the skew-symmetrical tensor~$w^{ik}$; hence
\[
\frac{d(H_{ik} - H_{ki})}{dt}
+ \left[\begin{alignedat}{2}
&v_{i}^{r} H_{rk} &&+ v_{k}^{r} H_{ir} \\
-&v_{k}^{r} H_{ri} &&+ v_{i}^{r} H_{kr}
\end{alignedat}\right] = 0.
\]
We shall now substitute the expression~\Eq{(33)} for~$H_{ik}$. Since, on
account of the symmetry of~$T_{ik}$,
\[
v_{k}^{r} H_{ir} (= v_{k}^{r} v_{i}^{s} T_{rs})
\]
\PageSep{53}
is also symmetrical in $i$~and~$k$, the two last terms of the sum in the
square brackets destroy one another. If we now put the symmetrical
tensor
\[
v_{i}^{r} v_{kr} = g_{rs} v_{i}^{r} v_{k}^{s} = (v\Com v)_{ik}
\]
we finally get our equations into the form
\[
\frac{d}{dt}(v_{ir} T_{k}^{r} - v_{kr} T_{i}^{r})
= (v\Com v)_{ir} T_{k}^{r} - (v\Com v)_{kr} T_{i}^{r}.
\]
It is well known that we may introduce a Cartesian co-ordinate
system composed of the three principal axes of inertia, so that in
these
\[
g_{ik} = \begin{cases}
1 & (i = k)\Add{,} \\
0 & (i \neq k)\Add{,}
\end{cases}
\quad\text{and}\quad
T_{ik} = 0\quad\text{(for $i \neq k$).}
\]
If we then write~$T_{1}$ in place of~$T_{1}^{1}$, and do the same for the remaining
indices, our equations in this co-ordinate system assume
the simple form
\[
(T_{i} + T_{k}) \frac{dv_{ik}}{dt} = (T_{k} - T_{i})(v\Com v)_{ik}.
\]
These are the differential equations for the components~$v_{ik}$ of the
unknown angular velocity---equations which, as is known, may be
solved in elliptic functions of~$t$. The principal moments of inertia~$T_{i}$
which occur here are connected with those,~$T_{i}^{*}$, given in accordance
with the usual definitions by the equations
\[
T_{1}^{*} = T_{2} + T_{3},\qquad
T_{2}^{*} = T_{3} + T_{1},\qquad
T_{3}^{*} = T_{1} + T_{2}.
\]
The above treatment of the problem of rotation may, in \Chg{contradistinction}{contra-distinction}
to the usual method, be transposed, word for word, from
three-dimensional space to multi-dimensional spaces. This is,
indeed, irrelevant in practice. On the other hand, the fact that we
have freed ourselves from the limitation to a definite dimensional
number and that we have formulated physical laws in such a way
that the dimensional number appears \Emph{accidental} in them, gives
us an assurance that we have succeeded fully in grasping them
mathematically.
The study of tensor-calculus\footnote
{\Chg{Note 4.}{\textit{Vide} \FNote{4}.}}
is, without doubt, attended by
conceptual difficulties---over and above the apprehension inspired
by indices, which must be overcome. From the formal aspect,
however, the method of reckoning used is of extreme simplicity;
it is much easier than, e.g., the apparatus of elementary vector-calculus.
There are two operations, multiplication and contraction;
then putting the components of two tensors with totally different
indices alongside of one another; the identification of an upper
\PageSep{54}
index with a lower one, and, finally, summation (not expressed)
over this index. Various attempts have been made to set up a
standard terminology in this branch of mathematics involving only
the vectors themselves and not their components, analogous to that
of vectors in vector analysis. This is highly expedient in the latter,
but very cumbersome for the much more complicated framework
of the tensor calculus. In trying to avoid continual reference to
the components we are obliged to adopt an endless profusion of
names and symbols in addition to an intricate set of rules for
carrying out calculations, so that the balance of advantage is considerably
on the negative side. An emphatic protest must be
entered against these orgies of formalism which are threatening
the peace of even the technical scientist.
\Section{7.}{Symmetrical Properties of Tensors}
It is obvious from the examples of the preceding paragraph that
symmetrical and skew-symmetrical tensors of the second order,
wherever they are applied, represent entirely different kinds of
quantities. Accordingly the character of a quantity is not in
general described fully, if it is stated to be a tensor of such and
such an order, but \Emph{symmetrical characteristics} have to be added.
A linear form of several series of variables is called \Emph{symmetrical}
if it remains unchanged after any two of these series of
variables are interchanged, but is called \Emph{skew-symmetrical} if this
converts it into its negative, i.e.\ reverses its sign. A symmetrical
linear form does not change if the series of variables are subjected
to any permutations among themselves; a skew-symmetrical one
does not change if an even permutation is carried out with the series
of variables, but changes its sign if the permutation is odd. The
co-efficients~$\Typo{\alpha}{a}_{ikl}$ of a symmetrical trilinear form (we purposely
choose three again as an example) satisfy the conditions
\[
%[** TN: Correctly set in the original!]
a_{ikl} = a_{kli} = a_{lik} = a_{kil} = a_{lki} = a_{ilk}.
\]
Of the co-efficients of a skew-symmetrical tensor only those which
have three different indices can be~$\neq 0$ and they satisfy the equations
\[
\Typo{\alpha}{a}_{ikl}
= \Typo{\alpha}{a}_{kli}
= \Typo{\alpha}{a}_{lik}
= -\Typo{\alpha}{a}_{kil}
= -\Typo{\alpha}{a}_{lki}
= -\Typo{\alpha}{a}_{ilk}.
\]
There can consequently be no (non-vanishing) skew-sym\-met\-ri\-cal
forms of more than~$n$ series of variables in a domain of $n$~variables.
Just as a symmetrical bilinear form may be entirely replaced
by the quadratic form which is derived from it by identifying
the two series of variables, so a symmetrical trilinear form is
uniquely determined by the cubical form of a single series of variables
\PageSep{55}
with the co-efficients~$\Typo{\alpha}{a}_{ikl}$, which is derived from the trilinear
form by the same process. If in a skew-symmetrical trilinear form
\index{Skew-symmetrical}%
% [** TN: Upright F in the original]
\[
F = \sum_{i\Com k\Com l} \Typo{\alpha}{a}_{ikl} \xi^{i} \eta^{k} \zeta^{l}
\]
we perform the $3!$~permutations on the series of variables $\xi$,~$\eta$,~$\zeta$,
and prefix a positive or negative sign to each according as the permutation
is even or odd, we get the original form six times. If
they are all added together, we get the following scheme for them:---
\[
F = \frac{1}{3!} \sum \Typo{\alpha}{a}_{ikl} \left\lvert
\begin{array}{@{}rrr@{}}
\xi^{i} & \xi^{k} & \xi^{l} \\
\eta^{i} & \eta^{k} & \eta^{l} \\
\zeta^{i} & \zeta^{k} & \zeta^{l} \\
\end{array}
\right\rvert\Add{.}
\Tag{(36)}
\]
In a linear form the property of being symmetrical or skew-symmetrical
is not destroyed if each series of variables is subjected
to the same linear transformation. Consequently, a meaning may
be attached to the terms \Emph{symmetrical} and \Emph{skew-symmetrical},
\Emph{co-variant} or \Emph{contra-variant} tensors. But these expressions have
no meaning in the domain of mixed tensors. We need spend no
further time on symmetrical tensors, but must discuss skew-symmetrical
co-variant tensors in somewhat greater detail as they have
\index{Co-variant tensors}%
a very special significance.
The components~$\xi^{i}$ of a displacement determine the direction of
a straight line (positive or negative) as well as its magnitude. If
$\xi^{i}$~and~$\eta^{i}$ are any two linearly independent displacements, and if
they are marked out from any arbitrary point~$O$, they trace out a
plane. The ratios of the quantities
\[
\xi^{i} \eta^{k} - \xi^{k} \eta^{i} = \xi^{ik}
\]
define the ``position'' of this plane (a ``direction'' of the plane) in
the same way as the ratios of the~$\xi^{i}$ fix the position of a straight
line (its ``direction''). The~$\xi^{ik}$ are each $= 0$ if, and only if, the two
displacements $\xi^{i}$,~$\eta^{i}$ are linearly dependent; in this case they do not
map out a two-dimensional manifold. When two linearly independent
displacements $\xi^{i}$~and~$\eta^{i}$ trace out a plane, a definite sense of
rotation is implied, viz.\ the sense of the rotation about~$O$ in the
plane which for a turn $< 180°$ brings~$\xi$ to coincide with~$\eta$; also a
definite measure (quantity), viz.\ the area of the parallelogram enclosed
by $\xi$~and~$\eta$. If we mark off two displacements $\xi$,~$\eta$ from an
arbitrary point~$O$, and two $\xi_{*}$\Add{,}~$\eta_{*}$ from an arbitrary point~$O_{*}$, then
the position, the sense of rotation, and the magnitude of the plane
marked out are identical in each if, and only if, the~$\xi^{ik}$'s of the one
pair coincide with those of the other, i.e.\
\[
\xi^{i} \eta^{k} - \xi^{k} \eta^{i}
= \xi_{*}^{i} \eta_{*}^{k} - \xi_{*}^{k} \eta_{*}^{i}\Add{.}
\]
\PageSep{56}
So that just as the~$\xi^{i}$'s determine the direction and length of a
straight line, so the~$\xi^{ik}$'s determine the sense and surface area of a
plane; the completeness of the analogy is evident.
To express this we may call the first configuration a \Emph{one-dimensional
space-element}, the second a \Emph{two-dimensional
\index{Line-element!Euclidean@{(in Euclidean geometry)}}%
\index{Space!element@{-element}}%
space-element}. Just as the square of the magnitude of a one-dimensional
space-element is given by the invariant
\[
\xi_{i} \xi^{i} = g_{ik} \xi^{i} \xi^{k} = Q(\xi)
\]
so the square of the magnitude of the two-dimensional space-element
is given, in accordance with the formulæ of analytical
geometry, by
\[
\tfrac{1}{2} \xi^{ik} \xi_{ik};
\]
for which we may also write
\begin{align*}
\xi_{i} \eta_{k} (\xi^{i}\eta^{k} - \xi^{k} \eta^{i})
&= (\xi_{i} \xi^{i}) (\eta^{k} \eta_{k}) - (\xi_{i} \eta^{i}) (\xi^{k} \eta_{k}) \\
&= Q(\xi) · Q(\eta) - Q^{2}(\xi\Com \eta).
\end{align*}
In the same sense the determinants
\[
\xi^{ikl} = \left\lvert
\begin{array}{@{}rrr@{}}
\xi^{i} & \xi^{k} & \xi^{l} \\
\eta^{i} & \eta^{k} & \eta^{l} \\
\zeta^{i} & \zeta^{k} & \zeta^{l} \\
\end{array}
\right\rvert
\]
which are derived from three independent displacements $\xi$,~$\eta$,~$\zeta$,
are the components of a \Emph{three-dimensional space-element}, the
magnitude of which is given by the square root of the invariant
\[
\tfrac{1}{3!} \xi^{ikl} \xi_{ikl}.
\]
In three-dimensional space this invariant is
\[
\xi_{123} \xi^{123} = g_{1i} g_{2k} g_{3l} \xi^{ikl} \xi^{123},
\]
and since $\xi^{ikl} = ±\xi^{123}$, according as $ikl$~is an even or an odd
permutation of~$123$, it assumes the value
\[
g · (\xi^{123})^{2}
\]
where $g$~is the determinant of the co-efficients~$g_{ik}$ of the fundamental
metrical form. The volume of the parallelepiped thus
becomes
\[
= \sqrt{g} · \left\lvert
\begin{array}{@{}rrr@{}}
\xi^{1} & \xi^{2} & \xi^{3} \\
\eta^{1} & \eta^{2} & \eta^{3} \\
\zeta^{1} & \zeta^{2} & \zeta^{3} \\
\end{array}
\right\rvert\quad
\settowidth{\TmpLen}{\text{(taking the absolute,)}}
\parbox{\TmpLen}{(taking the absolute,
i.e.\ positive value of
the determinants).}
\]
This agrees with the elementary formulæ of analytical geometry.
In a space of more than three dimensions we may similarly pass
on to four-dimensional space-elements,~etc.
Just as a co-variant tensor of the first order assigns a number
\PageSep{57}
\index{Linear equation!tensor}%
linearly (and independently of the co-ordinate system) to every
one-dimensional space-element (i.e.\ displacement), so a skew-symmetrical
co-variant tensor of the second order assigns a
number to every two-dimensional space-element, a skew-symmetrical
tensor of the third order to each three-dimensional
space-element, and so on: this is immediately evident from the form
in which \Eq{(36)}~is expressed. For this reason we consider it justifiable
to call the co-variant skew-symmetrical tensors simply \Emph{linear
tensors}. Among operations in the domain of linear tensors
we shall mention the two following ones:---
\begin{gather*}
a_{i} b_{k} - a_{k} b_{i} = c_{ik}\Add{,}
\Tag{(37)} \\
a_{i} b_{kl} - a_{k} b_{li} + a_{l} b_{ik} = c_{ikl}\Add{.}
\Tag{(38)}
\end{gather*}
The former produces a linear tensor of the second order from two
linear tensors of the first order; the latter produces a linear tensor
of the third order from one of the first and one of the second.
Sometimes conditions of symmetry more complicated than
those considered heretofore occur. In the realm of quadrilinear
forms $F(\xi, \eta, \xi', \eta')$ those play a particular part which satisfy the
conditions\Pagelabel{57}
\begin{gather*}
F(\eta\Com \xi\Com \xi'\Com \eta')
= F(\xi\Com \eta\Com \eta'\Com \xi')
= -F(\xi\Com \eta\Com \xi'\Com \eta')\Add{,}
\Tag{(39_{1})} \\
%
F(\xi'\Com \eta'\Com \xi\Com \eta)
= F(\xi\Com \eta\Com \xi'\Com \eta')\Add{,}
\Tag{(39_{2})} \\
%
F(\xi\Com \eta\Com \xi'\Com \eta')
+ F(\xi\Com \xi'\Com \eta'\Com \eta)
+ F(\xi\Com \eta'\Com \eta\Com \xi') = 0\Add{.}
\Tag{(39_{3})}
\end{gather*}
For it may be shown that for every quadratic form of an arbitrary
two-dimensional space-element
\[
\xi^{ik} = \xi^{i} \eta^{k} - \xi^{k} \eta^{i}
\]
there is one and only one quadrilinear form~$F$ which satisfies
these conditions of symmetry, and from which the above quadratic
form is derived by identifying the second pair of variables $\xi'$,~$\eta'$
with the first pair $\xi$,~$\eta$. We must consequently use co-variant
tensors of the fourth order having the symmetrical properties~\Eq{(39)}
if we wish to represent functions which stand in quadratic relationship
with an element of surface.
The \Emph{most general form of the condition of symmetry} for a
tensor~$F$ of the fifth order of which the first, second, and fourth
series of variables are contra-gredient, the third and fifth co-gredient
(we are taking a particular case) are
\[
\sum_{S} e_{S} F_{S} = 0
\]
in which $S$~signifies all permutations of the five series of variables
in which the contra-gredient ones are interchanged among themselves
\PageSep{58}
and likewise the co-gredient ones; $F_{S}$~denotes the form which
results from~$F$ after the permutation~$S$; $e_{S}$~is a system of definite
numbers, which are assigned to the permutations~$S$. The summation
is taken over all the permutations~$S$. The kind of
symmetry underlying a definite type of tensors expresses itself
in one or more of such conditions of symmetry.
\Section{8.}{Tensor Analysis. Stresses}
\index{Stresses!elastic}%
\index{Tensor!field!(general)}%
Quantities which describe how the state of a spatially extended
physical system varies from point to point have not a distinct value
but only one ``for each point'': in mathematical language they
are ``functions of the place or point''. According as we are dealing
with a scalar, vector, or tensor, we speak of a scalar, vector, or
\index{Scalar!field}%
tensor \Emph{field}.
Such a field is given if a scalar, vector, or tensor of the proper
type is assigned to every point of space or to a definite region of it.
If we use a definite co-ordinate system the value of the scalar
quantities or of the components of the vector or tensor quantities
respectively, appear in the co-ordinate system as functions of the
co-ordinates of a variable point in the region under consideration.
Tensor analysis tells us how, by differentiating with respect to
\index{Differentiation of tensors and tensor-densities}%
the space co-ordinates, a new tensor can be derived from the old
one in a manner entirely independent of the co-ordinate system.
This method, like tensor algebra, is of extreme simplicity. Only
one operation occurs in it, viz.\ \Emph{differentiation}.
If
\[
\phi = f(x_{1}\Com x_{2}\Com \dots\Com x_{n}) = f(x)
\]
denotes a given scalar field, the change of~$\phi$ corresponding to an
infinitesimal displacement of the variable point, in which its co-ordinates~$x_{i}$
suffer changes~$dx_{i}$ respectively, is given by the total
differential
\[
df = \frac{\dd f}{\dd x_{1}}\, dx_{1}
+ \frac{\dd f}{\dd x_{2}}\, dx_{2}
+ \dots
+ \frac{\dd f}{\dd x_{n}}\, dx_{n}.
\]
This formula signifies that if the~$\Delta x_{i}$ are first taken as the components
of a finite displacement and the~$\Delta f$ are the corresponding
changes in~$f$, then the difference between
\[
\Delta f\quad\text{and}\quad \sum_{i} \frac{\dd f}{\dd x_{i}}\, \Delta x_{i}
\]
{\Loosen does not only decrease absolutely to zero with the components of
the displacement, but also relatively to the amount of the displacement,
\PageSep{59}
the measure of which may be defined as $|\Delta x_{1}| + |\Delta x_{2}| + \dots + |\Delta x_{n}|$.
We link up the linear form}
\[
\sum_{i} \frac{\dd f}{\dd x_{i}}\, \xi^{i}
\]
in the variables~$\xi^{i}$ to this differential. If we carry out the same
construction in another co-ordinate system (with horizontal bars
over the co-ordinates), it is evident from the meaning of the term
differential that the first linear form passes into the second, if the~$\xi^{i}$'s
are subjected to the transformation which is contra-gredient
to the fundamental vectors. Accordingly
\[
\frac{\dd f}{\dd x_{1}},\quad
\frac{\dd f}{\dd x_{2}},\ \dots\Add{,}\quad
\frac{\dd f}{\dd x_{n}}
\]
are the co-variant components of a vector which arises from the
scalar field~$\phi$ in a manner independent of the co-ordinate system.
In ordinary vector analysis it occurs as the \Emph{gradient} and is
\index{Gradient}%
denoted by the symbol~$\grad \phi$.
This operation may immediately be transposed from a scalar
to any arbitrary tensor field. If, e.g., $f_{ik}^{h}(x)$~are components of a
tensor field of the third order, contra-variant with respect to~$h$,
but co-variant with respect to $i$~and~$k$, then
\[
f_{ik}^{h} \xi_{h} \eta^{i} \zeta^{k}
\]
is an invariant, if we take~$\xi_{h}$ as standing for the components of an
arbitrary but constant co-variant vector (i.e.\ independent of its
position), and $\eta^{i}$,~$\zeta^{i}$ each as standing for the components of a
similar contra-variant vector in turn. The change in this invariant
due to an infinitesimal displacement with components~$dx_{i}$ is
given by
\[
\frac{\dd f_{ik}^{h}}{\dd x_{l}}\, \xi_{h} \eta^{i} \zeta^{k}\, dx_{l}
\]
hence
\[
f_{ikl}^{h} = \frac{\dd f_{ik}^{h}}{\dd x_{l}}
\]
are the components of a tensor field of the fourth order, which
arises from the given one in a manner independent of the co-ordinate
system. \Emph{Just this is the process of differentiation};
as is seen, it raises the order of the tensor by~$1$. We have still to
remark that, on account of the circumstance that the fundamental
metrical tensor is independent of its position, one obtains the
components of the tensor just formed, for example, which are
contra-variant with respect to the index~$k$, by transposing the
\PageSep{60}
index~$k$ under the sign of differentiation to the top, viz.\ $\dfrac{\dd f^{hki}}{\dd x_{l}}$. The
change from co-variant to contra-variant is interchangeable with
differentiation. Differentiation may be carried out purely formally
by imagining the tensor in question multiplied by a vector having
the co-variant components
\[
\frac{\dd}{\dd x_{1}},\quad
\frac{\dd}{\dd x_{2}},\ \dots\Add{,}\quad
\frac{\dd}{\dd x_{n}}
\Tag{(40)}
\]
and treating the differential quotient~$\dfrac{\dd f}{\dd x_{i}}$ as the symbolic product
of $f$ and~$\dfrac{\dd}{\dd x_{i}}$. The symbolic vector~\Eq{(40)} is often encountered in
mathematical literature under the mysterious name ``nabla-vector''.
\Par{Examples.}---The vector with the co-variant components~$u_{i}$
gives rise to the tensor of the second order $\dfrac{\dd u_{i}}{\dd x_{k}} = u_{ik}$. From this
we form
\[
\frac{\dd \Typo{u^{i}}{u_{i}}}{\dd x_{k}} - \frac{\dd u_{k}}{\dd x_{i}}\Add{.}
\Tag{(41)}
\]
These quantities are the co-variant components of a linear tensor
of the second order. In ordinary vector analysis it occurs (with
the signs reversed) as ``\Emph{rotation}'' (rot, spin or \Emph{curl}). On the
\index{Curl}%
\index{Rotation!curl@{(or curl)}}%
other hand the quantities
\[
\tfrac{1}{2}\left(\frac{\dd u_{i}}{\dd x_{k}} + \frac{\dd u_{k}}{\dd x_{i}}\right)
\]
are the co-variant components of a symmetrical tensor of the
\index{Divergence@{Divergence (\emph{div})}}%
\index{Stresses!elastic}%
second order. If the vector~$u$ represents the velocity of continuously
extended moving matter as a function of its position, the
vanishing of this tensor at a point signifies that the immediate
neighbourhood of the point moves as a rigid body; it thus merits
the name \Emph{distortion tensor}. Finally by contracting~$u_{k}^{i}$ we get
\index{Distortion tensor}%
the scalar
\[
\frac{\dd u^{i}}{\dd x_{i}}
\]
which is known in vector analysis as ``\Emph{divergence}'' (div.).
By differentiating and contracting a tensor of the second order
having mixed components~$S_{i}^{k}$ we derive the vector
\[
\frac{\dd S_{i}^{k}}{\dd x_{k}}.
\]
If $v_{ik}$~are the components of a linear tensor field of the second
order, then, analogously to formula~\Eq{(38)} in which we substitute~$v$
\PageSep{61}
or~$b$ and the symbolic vector ``differentiation'' for~$a$, we get the
linear tensor of the third order with the components
\[
\frac{\dd v_{kl}}{\dd x_{i}} +
\frac{\dd v_{li}}{\dd x_{k}} +
\frac{\dd v_{ik}}{\dd x_{l}}\Add{.}
\Tag{(42)}
\]
Tensor~\Eq{(41)}, i.e.\ the curl, vanishes if $v_{i}$~is the gradient of a scalar
field; tensor~\Eq{(42)} vanishes if $v_{ik}$~is the curl of a vector~$u_{i}$.
\Par{Stresses.}---An important example of a tensor field is offered by
the stresses occurring in an elastic body; it is, indeed, from this
example that the name ``tensor'' has been derived. When tensile
or compressional forces act at the surface of an elastic body, whilst,
in addition, ``volume-forces'' (e.g.\ gravitation) act on various
portions of the matter within the body, a state of equilibrium establishes
itself, in which the forces of cohesion called up in the
matter by the distortion balance the impressed forces from without.
If we imagine any portion~$J$ of the matter cut out of the body and
suppose it to remain coherent after we have removed the remaining
portion, the impressed volume forces will not of themselves keep
this piece of matter in a state of equilibrium. They are, however,
balanced by the compressional forces acting on the surface~$\Omega$ of the
portion~$J$, which are exerted on it by the portion of matter removed.
We have actually, if we do not take the atomic (granular) structure
of matter into account, to imagine that the forces of cohesion are
only active in direct contact, with the consequence that the action
of the removed portion upon~$J$ must be representable by superficial
forces such as pressure: and indeed, if $\vS\, do$~is the pressure acting
on an element of surface~$do$ ($\vS$~here denotes the pressure per unit
surface), $\vS$~can depend only upon the place at which the element of
surface~$do$~happens to be and on the inward normal~$n$ of this element
of surface with respect to~$J$, which characterises the ``position'' of~$do$.
We shall write $\vS_{n}$ for~$\vS$ to emphasise this connection between
$\vS$ and~$n$. If $-n$~denotes the normal in a direction reversed to that
of~$n$, it follows from the equilibrium of a small infinitely thin disc,
that
\[
\vS_{-n} = -\vS_{n}\Add{.}
\Tag{(43)}
\]
We shall use Cartesian co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$. The compressional
forces per unit of area at a point, which act on an element
of surface situated at the same point, the inward normals of which
coincide with the direction of the positive $x_{1}$-,~$x_{2}$-, $x_{3}$-axis respectively
will be denoted by $\vS_{1}$,~$\vS_{2}$,~$\vS_{3}$. We now choose any
three positive numbers $\Typo{\alpha}{a}_{1}$,~$\Typo{\alpha}{a}_{2}$,~$\Typo{\alpha}{a}_{3}$, and a positive number~$\epsilon$, which is
to converge to the value~$0$ (whereas the~$\Typo{\alpha}{a}_{i}$ remain fixed). From
\PageSep{62}
the point~$O$ under consideration we mark off in the direction of
the positive co-ordinate axes the distances
\[
OP_{1} = \epsilon \Typo{\alpha}{a}_{1},\qquad
OP_{2} = \epsilon \Typo{\alpha}{a}_{2},\qquad
OP_{3} = \epsilon \Typo{\alpha}{a}_{3}
\]
and consider the infinitesimal tetrahedron $OP_{1}P_{2}P_{3}$ having $OP_{2}P_{3}$,
$OP_{3}P_{1}$, $OP_{1}P_{2}$ as walls and $P_{1}P_{2}P_{3}$~as its ``roof''. If $f$~is the
superficial area of the roof and $\Typo{\alpha}{a}_{1}$,~$\Typo{\alpha}{a}_{2}$,~$\Typo{\alpha}{a}_{3}$ are the direction cosines of
its inward normals~$n$, then the areas of the walls are
\[
-f · \Typo{\alpha}{a}_{1} (= \tfrac{1}{2} \epsilon^{2} \Typo{\alpha}{a}_{2}\Typo{\alpha}{a}_{3}),\qquad
-f · \Typo{\alpha}{a}_{2},\qquad
-f · \Typo{\alpha}{a}_{3}.
\]
The sum of the pressures on the walls and the roof becomes for
evanescent values of~$\epsilon$:
\[
f\bigl\{\vS_{n}
- (\Typo{\alpha}{a}_{1}\vS_{1}
+ \Typo{\alpha}{a}_{2}\vS_{2}
+ \Typo{\alpha}{a}_{3}\vS_{3})\bigr\}.
\]
The magnitude of~$f$ is of the order~$\epsilon^{2}$: but the volume force acting
upon the volume of the tetrahedron is only of the order of magnitude~$\epsilon^{3}$.
Hence, owing to the condition for equilibrium, we must
have
\[
\vS_{n} = (\Typo{\alpha}{a}_{1}\vS_{1}
+ \Typo{\alpha}{a}_{2}\vS_{2}
+ \Typo{\alpha}{a}_{3}\vS_{3}).
\]
With the help of~\Eq{(43)} this formula may be extended immediately
to the case in which the tetrahedron is situated in any of the remaining
7~octants. If we call the components of~$\vS_{i}$ with respect
to the co-ordinate axes $S_{i1}$,~$S_{i2}$,~$S_{i3}$, and if $\xi^{i}$,~$\eta^{i}$ are the components
of any two arbitrary displacements of length~$1$, then
\[
\sum_{i\Com k} S_{ik} \xi^{i} \eta^{k}
\Tag{(44)}
\]
is the component, in the direction~$\eta$, of the compressional force
which is exerted on an element of surface of which the inner
normal is~$\xi$. The bilinear form~\Eq{(44)} has thus a significance independent
of the co-ordinate system, and the~$S_{ik}$'s are the components
of a ``stress'' tensor field. We shall continue to operate
in rectangular co-ordinate systems so that we shall not have to
distinguish between co-variant and contra-variant quantities.
We form the vector~$\vS_{1}'$ having components $S_{1i}$,~$S_{2i}$,~$S_{3i}$. The
component of~$\vS_{1}'$ in the direction of the inward normal~$n$ of an
element of surface is then equal to the $x_{1}$-component of~$\vS_{n}$. The
$x_{1}$-component of the total pressure which acts on the surface~$\Omega$
of the detached portion of matter~$J$ is therefore equal to the surface
integral of the normal components of~$\vS_{1}'$ and this, by Gauss's
Theorem, is equal to the volume integral
\[
-\int_{J} \div \vS_{1}' · dV.
\]
\PageSep{63}
The same holds for the $x_{2}$~and the $x_{3}$~component. We have thus
to form the vector~$\vp$ having the components
\[
p_{i} = -\sum_{k} \frac{\delta S_{i}^{k}}{\delta x_{k}}
\]
(this is performed, as we know, according to an invariant law).
The compressional forces~$\vS$ are then equivalent to a volume force
having the direction and intensity given by $\vp$~per unit volume in
the sense that, for every dissociated portion of matter~$J$,
\[
\int_{\Omega} \vS_{n}\, do = \int_{J} \vp\, dV\Add{.}
\Tag{(45)}
\]
If $\vk$~is the impressed force per unit volume, the first condition of
equilibrium for the piece of matter considered coherent after being
detached is
\[
\int_{J} (\vp + \vk)\, dV = \Typo{0}{\0},
\]
and as this must hold for every portion of matter
\[
\vp + \vk = \Typo{0}{\0}\Add{.}
\Tag{(46)}
\]
If we choose an arbitrary origin~$O$ and if $\vr$~denote the radius
vector to the variable point~$P$, and the square bracket denote the
``vectorial'' product, the second condition for equilibrium, the
equation of moments, is
\[
\int_{\Omega} [\vr, \vS_{n}]\, do
+ \int_{J} [\vr, \vk]\, dV = \Typo{0}{\0},
\]
and since \Eq{(46)}~holds generally we must have, besides~\Eq{(45)},
\[
\int_{\Omega} [\vr, \vS_{n}]\, do
= \int_{J} [\vr, \vp]\, dV.
\]
{\Loosen The $x_{1}$~component of $[\vr, \vS_{n}]$ is equal to the component of $x_{2} \vS_{3}' - x_{3} \vS_{2}'$ in the direction of~$n$. Hence, by Gauss's theorem, the $x_{1}$~component
of the left-hand member is}
\[
- \int_{J} \div(x_{2} \vS_{3}' - x_{3} \vS_{2}')\, dV.
\]
Hence we get the equation
\[
\div(x_{2} \vS_{3}' - x_{3} \vS_{2}') = -(x_{2}p_{3} - x_{3}p_{2}).
\]
But the left-hand member
\begin{align*}
&= (x_{2} \div \vS_{3}' - x_{3} \div \vS_{2}')
+ (\vS_{3}' · \grad x_{2} - \vS_{2}' \Add{·} \grad x_{3}) \\
&= -(x_{2}p_{3} - x_{3}p_{2}) + (S_{23} - S_{32}).
\end{align*}
\PageSep{64}
Accordingly, if we form the $x_{2}$~and $x_{3}$~components in addition to
the $x_{1}$~component, this condition of equilibrium gives us
\[
S_{23} = S_{32},\qquad
S_{31} = S_{13},\qquad
S_{12} = S_{21},
\]
i.e.\ the symmetry of the \Emph{stress-tensor~$\vS$}. For an arbitrary displacement
having the components~$\xi^{i}$,
\[
\frac{\sum S_{ik} \xi^{i} \xi^{k}}{\sum g_{ik} \xi^{i} \xi^{k}}
\]
is the component of the pressure per unit surface for the component
in the direction~$\xi$, which acts on an element of surface placed at
right angles to this direction. (We may here again use any arbitrary
affine co-ordinate system.) \Emph{The stresses are fully equivalent
to a volume force} of which the density~$p$ is calculated
according to the invariant formulæ
\[
-p_{i} = \frac{\delta S_{i}^{k}}{\delta x_{k}}\Add{.}
\Tag{(47)}
\]
In the case of a pressure~$p$ which is equal in all directions
\[
S_{i}^{k} = p · \delta_{i}^{k},\qquad
p_{i} = -\frac{\delta p}{\delta x_{i}}.
\]
As a result of the foregoing reasoning we have formulated in
exact terms the conception of stress alone, and have discovered
how to represent it mathematically. To set up the fundamental
laws of the theory of elasticity it is, in addition, necessary to find
out how the stresses depend on the distortion brought about in
the matter by the impressed forces. There is no occasion for us to
discuss this in greater detail.
\Section{9.}{Stationary Electromagnetic Fields}
\index{Maxwell's!theory!(stationary case)}%
Hitherto, whenever we have spoken of mechanical or physical
things, we have done so for the purpose of showing in what manner
their spatial nature expresses itself: namely, that its laws manifest
themselves as invariant tensor relations. This also gave us an
opportunity of demonstrating the importance of the tensor calculus
by giving concrete examples of it. It enabled us to prepare
the ground for later discussions which will grapple with physical
theories in greater detail, both for the sake of the theories themselves
and for their important bearing on the problem of time. In
this connection the \Emph{theory of the electromagnetic field}, which
\index{Electromagnetic field}%
is the most perfect branch of physics at present known, will be of
the highest importance. It will here only be considered in so far
\PageSep{65}
as time does not enter into it, i.e.\ we shall confine our attention
to conditions which are stationary and invariable in time.
Coulomb's Law for electrostatics may be enunciated thus. If
any charges of electricity are distributed in space with the density~$\rho$
they exert a force
\[
\vK = e · \vE
\Tag{(48)}
\]
upon a point-charge~$e$, whereby
\[
\vE = -\int \frac{\rho · \vr}{4\pi r^{3}}\, dV\Add{.}
\Tag{(49)}
\]
$\vr$~here denotes the vector~$\Vector{OP}$ which leads from the ``point of emergence~$O$''
at which $\vE$~is to be determined, to the ``current point'' or
source, with respect to which the integral is taken: $r$~is its length
and $dV$~is the element of volume. The force is thus composed of
two factors, the charge~$e$ of the small testing body, which depends
on its condition alone, and of the ``intensity of field''~$\vE$, which on
\index{Electrical!intensity of field}%
\index{Field action of electricity!intensity of electrical}%
\index{Intensity of field}%
the contrary is determined solely by the given distribution of the
charges in space. We picture in our minds that even if we do
not observe the force acting on a testing body, an ``electric field''
is called up by the charges distributed in space, this field being
described by the vector~$\vE$; the action on a point-charge~$e$ expresses
itself in the force~\Eq{(48)}. We may derive~$\vE$ from a potential~$-\phi$
in accordance with the formulæ
\[
\vE = \grad\phi\Add{,}\qquad
-4\pi \phi = \int \frac{\rho}{r}\, dV\Add{.}
\Tag{(50)}
\]
From \Eq{(50)} it follows (1)~that $\vE$~is an irrotational (and hence lamellar)
vector, and (2)~that the flux of~$\vE$ through any closed surface is equal
to the charges enclosed by this surface, or that the electricity is the
source of the electric field; i.e.\ in formulæ
\[
\curl \vE = \Typo{0}{\0}\Add{,}\qquad
\div \vE = \rho\Add{.}
\Tag{(51)}
\]
Inversely, Coulomb's Law arises out of these simple differential
laws if we add the condition that the field~$\vE$ vanish at infinite
distances. For if we put $\vE = \grad\phi$ from the first of the equations~\Eq{(51)},
we get from the second, to determine~$\phi$, Poisson's equation
$\Delta\phi = \rho$, the solution of which is given by~\Eq{(50)}.
Coulomb's Law deals with ``\Emph{action at a distance}''. The
intensity of the field at a point is expressed by it \Erratum{independently of}{depending on}
the charges at all other points, near or far, in space. In contra-distinction
from this the far simpler formulæ~\Eq{(51)} express laws
relating to ``infinitely near'' action. As a \Typo{knowlege}{knowledge} of the values
of a function in an arbitrarily small region surrounding a point is
sufficient to determine the differential quotient of the function at
\PageSep{66}
the point, the values of $\rho$~and~$\vE$ at a point and in its immediate
neighbourhood are brought into connection with one another by~\Eq{(51)}.
We shall regard these laws of infinitely near action as the
true expression of the uniformity of action in nature, whereas we
look upon~\Eq{(49)} merely as a mathematical result following logically
from it. In the light of the laws expressed by~\Eq{(51)} which have
such a simple intuitional significance we believe that we \Emph{understand}
the source of Coulomb's Law. In doing this we do indeed
bow to dictates of the theory of knowledge. Even Leibniz formulated
the postulate of continuity, of infinitely near action, as a
general principle, and could not, for this reason, become reconciled
to Newton's Law of Gravitation, which entails action at a distance
and which corresponds fully to that of Coulomb. The mathematical
clearness and the simple meaning of the laws\Eq{(51)} are
additional factors to be taken into account. In building up the
theories of physics we notice repeatedly that once we have succeeded
in bringing to light the uniformity of a certain group of
phenomena it may be expressed in formulæ of perfect mathematical
harmony. After all, from the physical point of view, Maxwell's
theory in its later form bears uninterrupted testimony to the
stupendous fruitfulness which has resulted through passing from
the old idea of action at a distance to the modern one of infinitely
near action.
The field exerts on the charges which produce it a force of
which the density per unit volume is given by the formula
\[
\vp = \rho \vE\Add{.}
\Tag{(52)}
\]
This is the rigorous interpretation of the equation~\Eq{(48)}.
If we bring a test charge (on a small body) into the field, it
also becomes one of the field-producing charges, and formula~\Eq{(48)}
will lead to a correct determination of the field~$\vE$ existing before
the test charge was introduced, only if the test charge~$e$ is so weak
that its effect on the field is imperceptible. This is a difficulty
which permeates the whole of experimental physics, viz.\ that by
introducing a measuring instrument the original conditions which
are to be measured become disturbed. This is, to a large extent,
the source of the errors to the elimination of which the experimenter
has to apply so much ingenuity.
The fundamental law of mechanics: $\text{mass} × \text{acceleration} = \text{force}$,
\index{Mechanics!fundamental law of!Newton@{of Newton's}}%
tells us how masses move under the influence of given forces
(the initial velocities being given). Mechanics does not, however,
teach us what is force; this we learn from physics. \emph{The fundamental
law of mechanics is a blank form which acquires a concrete
\PageSep{67}
content only when the conception of force occurring in it is filled in
by physics.} The unfortunate attempts which have been made to
develop mechanics as a branch of science distinct in itself have, in
consequence, always sought help by resorting to an explanation in
\emph{words} of the fundamental law: force \Emph{signifies} $\text{mass} × \text{acceleration}$.
In the present case of electrostatics, i.e.\ for the particular
category of physical phenomena, we recognise what is force, and how
it is determined according to a definite law by~\Eq{(52)} from the phase-quantities
charge and field. If we regard the charges as being
given, the field equations~\Eq{(51)} give the relation in virtue of which
the charges determine the field which they produce. With regard
to the charges, it is known that they are bound to matter. The
modern theory of electrons has shown that this can be taken in a
perfectly rigorous sense. Matter, is composed of elementary quanta,
electrons, which have a definite invariable mass, and, in addition,
a definite invariable charge. Whenever new charges appear to
spring into existence, we merely observe the separation of positive
and negative elementary charges which were previously so close
together that the ``action at a distance'' of the one was fully compensated
by that of the other. In such processes, accordingly, just
as much positive electricity ``arises'' as negative. The laws thus
constitute a cycle. The distribution of the elementary quanta of
matter provided with charges fixed once and for all (and, in the
case of non-stationary conditions, also their velocities) determine
the field. The field exerts upon charged matter a ponderomotive
\index{Ponderomotive force!of the electric, magnetic and electromagnetic field}%
force which is given by~\Eq{(52)}. The force determines, in accordance
with the fundamental law of mechanics, the acceleration, and hence
the distribution and velocity of the matter at the following moment.
\Emph{We require this whole network of theoretical considerations
to arrive at an experimental means of verification},---if we
assume that what we directly observe is the motion of matter.
(Even this can be admitted only conditionally.) We cannot merely
test a single law detached from this theoretical fabric! The connection
between direct experience and the objective element behind
it, which reason seeks to grasp conceptually in a theory, is not so
simple that every single statement of the theory has a meaning
which may be verified by direct intuition. We shall see more and
more clearly in the sequel that Geometry, Mechanics, and Physics
form an inseparable theoretical whole in this way. We must
never lose sight of this totality when we enquire whether these
sciences interpret rationally the reality which proclaims itself
in all subjective experiences of consciousness, and which itself
transcends consciousness: that is, truth forms a \Emph{system}. For the
\PageSep{68}
rest, the physical world-picture here described in its first outlines
is characterised by the dualism of \Emph{matter} and \Emph{field}, between
\index{Matter}%
which there is a reciprocal action. Not till the advent of the
theory of relativity was this dualism overcome, and, indeed, in
favour of a physics based solely on fields (cf.\ §\,24).
The ponderomotive force in the electric field was traced back
\index{Field action of electricity!general@{(general conception)}}%
\index{Force!(electric)}%
\index{Force!(ponderomotive, of electrical field)}%
to stresses even by Faraday. If we use a rectangular system of
co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$ in which $E_{1}$,~$E_{2}$,~$E_{3}$ are the components of
the electrical intensity of field, the $x_{i}$~component of the force-density
is
\[
p_{i} = \rho E_{i}
= E\left(\frac{\dd E_{1}}{\dd x_{1}}
+ \frac{\dd E_{2}}{\dd x_{2}}
+ \frac{\dd E_{3}}{\dd x_{3}}\right).
\]
By a simple calculation which takes account of the irrotational
property of~$\vE$ we discover from this that the components~$p_{i}$ of the
force-density are derived by the formulæ~\Eq{(47)} from the stress tensor,
the components~$S_{ik}$ of which are tabulated in the following quadratic
scheme
\[
\left\lvert
\begin{array}{@{}ccc@{}}
\frac{1}{2}(E_{2}^{2} + E_{3}^{2} - E_{1}^{2}) & -E_{1}E_{2} & -E_{1}E_{3} \\
-E_{2}E_{1} & \frac{1}{2}(E_{3}^{2} + E_{1}^{2} - E_{2}^{2}) & -E_{2}E_{3} \\
-E_{3}E_{1} & -E_{3}E_{2} & \frac{1}{2}(E_{1}^{2} + E_{2}^{2} - E_{3}^{2}) \\
\end{array}\right\rvert
\Tag{(53)}
\]
We observe that the condition of symmetry $S_{ki} = S_{ik}$ is fulfilled. It
is, above all, important to notice that the components of the stress
tensor at a point depend only on the electrical intensity of field at
this point. (They, moreover, depend only on the \Emph{field}, and not on
the charge.) Whenever a force~$p$ can be retraced by~\Eq{(47)} to stresses~$S$,
which form a symmetrical tensor of the second order only dependent
on the values of the phase-quantities describing the physical
state at the point in question, we shall have to regard these stresses
as the primary factors and the actions of the forces as their consequent.
The mathematical justification for this point of view is
brought to light by the fact that the force~$p$ results from differentiating
the stress. Compared with forces, stresses are thus, so to
speak, situated on the next lower plane of differentiation, and yet
do not depend on the whole series of values traversed by the phase-quantities,
as would be the case for an arbitrary integral, but only
on its value at the point under consideration. It further follows
from the fact that the electrostatic forces which charged bodies
exert on one another can be retraced to a symmetrical stress tensor,
that the resulting total force as well as the resulting couple vanishes
(because the integral taken over the whole space has a divergence
$= 0$). This means that an isolated system of charged masses
\PageSep{69}
which is initially at rest cannot of itself acquire a translational or
rotational motion as a whole.
The tensor~\Eq{(53)} is, of course, independent of the choice of co-ordinate
system. If we introduce the square of the value of the
field intensity
\[
|E|^{2} = E_{i} E^{i}
\]
then we have
\[
S_{ik} = \tfrac{1}{2}g_{ik} |E|^{2} - E_{i} E_{k}.
\]
These are the co-variant stress components not only in a Cartesian
but also in any arbitrary affine co-ordinate system, if $E_{i}$ are the co-variant
components of the field intensity. The physical significance
of these stresses is extremely simple. If, for a certain point, we
use rectangular co-ordinates, the $x_{1}$~axis of which points in the
direction~$\vE$: then
\[
E_{1} = |E|\Add{,}\qquad
E_{2} = 0\Add{,}\qquad
E_{3} = 0\Add{;}
\]
we thus find them to be composed of a tension having the intensity
$\frac{1}{2} |E|^{2}$ in the direction of the lines of force, and of a pressure of
the same intensity acting perpendicularly to them.
\Emph{The fundamental laws of electrostatics may now be summarised
in the following invariant tensor form}:---
\[
\left.
\begin{aligned}
\Inum{(I)}\vphantom{\dfrac{\dd E}{\dd E}}& \\
\Inum{(II)}\vphantom{\dfrac{\dd E}{\dd E}}& \\
\Inum{(III)}
\end{aligned}\quad
\begin{gathered}
\frac{\dd E_{i}}{\dd x_{k}} - \frac{\dd E_{k}}{\dd x_{i}} = 0,
\text{ or }
E_{i} = \frac{\dd \phi}{\dd x_{i}}\text{ respectively;} \\
\frac{\dd E^{i}}{\dd x_{i}} = \rho; \\
S_{ik} = \tfrac{1}{2}g_{ik}|E|^{2} - E_{i} E_{k}.
\end{gathered}
\right\}
\Tag{(54)}
\]
A system of discrete point-charges $e_{1}$,~$e_{2}$, $e_{3}$,~\dots\ has potential
energy
\[
U = \frac{1}{8\pi} \sum_{i \neq k} \frac{e_{i} e^{k}}{r_{ik}}
\]
in which $r_{ik}$~denotes the distance between the two charges $e_{i}$ and~$e_{k}$.
This signifies that the virtual work which is performed by the
forces acting at the separate points (owing to the charges at the
remaining points) for an infinitesimal displacement of the points
is a total differential, viz.~$\delta U$. For continuously distributed charges
this formula resolves into
\[
U = \iint \frac{\rho(P) \rho(P')}{8\pi r_{PP'}}\, dV\, dV'
\]
in which both volume integrations with respect to $P$ and~$P'$ are to
\PageSep{70}
be taken over the whole space, and $r_{PP'}$~denotes the distance between
these two points. Using the potential~$\phi$ we may write
\[
U = -\tfrac{1}{2} \int \rho\phi\, dV.
\]
The integrand is $\phi · \div\vE$. In consequence of the equation
\[
\div(\phi\vE) = \phi · \div\vE + \vE \grad\phi
\]
and of Gauss's theorem, according to which the integral of $\div(\phi\vE)$
taken over the whole space is equal to~$0$, we have
\[
-\int \rho\phi\, dV = \int (\vE \grad\phi)\, dV
= \int |E|^{2}\, dV;
\]
i.e.\
\[
U = \int \tfrac{1}{2} |\vE|^{2}\, dV\Add{.}
\Tag{(55)}
\]
This representation of the energy makes it directly evident that
the energy is a \Emph{positive} quantity. If we trace the forces back to
stresses, we must picture these stresses (like those in an elastic
body) as being everywhere associated with positive potential energy
of strain. The seat of the energy must hence be sought in the field.
Formula~\Eq{(55)} gives a fully satisfactory account of this point. It
tells us that the energy associated with the strain amounts to $\frac{1}{2}|E|^{2}$
per unit volume, and is thus exactly equal to the tension and the
pressure which are exerted along and perpendicularly to the lines
of force. The deciding factor which makes this view permissible is
again the circumstance that the value obtained for the energy-density
\index{Energy-density!(in the electric field)}%
depends solely on the value, \Emph{at the point in question}, of
the phrase-quantity~$\vE$ which characterises the field. Not only the
field as a whole, but every portion of the field has a definite
amount of potential energy $= \int \frac{1}{2}|E|^{2}\, dV$. In statics, it is only the
total energy which comes into consideration. Only later, when
we pass on to consider variable fields, shall we arrive at irrefutable
confirmation of the correctness of this view.
In the case of conductors in a statical field the charges collect
on the outer surface and there is no field in the interior. The
equations~\Eq{(51)} then suffice to determine the electrical field in free
space in the ``æther''. If, however, there are non-conductors,
dielectrics in the field, the phenomenon of \Emph{dielectric polarisation}
\index{Dielectric}%
\index{Displacement current!dielectric}%
(displacement) must be taken into consideration. Two charges
$+e$ and~$-e$ at the points $P_{1}$~and $P_{2}$ respectively, ``source and
sink'' as we shall call them, produce a field, which arises from
the potential
\[
\frac{e}{4\pi} \left(\frac{1}{r_{1}} - \frac{1}{r_{2}}\right)
\]
\PageSep{71}
in which $r_{1}$~and~$r_{2}$ denote the distances of the points $P_{1}$,~$P_{2}$ from
the origin,~$O$. Let the product of~$e$ and the vector~$\Vector{P_{1}P_{2}}$ be called
the moment~$\vm$ of the ``source and sink'' pair. If we now suppose
the two charges to approach one another in a definite direction at
a point~$P$, the charge increasing simultaneously in such a way
that the moment~$\vm$ remains constant, we get, in the limit, a
``doublet'' of moment~$\vm$, the potential of which is given by
\[
\frac{\vm}{4\pi} \grad_{P} \frac{1}{r}.
\]
The result of an electric field in a dielectric is to give rise to
\index{Displacement current!electrical}%
these doublets in the separate elements of volume: this effect is
known as \Emph{polarisation}. If $\vm$~is the electric moment of the
\index{Polarisation}%
doublets per unit volume, then, instead of~\Eq{(50)}, the following
formula holds for the potential
\[
-4\pi \phi
= \int \frac{\rho}{r}\, dV + \int \vm · \grad_{P} \frac{1}{r}\, dv\Add{.}
\Tag{(56)}
\]
From the point of view of the theory of electrons this circumstance
\index{Atom, Bohr's}%
\index{Bohr's model of the atom}%
becomes immediately intelligible. Let us, for example, imagine an
atom to consist of a positively charged ``nucleus'' at rest, around
which an oppositely charged electron rotates in a circular path.
The mean position of the electron for the mean time of a complete
revolution of the electron round the nucleus will then
coincide with the position of the nucleus, and the atom will appear
perfectly neutral from without. But if an electric field acts, it
exerts a force on the negative electron, as a result of which its
%[** TN: [sic] excentrically]
path will lie excentrically with respect to the atomic nucleus, e.g.\
will become an ellipse with the nucleus at one of its foci. In the
mean, for times which are great compared with the time of revolution
of the electron, the atom will act like a doublet; or if we
treat matter as being continuous we shall have to assume continuously
distributed doublets in it. Even before entering upon
an exact atomistic treatment of this idea we can say that, at least
to a first approximation, the moment~$\vm$ per unit volume will be
proportional to the intensity~$\vE$ of the electric field: i.e.\ $\vm = k\vE$,
in which $k$~denotes a constant characteristic of the matter, which
is dependent on its chemical constitution, viz.\ on the structure of
its atoms and molecules.
Since
\[
\div \left(\frac{\vm}{r}\right)
= \vm \grad \frac{1}{r} + \frac{\div \vm}{r}
\]
we may replace equation~\Eq{(56)} by
\[
-4\pi \phi = \int \frac{\rho - \div\vm}{r}\, dV.
\]
\PageSep{72}
From this we get for the field intensity $\vE = \grad\phi$
\[
\div \vE = \rho - \div \vm.
\]
If we now introduce the ``electric displacement''
\[
\vD = \vE + \vm
\]
the fundamental equations become:
\[
\curl \vE = \Typo{0}{\0},\qquad
\div \vD = \rho\Add{.}
\Tag{(57)}
\]
They correspond to equations~\Eq{(51)}; in one of them the intensity~$\vE$
of field now occurs, in the other $\vD$~the electric displacement.
With the above assumption $\vm = k\vE$ we get the law of matter
\[
\vD = \epsilon\vE
\Tag{(58)}
\]
if we insert the constant $\epsilon = 1 + k$, characteristic of the matter,
called the \Emph{dielectric constant}.
\index{Dielectric!constant}%
These laws are excellently confirmed by observation. The
influence of the intervening medium which was experimentally
proved by Faraday, and which expresses itself in them, has been
of great importance in the development of the theory of action by
contact. We may here pass over the corresponding extension of
the formulæ for stress, energy, and force.
It is clear from the mode of derivation that \Eq{(57)}~and~\Eq{(58)} are
not rigorously valid laws, since they relate only to mean values and
are deduced for spaces containing a great number of atoms and for
times which are great compared with the times of revolution of the
electrons round the atom. \Emph{We still look upon~\Eq{(51)} as expressing
the physical laws exactly.} Our objective here and
in the sequel is above all to derive the strict physical laws. But if
we start from phenomena, such ``phenomenological laws'' as \Eq{(57)}~and~\Eq{(58)}
are necessary stages in passing from the results of direct
observation to the exact theory. In general, it is possible to work
out such a theory only by starting in this way. The validity of
the theory is then established if, with the aid of definite ideas
about the atomic structure of matter, we can again arrive at the
phenomenological laws by using mean value arguments. If the
atomic structure is known, this process must, in addition, yield the
values of the constants occurring in these laws and characteristic
of the matter in question (such constants do not occur in exact
physical laws). Since laws of matter such as~\Eq{(58)}, which only take
the influence of massed matter into account, certainly fail for events
in which the fine structure of matter cannot be neglected, the
range of validity of the phenomenological theory must be furnished
by an atomistic theory of this kind, as must also those laws which
have to be substituted in its place for the region beyond this range.
\PageSep{73}
In all this the electron theory has met with great success, although,
in view of the difficulty of the task, it is far from giving a complete
statement of the more detailed structure of the atom and its inner
mechanism.
In the first experiments with permanent magnets, magnetism
appears to be a mere repetition of electricity: here Coulomb's Law
\index{Coulomb's Law}%
holds likewise! A characteristic difference, however, immediately
asserts itself in the fact that positive and negative magnetism cannot
be dissociated from one another. There are no sources, but
only doublets in the magnetic field. Magnets consist of infinitely
small elementary magnets, each of which itself contains positive
and negative magnetism. The amount of magnetism in every
portion of matter is \textit{de~facto} nil; this would appear to mean that
there is really no such thing as magnetism. The explanation of
this was furnished by Oersted's discovery of the magnetic action of
electric currents. The exact quantitative formulation of this action
as expressed by Biot and Savart's Law leads, just like Coulomb's
\index{Biot and Savart's Law}%
Law, to two simple laws of action by contact. If $\vs$~denotes the
density of the electric current, and $\vH$~the intensity of the magnetic
field, then
\[
\curl \vH = \vs,\qquad
\div \vH = 0\Add{.}
\Tag{(59)}
\]
The second equation asserts the non-existence of sources in the
\index{Electrostatic potential}%
magnetic field. Equations~\Eq{(59)} are exactly analogous to~\Eq{(51)} if div
and curl be interchanged. These two operations of vector analysis
correspond to one another in exactly the same way as do scalar and
vectorial multiplication in vector algebra (div denotes scalar, curl
vectorial, multiplication by the symbolic vector ``differentiation'').
The solution of the equations~\Eq{(59)} vanishes for infinite distances;
for a given distribution of current it is given by
\[
%[** TN: Bracket notation for cross product]
\vH = \int \frac{[\vs\Com \vr]}{4\pi r^{3}}\, dV\Add{,}
\Tag{(60)}
\]
which is exactly analogous to~\Eq{(49)} and is, indeed, the expression of
Biot and Savart's Law. This solution may be derived from a
``vector potential''\Typo{---$\vf$}{ $-\vf$} in accordance with the formulæ
\[
\vH = -\curl \vf\Add{,}\qquad
-4\pi \vf = \int \frac{\vs}{r}\, dV.
\]
Finally the formula for the density of force in the magnetic field is
\index{Energy-density!(in the magnetic field)}%
\index{Force!(ponderomotive, of magnetic field)}%
\index{Ponderomotive force!of the electric, magnetic and electromagnetic field}%
\[
\vp = [\vs\Com \vH]
\Tag{(61)}
\]
corresponding exactly with~\Eq{(52)}\Add{.}
There is no doubt that these laws give us a true statement of
\PageSep{74}
magnetism. They are not a repetition but an exact counterpart
\index{Magnetism}%
of electrical laws, and bear the same relation to the latter as
vectorial products to scalar products. From them it may be
proved mathematically that a small circular current acts exactly
like a small elementary magnet thrust through it perpendicularly
to its plane. Following Ampère we have thus to imagine the
magnetic action of magnetised bodies to depend on \Emph{molecular
currents}; according to the electron theory these are straightway
\index{Molecular currents}%
given by the electrons circulating in the atom.
The force~$\vp$ in the magnetic field may also be traced back to
stresses, and we find, indeed, that we get the same values for the
stress components as in the electrostatic field: we need only
replace $\vE$ by~$\vH$. Consequently we shall use the corresponding
value $\frac{1}{2}\vH^{2}$ for the density of the potential energy contained in the
\index{Potential!vector-}%
field. This step will only be properly justified when we come to
the theory of fields varying with the time.
It follows from~\Eq{(59)} that the current distribution is free of
sources: $\div \vs = 0$. The current field can therefore be entirely
divided into current tubes all of which again merge into themselves,
i.e.\ are continuous. The same total current flows through every
cross-section of each tube. In no wise does it follow from the
laws holding in a stationary field, nor does it come into consideration
for such a field, that this current is an electric current in the
ordinary sense, i.e.\ that it is composed of electricity in motion;
this is, however, without doubt the case. In view of this fact the
law $\div \vs = 0$ asserts that electricity is neither created nor destroyed.
It is only because the flux of the current vector through a closed
\index{Vector!potential}%
surface is nil that the density of electricity remains everywhere
unchanged---so that electricity is neither created nor destroyed.
(We are, of course, dealing with stationary fields exclusively.)
The expression \Emph{vector potential}~$\vf$, introduced above, also satisfies
the equation $\div \vf = 0$.
Being an electric current, $\vs$~is without doubt a vector in the
true sense of the word. It then follows, however, from the Law of
Biot and Savart that \Emph{$\vH$~is not a vector but a linear tensor of
the second order}. Let its components in any co-ordinate system
(Cartesian or even merely affine) be~$H_{ik}$. The vector potential~$\vf$ is
a true vector. If $\phi_{i}$~are its co-variant components and $s^{i}$~the
contra-variant components of the current-density (the current is
like velocity fundamentally a contra-variant vector), the following
table gives us the final form (independent of the dimensional
number) of \Emph{the laws which hold in the magnetic field produced
by a stationary electric current}.
\PageSep{75}
\begin{gather*}
\frac{\dd H_{kl}}{\dd x_{i}} +
\frac{\dd H_{li}}{\dd x_{k}} +
\frac{\dd H_{ik}}{\dd x_{l}} = 0\Add{,}
\Tag{\Chg{(62, I)}{(62_{1})}}\displaybreak[0] \\
H_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
\quad\text{respectively} \\
\frac{\dd H_{ik}}{\dd x_{k}} = s^{i}\Add{.}
\Tag{\Chg{(62, II)}{(62_{2})}}
\end{gather*}
The stresses are determined by:
\index{Field action of electricity!magnetic@{of magnetic}}%
\index{Induction, magnetic}%
\index{Maxwell's!stresses}%
\index{Stresses!Maxwell's}%
\[
S_{i}^{k} = H_{ir} H^{kr} - \tfrac{1}{2} \delta_{i}^{k}|H|^{2}
\Tag{\Chg{(62, III)}{(62_{3})}}
\]
in which $|H|$~signifies the strength of the magnetic field:
\index{Magnetic!induction}%
\index{Magnetic!intensity of field}%
\index{Magnetic!permeability}%
\index{Permeability, magnetic}%
\[
|H|^{2} = H_{ik} H^{ik}.
\]
The stress tensor is symmetrical, since
\[
H_{ir} H_{k}^{r} = H_{i}^{r} H_{kr} = g^{rs} H_{ir} H_{ks}.
\]
The components of the force-density are
\[
p_{i} = H_{k}^{i} s^{k}\Add{.}
\Tag{\Chg{(62, IV)}{(62_{4})}}
\]
The energy-density $= \frac{1}{2}|H|^{2}$.
These are the laws that hold for the field in empty space. We
regard them as being exact physical laws which are generally valid,
as in the case of electricity. For a phenomenological theory it is,
however, necessary to take into consideration the \Emph{magnetisation},
\index{Magnetisation}%
a phenomenon analogous to dielectric polarisation. Just as $\vD$~occurred
in conjunction with~$\vE$, so the ``magnetic induction''~$\vB$
associates itself with the intensity of field~$\vH$. The laws
\[
\curl \vH = \vs,\qquad
\div \vB = 0
\]
hold in the field, as does the law which takes account of the
magnetic character of the matter
\[
\vB = \mu \vH\Add{.}
\Tag{(63)}
\]
The constant~$\mu$ is called magnetic permeability. But whereas the
single atom only becomes polarised by the action of the intensity
of the electrical field (i.e.\ becomes a doublet), (this takes place
in the direction of the field intensity), the atom is from the outset
an elementary magnet owing to the presence of rotating electrons
in it (at least, in the case of para- and ferro-magnetic substances).
All these elementary magnets, however, neutralise one another's
effects, as long as they are irregularly arranged and all positions
of the electronic orbits occur equally frequently on the average.
The imposed magnetic force merely fulfils the function of \Emph{directing}
the existing doublets. It evidently is due to this fact that the
range within which \Eq{(63)}~holds is much less than the corresponding
\PageSep{76}
range of~\Eq{(63)}. Permanent magnets and ferro-magnetic bodies
(iron, cobalt, nickel) are, above all, not subject to it.
In the phenomenological theory there must be added to the
laws already mentioned that of \Emph{Ohm}:\Pagelabel{76}
\[
\vs = \sigma \vE\qquad
(\sigma = \text{conductivity}).
\index{Conductivity}%
\]
It asserts that the current follows the fall of potential and is
proportional to it for a given conductor. Corresponding to Ohm's
Law we have in the atomic theory the fundamental law of mechanics,
according to which the motion of the ``free'' electrons is determined
by the electric and magnetic forces acting on them which thus
produce an electric current. Owing to collisions with the molecules
no permanent acceleration can come about, but (just as in the case
of a heavy body which is falling and experiences the resistance of
the air) a mean limiting velocity is reached, which may, to a first
approximation at least, be put proportional to the driving electric
force~$\vE$. In this way Ohm's Law acquires a meaning.
\index{Ohm's Law}%
If the current is produced by a voltaic cell or an accumulator,
\index{Electromotive force}%
the chemical action which takes place maintains a constant difference
of potential, the ``\Chg{electro-motive}{electromotive} force,'' between the two
ends of the conducting wire. Since the events which occur in the
contrivance producing the current can obviously be understood
only in the light of an atomic theory, it leads to the simplest result
phenomenologically to represent it by means of a cross-section
taken through the conducting circuit at each end, beyond which
the potential makes a sudden jump equal to the electromotive
force.
This brief survey of Maxwell's theory of stationary fields will
suffice for what follows. We have not the space here to enlarge
upon details and concrete applications.
\PageSep{77}
\Chapter{II}
{The Metrical Continuum}
\Section[Note on Non-Euclidean Geometry]
{10.}{Note on Non-Euclidean Geometry\protect\footnotemark}
\index{Asymptotic straight line}%
\index{Non-Euclidean!geometry}%
\footnotetext{\Chg{Note 1.}{\textit{Vide} \FNote{1}.}}
\First{Doubts} as to the validity of Euclidean geometry seem to
have been raised even at the time of its origin, and are not,
as our philosophers usually assume, outgrowths of the
hypercritical tendency of modern mathematicians. These doubts
have from the outset hovered round the fifth postulate. The substance
of the latter is that in a plane containing a given straight
line~$g$ and a point~$P$ external to the latter (but in the plane) there
is only one straight line through~$P$ which does not intersect~$g$: it
is called the straight line parallel to~$P$. Whereas the remaining
axioms of Euclid are accepted as being self-evident, even the
earliest exponents of Euclid have endeavoured to prove this
theorem from the remaining axioms. Nowadays, knowing that
this object is unattainable, we must look upon these reflections
and efforts as the beginning of ``non-Euclidean'' geometry, i.e.\ of
the construction of a geometrical system which can be developed
logically by accepting all the axioms of Euclid, except the postulate
of parallels. A report of Proclus (\AD.~5) about these attempts
has been handed down to posterity. Proclus utters an emphatic
warning against the abuse that may be practised by calling propositions
self-evident. This warning cannot be repeated too often;
on the other hand, we must not fail to emphasise the fact that, in
spite of the frequency with which this property is wrongfully used,
the ``self-evident'' property is the final root of all knowledge, including
empirical knowledge. Proclus insists that ``asymptotic
lines'' may exist.
We may picture this as follows. Suppose a straight line~$g$ be
given in a plane, also a point~$P$ outside it in the plane, and a
straight line~$s$ passing through~$P$ and which may be rotated about~$P$.
\Figure{2}
Let $s$~be perpendicular to~$\Typo{P}{g}$ initially. If we now rotate~$s$, the
point of intersection of $s$~and~$g$ glides along~$g$, e.g.\ to the right, and
if we continue turning, a definite moment arrives at which this
point of intersection just vanishes to infinity; $s$~then occupies the
\PageSep{78}
\index{Asymptotic straight line}%
position of an ``asymptotic'' straight line. If we continue turning,
Euclid assumes that, at even this same moment, a point of intersection
already appears on the left. Proclus, on the other hand,
points out the possibility that one may perhaps have to turn~$s$
through a further definite angle before a point of intersection arises
to the left. We should then have two ``asymptotic'' straight lines,
one to the right, viz.~$s'$, and the other to the left, viz.~$s''$. If the
straight line~$s$ through~$P$ were then situated in the angular space
between $s''$ and~$s'$ (during the rotation just described) it would cut~$g$;
if it lay between $s'$ and~$s''$, it would \Emph{not} intersect~$g$. There must
be at least \Emph{one} non-intersecting straight line; this follows from the
other axioms of Euclid. I shall recall a familiar figure of our early
studies in plane geometry, consisting of the straight line~$h$ and two
straight lines $g$ and~$g'$ which intersect~$h$ at $A$~and~$A'$ and make
equal angles with it, $g$~and $g'$ are each divided into a right and a
left half by their point of intersection with~$h$. Now, if $g$~and~$g'$
had a common point~$s$ to the right of~$h$, then, since $BAA'B'$~is congruent
\Figure{3}
with $C'A'AC$ (\textit{vide} \Fig{3}), there would also be a point of
intersection~$S^{*}$ to the left of~$h$. But this is impossible since there
is only one straight line that passes through two given points
$S$~and~$S^{*}$.
Attempts to prove Euclid's postulate were continued by Arabian
\index{Parallels, postulate of}%
and western mathematicians of the Middle Ages. Passing straight
to a more recent period we shall mention the names of only the
last eminent forerunners of non-Euclidean geometry, viz.\ the Jesuit
father Saccheri (beginning of the eighteenth century) and the
mathematicians Lambert and Legendre. Saccheri was aware that
the question whether the postulate of parallels is valid is equivalent
to the question whether the sum of the angles of a triangle are
equal to or less than~$180°$. If they amount to~$180°$ in \Emph{one} triangle,
then they must do so in every triangle and Euclidean geometry holds.
If the sum is $< 180°$ in one triangle then it is $< 180°$ in every
triangle. That they cannot be $> 180°$ is excluded for the same
reason for which we just now concluded that not all the straight
lines through~$P$ can cut the fixed straight line~$g$. Lambert discovered
\PageSep{79}
\index{Bolyai's geometry}%
\index{Lobatschefsky's geometry}%
that if we assume the sum of the three angles to be $< 180°$
there must be a unique length in geometry. This is closely related
\index{Geometry!non-Euclidean (Bolyai-Lobatschefsky)}%
to an observation which Wallis had previously made that there can
be no similar figures of different sizes in non-Euclidean geometry
(just as in the case of the geometry of the surface of a rigid sphere).
Hence if there is such a thing as ``form'' independent of size,
Euclidean geometry is justified in its claims. Lambert, moreover,
deduced a formula for the area of a triangle, from which it is clear
that, in the case of non-Euclidean geometry, this area cannot increase
beyond all limits. It appears that the researches of these
men has gradually spread the belief in wide circles that the postulate
of parallels cannot be proved. At that time this problem
occupied many minds. D'Alembert pronounced it a scandal of
geometry that it had not yet been decisively settled. Even the
authority of Kant, whose philosophic system claims Euclidean
geometry as \textit{a~priori} knowledge representing the content of pure
space-intuition in adequate judgments, did not succeed in settling
these doubts permanently.
Gauss also set out originally to prove the axiom of parallels, but
he early gained the conviction that this was impossible and thereupon
developed the principles of a non-Euclidean geometry, for
which the axioms of parallels does not hold, to such an extent that,
from it, the further development could be carried out with the
same ease as for Euclidean geometry. He did not make his investigations
known for, as he later wrote in a private letter, he
feared ``the outcry of the B\oe{}otians''; for, he said, there were only
a few people who understood what was the true essence of these
questions. Independently of Gauss, Schweikart, a professor of
jurisprudence, gained a full insight into the conditions of non-Euclidean
geometry, as is evident from a concise note addressed to
Gauss. Like the latter he considered it in no wise self-evident, and
established that Euclidean geometry is valid in our actual space.
His nephew Taurinus whom he encouraged to study these questions
was, in contrast to him, a believer of Euclidean geometry, but we
are nevertheless indebted to Taurinus for the discovery of the fact
that the formulæ of spherical trigonometry are real on a sphere
which has an imaginary radius $= \sqrt{-1}$, and that through them a
geometrical system is constructed along analytical lines which
satisfies all the axioms of Euclid except the fifth postulate.
For the general public the honour of discovering and elaborating
%[** TN: "Lobatschefsky" elsewhere; retaining original text]
non-Euclidean geometry must be shared between Nikolaj
Iwanowitsch Lobatschefskij (1793--1856), a Russian professor of
mathematics at Kasan, and Johann Bolyai (1802--1860), a
\PageSep{80}
\index{Bolyai's geometry}%
\index{Klein's model}%
\index{Lobatschefsky's geometry}%
Hungarian officer in the Austrian army. The ideas of both
assumed a tangible form in~1826. The chief manuscript of both,
by which the public were informed of their discovery and which
offered an argument of the new geometry in the manner of Euclid,
\index{Geometry!non-Euclidean (Bolyai-Lobatschefsky)}%
had its origin in 1830--1831. The discussion by Bolyai is particularly
clear, inasmuch as he carries the argument as far as
possible without making an assumption as to the validity or non-validity
of the fifth postulate, and only afterwards derives the
theorems of Euclidean and non-Euclidean geometry from the
\index{Non-Euclidean!plane!(Klein's model)}%
theorems of his ``absolute'' geometry according to whether one
decides in favour of or against Euclid.
Although the structure was thus erected, it was by no means
definitely decided whether, in absolute geometry, the axiom of
parallels would not after all be shown to be a dependent theorem.
The strict proof that \Emph{non-Euclidean geometry is absolutely
consistent in itself} had yet to follow. This resulted almost of
itself in the further development of non-Euclidean geometry. As
often happens, the simplest way of proving this was not discovered
at once. It was discovered by Klein as late as~1870 and depends
on the construction of a \Emph{Euclidean model} for non-Euclidean
geometry (\Chg{\textit{v.}\ Note~2}{\textit{vide} \FNote{2}}). Let us confine our attention to the plane!
\index{Plane!(non-Euclidean)}%
In a Euclidean plane with rectangular co-ordinates $x$~and~$y$ we
shall draw a circle~$U$ of radius unity with the origin as centre.
Introducing homogeneous co-ordinates
\[
x = \frac{x_{1}}{x_{3}},\qquad
y = \frac{x_{2}}{x_{3}}
\]
(so that the position of a point is defined by the ratio of three
numbers, i.e.\ $x_{1}: x_{2}: x_{3}$), the equation to the circle becomes
\[
-x_{1}^{2} - x_{2}^{2} + x_{3}^{2} = 0.
\]
Let us denote the quadratic form on the left by~$\Omega(x)$ and the corresponding
symmetrical bilinear form of two systems of value,
$x_{i}\Com x_{i}'$ by~$\Omega(x\Com x')$. A transformation which assigns to every point~$x$
a transformed point~$x'$ according to the linear formulæ
\[
x_{i}' = \sum_{k=1}^{3} \Chg{\alpha_{ik}}{\alpha_{i}^{k}} x_{k}\qquad
(|\Chg{\alpha_{ik}}{\alpha_{i}^{k}}| \neq 0)
\]
is called, as we know, a collineation (affine transformations are a
special class of collineations). It transforms every straight line,
point for point, into another straight line and leaves the cross-ratio
of four points on a straight line unaltered. We shall now set up a
little dictionary by which we translate the conceptions of Euclidean
\PageSep{81}
geometry into a new language, that of non-Euclidean geometry;
we use inverted commas to distinguish its words. The vocabulary
of this dictionary is composed of only three words.
The word ``point'' is applied to any point on the inside of~$U$
(\Fig{4}).
A ``straight line'' signifies the portion of a straight line lying
wholly in~$U$. The collineations which transform the circle~$U$ into
itself are of two kinds; the first leaves
\Figure{4}
the sense in which $U$~is described
unaltered, whereas the second reverses
it. The former are called ``congruent''
\index{Congruent}%
transformations; two figures
composed of points are called ``congruent''
if they can be transformed
into one another by such a transformation.
All the axioms of Euclid except
the postulate of parallels hold for
these ``points,'' ``straight lines,'' and
the conception ``congruence''. A
whole sheaf of ``straight lines'' passing through the ``point''~$P$
which do not cut the one ``straight line''~$g$ is shown in \Fig{4}.
This suffices to prove the consistency of non-Euclidean geometry,
for things and relations are shown for which all the theorems
of Euclidean geometry are valid provided that the appropriate
nomenclature be adopted. It is evident, without further explanation,
that Klein's model is also applicable to spatial geometry.
We now determine the non-Euclidean distance between two
``points'' in this model, viz.\ between
\[
A = (x_{1}: x_{2}: x_{3})
\text{ and }
A' = (x_{1}': x_{2}': x_{3}').
\]
Let the straight line~$AA'$ cut the circle~$U$ in the two points, $B_{1}$,~$B_{2}$.
The homogeneous co-ordinates~$y_{i}$ of these two points are of
the form
\[
y_{i} = \lambda x_{i} + \lambda' x_{i}'
\]
and the corresponding ratio of the parameters, $\lambda: \lambda'$, is given by
the equation $\Omega(y) = 0$, viz.\
\[
\frac{\lambda}{\lambda'}
= \frac{-\Omega(x\Com x') ± \sqrt{\Omega^{2}(x\Com x') - \Omega(x)\Omega(x')}}{\Omega(x)}.
\]
Hence the cross-ratio of the four points, $A\Com A'\Com B_{1}\Com B_{2}$ is
\[
[AA']
= \frac{\Omega(x\Com x') + \sqrt{\Omega^{2}(x\Com x') - \Omega(x)\Omega(x')}}
{\Omega(x\Com x') - \sqrt{\Omega^{2}(x\Com x') - \Omega(x)\Omega(x')}}.
\]
\PageSep{82}
This quantity which depends on the two arbitrary ``points,'' $A\Com A'$,
is not altered by a ``congruent'' transformation. If $A\Com A'\Com A''$ are
any three ``points'' lying on a ``straight line'' in the order
written, then
\[
[AA''] = [AA'] · [A'A''].
\]
The quantity
\[
\tfrac{1}{2} \log [AA'] = \Bar{AA'} = r
\]
has thus the functional property
\[
\Bar{AA'} + \Bar{A'A''} = \Bar{AA''}.
\]
As it has the same value for ``congruent'' distances~$AA'$ too, we
must regard it as the non-Euclidean distance between the two
points, $A\Com A'$. Assuming the logs to be taken to the base~$e$, we get
an absolute determination for the unit of measure, as was recognised
by Lambert. The definition may be written in the shorter
form:
\begin{gather*}
\cosh r = \frac{\Omega(x\Com x')}{\sqrt{\Omega(x) · \Omega(x')}}
\Tag{(1)} \\
\text{(cosh denotes the hyperbolic cosine).}
\end{gather*}
This measure-determination had already been enunciated before
Klein by Cayley\footnote
{\textit{Vide} \FNote{3}.}
who referred it to an arbitrary real or imaginary
conic section $\Omega(x) = 0$: he called it the ``projective measure-determination''.
But it was reserved for Klein to recognise that
in the case of a real conic it leads to non-Euclidean geometry.
It must not be thought that Klein's model shows that the non-Euclidean
plane is finite. On the contrary, using non-Euclidean
\index{Plane!(Klein's model)}%
measures I can mark off the same distance on a ``straight line''
an infinite number of times in succession. It is only by using
\Emph{Euclidean} measures in the \Emph{Euclidean} model that the distances
of these ``\Chg{equi-distant}{equidistant}'' points becomes smaller and smaller. For
non-Euclidean geometry the bounding circle~$U$ represents unattainable,
infinitely distant, regions.
If we use an imaginary conic, Cayley's measure-determination
\index{Cayley's measure-determination}%
leads to ordinary spherical geometry, such as holds on the surface
of a sphere in Euclidean \Erratum{geometry}{space}. Great circles take the place
of straight lines in it, but every pair of points at the end of the
same diameter must be regarded as a single ``point,'' in order that
two ``straight lines'' may only intersect at one ``point''. Let us
project the points on the sphere by means of (straight) rays from
the centre on to the tangential plane at a point on the surface of
the sphere, e.g.\ the south pole. Two diametrically opposite points
will then coincide on the tangential plane as a result of the transformation.
\PageSep{83}
We must, in addition, as in projective geometry, furnish
this plane with an infinitely distant straight line; this is given by
the projection of the equator. We shall now call two figures in this
plane ``congruent'' if their projections (through the centre) on to
the surface of the sphere are congruent in the ordinary Euclidean
sense. Provided this conception of ``congruence'' is used, a non-Euclidean
geometry, in which all the axioms of Euclid except the
fifth postulate are fulfilled, holds in this plane. Instead of this
postulate we have the fact that each pair of straight lines, without
exception, intersects, and, in accordance with this, the sum of the
angles in a triangle $> 180°$. This seems to conflict with the
Euclidean proof quoted above. The apparent contradiction is explained
by the circumstance that in the present ``spherical'' geometry
\index{Spherical!geometry}%
the straight line is closed, whereas Euclid, although he does not
explicitly state it in his axioms, tacitly assumes that it is an open
line, i.e.\ that each of its points divides it into two parts. The
deduction that the hypothetical point of intersection~$S$ on the
``right-hand'' side is different from that~$S^{*}$ on the ``left-hand''
side is rigorously true only if this ``openness'' be assumed.
Let us mark out in space a Cartesian co-ordinate system
$x_{1}$,~$x_{2}$,~$x_{3}$, having its origin at the centre of the sphere and the line
connecting the north and south poles as its $x_{3}$~axis, the radius of
the sphere being the unit of length. If $x_{1}$,~$x_{2}$,~$x_{3}$ are the co-ordinates
of any point on the sphere, i.e.\
\[
\Omega(x) \equiv x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 1
\]
then $\dfrac{x_{1}}{x_{3}}$~and~$\dfrac{x_{2}}{x_{3}}$ are respectively the first and second co-ordinate of
the transformed point in our plane $x_{3} = 1$, i.e.\ $x_{1}: x_{2}: x_{}3$ is the
ratio of the homogeneous co-ordinates of the transformed point.
Congruent transformations of the sphere are linear transformations
which leave the quadratic form~$\Omega(x)$ invariant. The ``congruent''
transformations of the plane in terms of our ``spherical'' geometry
are thus given by such linear transformations of the homogeneous
co-ordinates as convert the equation $\Omega(x) = 0$, which signifies an
imaginary conic, into itself. This proves the statement made
above concerning the relationship between spherical geometry and
Cayley's measure-relation. This agreement is expressed in the
formula for the distance~$r$ between two points $A$,~$A'$, which is here
\[
\cos r = \frac{\Omega(x\Com x')}{\sqrt{\Omega(x) \Omega(x')}}\Add{.}
\Tag{(2)}
\]
At the same time we have confirmed the discovery of Taurinus
\PageSep{84}
that Euclidean geometry is identical with non-Euclidean geometry
\index{Geometry!Riemann's}%
on a sphere of radius~$\sqrt{-1}$.
Euclidean geometry occupies an intermediate position between
that of Bolyai-Lobatschefsky and spherical geometry. For if we
make a real conic section change to a degenerate one, and thence
to an imaginary one, we find that the plane with its corresponding
Cayley measure-relation is at first Bolyai-Lobatschefskyan, then
Euclidean, and finally spherical.
\Section{11.}{The Geometry of Riemann}
\index{Continuum}%
\index{Riemann's!geometry}%
The next stage in the development of non-Euclidean geometry
that concerns us chiefly is that due to Riemann. It links up with
the foundations of Differential Geometry, in particular with that
of the theory of surfaces as set out by Gauss in his \Title{Disquisitiones
circa superficies curvas}.
\Emph{The most fundamental property of space is that its
points form a three-dimensional manifold.} What does this
convey to us? We say, for example, that ellipses form a two-dimensional
manifold (as regards their size and form, i.e.\ considering
congruent ellipses similar, non-congruent ellipses as
dissimilar), because each separate ellipse may be distinguished in
the manifold by two given numbers, the lengths of the semi-major
and semi-minor axis. The difference in the conditions of equilibrium
of an ideal gas which is given by two independent variables, such
as pressure and temperature, form a two-dimensional manifold,
likewise the points on a sphere, or the system of pure tones (in
terms of intensity and pitch). According to the physiological
theory which states that the sensation of colour is determined by
the combination of three chemical processes taking place on the
retina (the black-white, red-green, and the yellow-blue process,
each of which can take place in a definite direction with a definite
intensity), colours form a three-dimensional manifold with respect
to quality and intensity, but colour qualities form only a two-dimensional
manifold. This is confirmed by Maxwell's familiar
construction of the colour triangle. The possible positions of a
rigid body form a six-dimensional manifold, the possible positions
of a mechanical system having $n$~degrees of freedom constitute,
in general, an $n$-dimensional manifold. \Emph{The characteristic of
an $n$-dimensional manifold is that each of the elements
composing it} (in our examples, single points, conditions of a gas,
colours, tones) \Emph{may be specified by the giving of $n$~quantities,
the ``co-ordinates,'' which are continuous functions within
the manifold.} This does not mean that the whole manifold with
\PageSep{85}
\index{Continuum}%
all its elements must be represented in a single and reversible
manner by value systems of $n$~co-ordinates (e.g.\ this is impossible
in the case of the sphere, for which $n = 2$); it signifies only that
if $P$~is an arbitrary element of the manifold, then in every case
a certain domain surrounding the point~$P$ must be representable
singly and reversibly by the value system of $n$~co-ordinates. If $x_{i}$~is
a system of $n$~co-ordinates, $x_{i}'$~another system of $n$~co-ordinates,
then the co-ordinate values $x_{i}$,~$x_{i}'$ of the same element will in
general be connected with one another by relations
\[
x_{i} = f_{i}(x_{1}', x_{2}', \dots\Add{,} x_{n}')\qquad
(i = 1, 2, \dots\Add{,} n)
\Tag{(3)}
\]
which can be resolved into terms of~$x_{i}'$ and in which the~$f_{i}$'s are
continuous functions of their arguments. As long as nothing more
is known about the manifold, we cannot distinguish any one co-ordinate
system from the others. For an analytical treatment of
arbitrary continuous manifolds we thus require a theory of invariance
with regard to arbitrary transformation of co-ordinates,
such as~\Eq{(3)}, whereas for the development of affine geometry in the
preceding chapter we used only the much more special theory of
invariance for the case of \Emph{linear} transformations.
Differential geometry deals with curves and surfaces in three-dimensional
Euclidean space; we shall here consider them mapped
out in Cartesian co-ordinates $x$,~$y$,~$z$. A \Emph{curve} is in general a one-dimensional
\index{Curve}%
point-manifold; its separate points can be distinguished
from one another by the values of a parameter~$u$. If the point~$u$
on the curve happens to be at the point $x$,~$y$,~$z$ in space, then $x$,~$y$,~$z$
will be certain continuous functions of~$u$:
\[
x = x(u),\qquad
y = y(u),\qquad
z = z(u)
\Tag{(4)}
\]
and \Eq{(4)}~is called the ``parametric'' representation of the curve. If
we interpret~$u$ as the time, then \Eq{(4)}~is the law of motion of a point
which traverses the given curve. The curve itself does not, however,
determine singly the parametric representation~\Eq{(4)} of the
curve; the parameter~$u$ may, indeed, be subjected to any arbitrary
continuous transformation.
A two-dimensional point-manifold is called a \Emph{surface}. Its
\index{Surface}%
points can be distinguished from one another by the values of two
parameters $u_{1}$,~$u_{2}$. It may therefore be represented parametrically
in the form
\[
x = x(u_{1}, u_{2}),\qquad
y = y(u_{1}, u_{2}),\qquad
z = z(u_{1}, u_{2})\Add{.}
\Tag{(5)}
\]
The parameters $u_{1}$,~$u_{2}$ may likewise undergo any arbitrary continuous
transformation without affecting the represented curve.
We shall assume that the functions~\Eq{(5)} are not only continuous
\PageSep{86}
\index{Co-ordinates, curvilinear!Gaussian@{(or Gaussian)}}%
but have also continuous differential co-efficients. Gauss, in his
general theory, starts from the form~\Eq{(5)} of representing any
surface; the parameters $u_{1}$,~$u_{2}$ are hence called the Gaussian (or
curvilinear) co-ordinates on the surface. For example, if, as in
the preceding section, we project the points of the surface of the
unit sphere in a small region encircling the origin of the co-ordinate
system on to the tangent plane $z = 1$ at the south pole, and if we
make $x$,~$y$,~$z$ the co-ordinates of any arbitrary point on the sphere,
$u_{1}$~and~$u_{2}$ being respectively the $x$~and $y$ co-ordinates of the point
of projection in this plane, then
\[
x = \frac{u_{1}}{\sqrt{1 + u_{1}^{2} + u_{2}^{2}}}\Add{,}\
y = \frac{u_{2}}{\sqrt{1 + u_{1}^{2} + u_{2}^{2}}}\Add{,}\
z = \frac{1}{\sqrt{1 + u_{1}^{2} + u_{2}^{2}}}\Add{.}
\Tag{(6)}
\]
This is a parametric representation of the sphere. It does not,
however, embrace the whole sphere, but only a certain region
round the south pole, viz.\ the part from the south pole to the
equator, \Erratum{including}{excluding} the latter. Another illustration of a parametric
representation is given by the geographical co-ordinates, latitude
and longitude.
In thermodynamics we use a graphical representation consisting
of a plane on which two rectangular co-ordinate axes are drawn,
and in which the state of a gas as denoted by its pressure~$p$ and
temperature~$\theta$ is represented by a point having the rectangular
co-ordinates $p$,~$\theta$. The same procedure may be adopted here.
With the point $u_{1}$,~$u_{2}$ on the surface, we associate a point in the
``representative'' plane having the rectangular co-ordinates $u_{1}$,~$u_{2}$.
The formulæ~\Eq{(5)} do not then represent only the surface, but also at
the same time a definite continuous \Emph{representation} of this surface
on the $u_{1}$,~$u_{2}$ plane. Geographical maps are familiar instances of
such representations of curved portions of surface by means of
planes. A curve on a surface is given mathematically by a parametric
representation
\[
u_{1} = u_{1}(t),\qquad
u_{2} = u_{2}(t)\Add{,}
\Tag{(7)}
\]
whereas a portion of a surface is given by a ``mathematical region''
expressed in the variables $u_{1}$,~$u_{2}$, and which must be characterised
by inequalities involving $u_{1}$~and~$u_{2}$; i.e.\ graphically by means of
the representative curve or the representative region in the $u_{1}$-$u_{2}$-plane.
If the representative plane be marked out with a network
of co-ordinates in the manner of squared paper, then this becomes
transposed, through the representation, to the curved surface as a
net consisting of meshes having the form of little parallelograms,
and composed of the two families of ``co-ordinate lines'' $u_{1} = \text{const.}$,
$u_{2} = \text{const.}$, respectively. If the meshes be made sufficiently fine
\PageSep{87}
it becomes possible to map out any given figure of the representative
plane on the curved surface.
The distance~$ds$ between two infinitely near points of the surface,
namely,
\[
(u_{1}, u_{2})
\quad\text{and}\quad
(u_{1} + du_{1}, u_{2} + du_{2})
\]
is determined by the expression
\[
ds^{2} = dx^{2} + dy^{2} + dz^{2}
\]
if we set
\[
dx = \frac{\dd x}{\dd u_{1}}\, du_{1} + \frac{\dd x}{\dd u_{2}}\, du_{2}
\Tag{(8)}
\]
in it, with corresponding expressions for $dy$~and~$dz$. We then get
a quadratic differential form for~$ds^{2}$ thus:
\[
ds^{2} = \sum_{i,k=1}^{2} g_{ik}\, du_{i}\, du_{k}\qquad
(g_{ki} = g_{ik})
\Tag{(9)}
\]
in which the co-efficients are
\[
g_{ik}
= \frac{\dd x}{\dd u_{i}}\, \frac{\dd x}{\dd u_{k}}
+ \frac{\dd y}{\dd u_{i}}\, \frac{\dd y}{\dd u_{k}}
+ \frac{\dd z}{\dd u_{i}}\, \frac{\dd z}{\dd u_{k}}
\]
and are not, in general, functions of $u_{1}$~and~$u_{2}$.
In the case of the parametric representation of the sphere~\Eq{(6)} we
have
\[
ds^{2} = \frac{(1 + u_{1}^{2} + u_{2}^{2}) (du_{1}^{2} + du_{2}^{2}) - (u_{1}\, du_{1} + u_{2}\, du_{2})^{2}}
{(1 + u_{1}^{2} + u_{2}^{2})^{2}}\Add{.}
\Tag{(10)}
\]
Gauss was the first to recognise that the metrical groundform is
the determining factor for \Emph{geometry on surfaces}. The lengths of
\index{Geometry!surface@{on a surface}}%
curves, angles, and the size of given regions on the surface depend
on it alone. The geometries on two different surfaces is accordingly
identical if, for a representation in appropriate parameters,
the co-efficients~$g_{ik}$ of the metrical groundform coincide in value.
\Proof.---The length of any arbitrary curve, given by~\Eq{(7)}, on the
surface is furnished by the integral
\[
\int ds
= \int \sqrt{\sum_{i\Com k} g_{ik}\, \frac{du_{i}}{dt}\, \frac{du_{k}}{dt}} · dt.
\]
If we fix our attention on a definite point $P^{0} = (u_{1}^{0}, u_{2}^{0})$ on the
surface and use the relative co-ordinates
\[
u_{i} - u_{i}^{0} = du_{i}\Add{,}\qquad
x - x^{0} = dx\Add{,}\qquad
y - y^{0} = dy\Add{,}\qquad
z - z^{0} = dz
\]
for its immediate neighbourhood, then equation~\Eq{(8)}, in which the
derivatives are to be taken for the point~$P^{0}$, will hold more exactly
the smaller $du_{1}$,~$du_{2}$, are taken; we say that it holds for ``infinitely
\PageSep{88}
small'' values $du_{1}$~and~$du_{2}$. If we add to these the analogous
equations for $dy$ and~$dz$, then they express that the immediate
neighbourhood of~$P^{0}$ is a plane, and that $du_{1}$,~$du_{2}$ are affine co-ordinates
on it.\footnote
{We here assume that the determinants of the second order which can be
formed from the table of co-efficients of these equations,
\[
\left\lvert\begin{array}{@{}ccc@{}}
\dfrac{\dd x}{\dd u_{1}} & \dfrac{\dd y}{\dd u_{1}} & \dfrac{\dd z}{\dd u_{1}} \\
\dfrac{\dd x}{\dd u_{2}} & \dfrac{\dd y}{\dd u_{2}} & \dfrac{\dd z}{\dd u_{2}} \\
\end{array}
\right\rvert,
\]
do not all vanish. This condition is fulfilled for the regular points of the
surface, at which there is a tangent plane. The three determinants are identically
equal to~$0$, if, and only if, the surface degenerates to a curve, i.e.\ the
functions $x$,~$y$,~$z$ of $u_{1}$~and~$u_{2}$ actually depend only on one parameter, a
function of $u_{1}$~and~$u_{2}$.}
Accordingly we may apply the formulæ of affine
geometry to the region immediately adjacent to~$P^{0}$. For the angle~$\theta$
between two line-elements or infinitesimal displacements having
the components $du_{1}$,~$du_{2}$ and $\delta u_{1}$,~$\delta u_{2}$ respectively, we get
\[
\cos \theta = \frac{Q(d\Com \delta)}{\sqrt{Q(d\Com d) Q(\delta\Com \delta)}}
\]
in which $Q(d\Com \delta)$ stands for the symmetrical bilinear form
\[
\sum_{i\Com k} g_{ik}\, du_{i}\, \delta u_{k}
\text{ corresponding to~\Eq{(9)}.}
\]
The area of the infinitesimal parallelogram marked out by these
\index{Parallelogram}%
two displacements is found to be
\[
\sqrt{g} \left\lvert\begin{array}{@{}cc@{}}
du_{1} & du_{2} \\
\delta u_{1} & \delta u_{2} \\
\end{array}\right\rvert
\]
in which $g$~denotes the determinant of the~$g_{ik}$'s. The area of a
curved portion of surface is accordingly given by the integral
\[
\iint \sqrt{g}\, du_{1}\, du_{2}
\]
taken over the corresponding part of the representative plane.
This proves Gauss' statement. The values of the expressions
obtained are of course independent of the choice of parametric
representation. This invariance with respect to arbitrary transformations
of the parameters can easily be confirmed analytically.
All the geometric relations holding on the surface can be studied
on the representative plane. The geometry of this plane is the
same as that of the curved surface if we agree to accept the distance~$ds$
of two infinitely near points as expressed by~\Eq{(9)} and \Emph{not} by
Pythagoras' formula
\[
ds^{2} = du_{1}^{2} + du_{2}^{2}.
\]
\PageSep{89}
The geometry of the surface deals with the inner measure
relations of the surface that belong to it independently of the
manner in which it is embedded in space. They are the relations
that can be determined by \Emph{measurements carried out on the
surface itself}. Gauss in his investigation of the theory of surfaces
started from the practical task of surveying Hanover geodetically.
The fact that the earth is not a plane can be ascertained by
measuring a sufficiently large portion of the earth's surface. Even
if each single triangle of the network is taken too small for the
deviation from a plane to come into consideration, they cannot be
put together to form a closed net on a plane in the way they do on
the earth's surface. To show this a little more clearly let us draw
a circle~$C$ on a sphere of radius unity (the earth), having its centre~$P$
on the surface of the sphere. Let us further draw radii of this
circle, i.e.\ arcs of great circles of the sphere radiating from~$P$ and
%[** TN: Large parentheses in the original]
ending at the circumference of~$C$ (let these arcs be $< \dfrac{\pi}{2}$). By
carrying out measurements on the sphere's surface we can now
ascertain that these radii starting out in all directions are the
shortest lines connecting~$P$ to the circle~$C$, and that they are all of
the same length~$r$; by measurement we find the closed curve~$C$ to
be of length~$s$. If we were dealing with a plane we should infer
from this that the ``radii'' are straight lines and hence the curve~$C$
would be a circle and we should expect $s$ to be equal to~$2\pi r$.
Instead of this, however, we find that $s$~is less than the value given
by the above formula, for in the actual case $s = 2\pi \sin r$. We
thus discover by measurements carried out on the surface of the
sphere that this surface is not a plane. If, on the other hand, we
draw figures on a sheet of paper and then roll it up, we shall find
the same values for measurements of these figures in their new
condition as before, provided that no distortion has occurred through
rolling up the paper. The same geometry will hold on it now as
on the plane. It is impossible for me to ascertain that it is curved
by carrying out geodetic measurements. Thus, in general, the
same geometry holds for two surfaces that can be transformed into
one another without distortion or tearing.
The fact that plane geometry does not hold on the sphere means
analytically that it is impossible to convert the quadratic differential
form~\Eq{(10)} by means of a transformation
%[** TN: Omitted vertical bar between equation pairs]
\begin{align*}
u_{1} &= u_{1}(u_{1}'\Com u_{2}') & u_{1}' &= u_{1}'(u_{1}\Com u_{2}) \\
u_{2} &= u_{2}(u_{1}'\Com u_{2}') & u_{2}' &= u_{2}'(u_{1}\Com u_{2})
\end{align*}
into the form
\[
(du_{1}')^{2} + (du_{2}')^{2}.
\]
\PageSep{90}
We know, indeed, that it is possible to do this for each point by a
linear transformation of the differentials, viz.\ by
\[
du_{i}' = \alpha_{i1}\, du_{1} + \alpha_{i2}\, du_{2}\qquad
(i = 1, 2)\Add{,}
\Tag{(11)}
\]
but it is impossible to choose the transformation of the differentials
at each point so that the expressions~\Eq{(11)} become \Emph{total} differentials
for $du_{1}'$,~$du_{2}'$.
Curvilinear co-ordinates are used not only in the theory of
surfaces but also in the treatment of space problems, particularly in
mathematical physics in which it is often necessary to adapt the
co-ordinate system to the bodies presented, as is instanced in the
case of cylindrical, spherical, and elliptic co-ordinates. The square
of the distance,~$ds^{2}$, between two infinitely near points in space, is
always expressed by a quadratic form
\[
\sum_{i,k=1}^{3} g_{ik}\, dx_{i}\, dx_{k}
\Tag{(12)}
\]
in which $x_{1}$,~$x_{2}$,~$x_{3}$ are any arbitrary co-ordinates. If we uphold
Euclidean geometry, we express the belief that this quadratic form
can be brought by means of some transformation into one which
has constant co-efficients.
These introductory remarks enable us to grasp the full meaning
of the ideas developed fully by Riemann in his inaugural address,
``Concerning the Hypotheses which lie at the Base of Geometry''.\footnote
{\textit{Vide} \FNote{4}.}
It is evident from Chapter~I that Euclidean geometry holds for a
three-dimensional \Emph{linear} point-configuration in a four-dimensional
Euclidean space; but curved three-dimensional spaces, which exist
in four-dimensional space just as much as curved surfaces occur in
three-dimensional space, are of a different type. Is it not possible
that our three-dimensional space of ordinary experience is curved?
Certainly. It is not embedded in a four-dimensional space; but it
is conceivable that its inner measure-relations are such as cannot
occur in a ``plane'' space; it is conceivable that a very careful
geodetic survey of our space carried out in the same way as the
above-mentioned survey of the earth's surface might disclose that it
is not plane. We shall continue to regard it as a three-dimensional
manifold, and to suppose that infinitesimal line elements may be
compared with one another in respect to length independently of
their position and direction, and that the square of their lengths,
the distance between two infinitely near points, may be expressed
by a quadratic form~\Eq{(12)}, any arbitrary co-ordinates~$x_{i}$ being used.
(There is a very good reason for this assumption; for, since every
transformation from one co-ordinate system to another entails
\PageSep{91}
\Emph{linear} transformation-formulæ for the co-ordinate differentials, a
quadratic form must always again pass into a quadratic form as a
result of the transformation.) We no longer assume, however,
that these co-ordinates may in particular be chosen as affine co-ordinates
such that they make the co-efficients~$g_{ik}$ of the groundform
become constant.
The transition from Euclidean geometry to that of Riemann is
founded in principle on the same idea as that which led from
physics based on action at a distance to physics based on infinitely
near action. We find by observation, for example, that the current
flowing along a conducting wire is proportional to the difference of
potential between the ends of the wire (Ohm's Law). But we are
firmly convinced that this result of measurement applied to a long
wire does not represent a physical law in its most general form;
we accordingly deduce this law by reducing the measurements obtained
to an infinitely small portion of wire. By this means we
arrive at the expression (Chap.~I, \Pageref[p.]{76}) on which Maxwell's theory
is founded. Proceeding in the reverse direction, we derive from
this differential law by mathematical processes the integral law,
which we observe directly, on the supposition \Emph{that conditions are
everywhere similar} (homogeneity). We have the same circumstances
\index{Homogeneity!of space}%
here. The fundamental fact of Euclidean geometry is that
the square of the distance between two points is a quadratic form
of the relative co-ordinates of the two points (\emph{Pythagoras' Theorem}).
\index{Pythagoras' Theorem}%
\emph{But if we look upon this law as being strictly valid only for the
case when these two points are infinitely near, we enter the domain of
Riemann's geometry.} This at the same time allows us to dispense
with defining the co-ordinates more exactly since Pythagoras' Law
expressed in this form (i.e.\ for infinitesimal distances) is invariant
for arbitrary transformations. We pass from Euclidean ``finite''
geometry to Riemann's ``infinitesimal'' geometry in a manner
exactly analogous to that by which we pass from ``finite'' physics
to ``infinitesimal'' (or ``contact'') physics. Riemann's geometry
is Euclidean geometry formulated to meet the requirements of continuity,
and in virtue of this formulation it assumes a much more
general character. Euclidean finite geometry is the appropriate
instrument for investigating the straight line and the plane, and
the treatment of these problems directed its development. As
soon as we pass over to differential geometry, it becomes natural
and reasonable to start from the property of infinitesimals set out
by Riemann. This gives rise to no complications, and excludes
all speculative considerations tending to overstep the boundaries
of geometry. In Riemann's space, too, a surface, being a two-dimensional
\PageSep{92}
manifold, may be represented parametrically in the
form $x_{i} = x_{i}(u_{1}, u_{2})$. If we substitute the resulting differentials,
\[
\Typo{dx}{dx_{i}}
= \frac{\dd x_{i}}{\dd \Typo{u_{i}}{u_{1}}} · du_{1}
+ \frac{\dd x_{i}}{\dd u_{2}} · du_{2}
\]
in the metrical groundform~\Eq{(12)} of Riemann's space, we get for the
square of the distance between two infinitely near surface-points a
quadratic differential form in $du_{1}$,~$du_{2}$ (as in Euclidean space).
The measure-relations of three-dimensional Riemann space may be
applied directly to any surface existing in it, and thus converts it
into a two-dimensional Riemann space. Whereas from the Euclidean
standpoint space is assumed at the very outset to be of a much
simpler character than the surfaces possible in it, viz.\ to be rectangular,
Riemann has generalised the conception of space just
sufficiently far to overcome this discrepancy. \Emph{The principle of
gaining knowledge of the external world from the behaviour
of its infinitesimal parts} is the mainspring of the theory of
knowledge in infinitesimal physics as in Riemann's geometry, and,
indeed, the mainspring of all the eminent work of Riemann, in
particular, that dealing with the theory of complex functions. The
question of the validity of the ``fifth postulate,'' on which historical
development started its attack on Euclid, seems to us nowadays
to be a somewhat accidental point of departure. The knowledge
that was necessary to take us beyond the Euclidean view was, in
our opinion, revealed by Riemann.
We have yet to convince ourselves that the geometry of Bolyai
and Lobatschefsky as well as that of Euclid and also spherical
geometry (Riemann was the first to point out that the latter was
a possible case of non-Euclidean geometry) are all included as
particular cases in Riemann's geometry. We find, in fact, that if
we denote a point in the Bolyai-Lobatschefsky plane by the rectangular
co-ordinates $u_{1}\Com u_{2}$ of its corresponding point in Klein's
model the distance~$ds$ between two infinitely near points is by~\Eq{(1)}
\[
ds^{2} = \frac{(1 - u_{1}^{2} - u_{2}^{2}) (du_{1}^{2} + du_{2}^{2}) + (u_{1}\, du_{1} + u_{2}\, du_{2})^{2}}
{(1 - u_{1}^{2} - u_{2}^{2})^{2}}\Add{.}
\Tag{(13)}
\]
By comparing this with~\Eq{(10)} we see that the Theorem of Taurinus
is again confirmed. The metrical groundform of three-dimensional
non-Euclidean space corresponds exactly to this expression.
%[** TN: Moved to top of preceding paragraph]
\WrapFigure{1in}{5}
If we can find a curved surface in Euclidean space for which formula~\Eq{(13)}
holds, provided appropriate Gaussian co-ordinates $u_{1}$,~$u_{2}$
be chosen, then the geometry of Bolyai and Lobatschefsky is valid
on it. Such surfaces can actually be constructed; the simplest is
the surface of revolution derived from the tractrix. The tractrix
\PageSep{93}
\index{Tractrix}%
is a plane curve of the shape shown in \Fig{5}, with one vertex and
\index{Plane!(Beltrami's model)}%
one asymptote. It is characterised geometrically by the property
that any tangent measured from the point of contact to the point
of intersection with the asymptote is of constant length. Suppose
the curve to revolve about its asymptote as axis. Non-Euclidean
\index{Non-Euclidean!plane!(Beltrami's model)}%
geometry holds on the surface generated. This Euclidean model
of striking simplicity was first mentioned by Beltrami (\textit{vide} \FNote{5}).
There are certain shortcomings in it; in the first place the form in
which it is presented confines it to two-dimensional geometry;
secondly, each of the two halves of the surface of revolution into
which the sharp edge divides it represents only a part of the non-Euclidean
plane. Hilbert proved rigorously that there cannot be
a surface free from singularities in Euclidean space which pictures
the whole of Lobatschefsky's plane (\textit{vide} \FNote{6}). Both of these
weaknesses are absent in the elementary geometrical
model of Klein.
So far we have pursued a speculative train of
thought and have kept within the boundaries of mathematics.
There is, however, a difference in demonstrating
the consistency of non-Euclidean geometry and
\Emph{inquiring whether it or Euclidean geometry holds
in actual space}. To decide this question Gauss long
ago measured the triangle having for its vertices Inselsberg,
Brocken, and Hoher Hagen (near Göttingen),
using methods of the greatest refinement, but the
deviation of the sum of the angles from~$180°$ was found to lie
within the limits of errors of observation. Lobatschefsky concluded
from the very small value of the parallaxes of the stars
that actual space could differ from Euclidean space only by an
extraordinarily small amount. Philosophers have put forward
the thesis that the validity or non-validity of Euclidean geometry
cannot be proved by empirical observations. It must in fact
be granted that in all such observations essentially physical assumptions,
such as the statement that the path of a ray of light is
a straight line and other similar statements, play a prominent part.
This merely bears out the remark already made above that it is
only the whole composed of geometry and physics that may be
tested empirically. Conclusive experiments are thus possible only
if physics in addition to geometry is worked out for Euclidean
space \Emph{and} generalised Riemann space. We shall soon see that
without making artificial limitations we can easily translate the
laws of the electromagnetic field, which were originally set up on
the basis of Euclidean geometry, into terms of Riemann's space.
\PageSep{94}
Once this has been done there is no reason why experience should
not decide whether the special view of Euclidean geometry or the
more general one of Riemann geometry is to be upheld. It is
clear that at the present stage this question is not yet ripe for
discussion.
%[** TN: Height-dependent coersion]
\enlargethispage{\baselineskip}
{\Loosen In this concluding paragraph we shall once again present the
foundations of Riemann's geometry in the form of a résumé, in
which we do not restrict ourselves to the dimensional number
$n = 3$.}
\emph{An $n$-dimensional Riemann space is an $n$-dimensional manifold,
not of an arbitrary nature, but one which derives its measure-relations
from a definitely positive quadratic differential form.} The two
principal laws according to which this form determines the metrical
quantities are expressed in \Eq{(1)}~and~\Eq{(2)} in which the~$x_{i}$'s denote any
co-ordinates whatsoever.
1. If $g$~is the determinant of the co-efficients of the groundform,
then the size of any portion of space is given by the integral
\[
\int \sqrt{g}\, dx_{1}\, dx_{2} \dots dx_{n}
\Tag{(14)}
\]
which is to be taken over the mathematical region of the variables~$x_{i}$,
which corresponds to the portion of space in question.
2. If $Q(d\Com \delta)$ denote the symmetrical bilinear form, corresponding
\index{Non-Euclidean!plane!(metrical groundform of)}%
\index{Plane!(metrical groundform)}%
to the quadratic groundform, of two line elements $d$~and~$\delta$
situated at the same point, then the angle~$\theta$ between them is
given by
\[
\cos\theta = \frac{Q(d\Com \delta)}{\sqrt{Q(d\Com d) · Q(\delta\Com \delta)}}\Add{.}
\Tag{(15)}
\]
{\Loosen An $m$-dimensional manifold existing in $n$-dimensional space
($1 \leq m \leq n$) is given in parametric terms by}
\[
x_{i} = x_{i}(u_{1}\Com u_{2}\Com \dots\Com u_{m})\qquad
(i = 1, 2, \dots\Add{,} n).
\]
By substituting the differentials
\[
dx_{i}
= \frac{\dd x_{i}}{\dd u_{1}} · du_{1}
+ \frac{\dd x_{i}}{\dd u_{2}} · du_{2}
+ \dots
+ \frac{\dd x_{i}}{\dd u_{m}} · du_{m}
\]
in the metrical groundform of space we get the metrical groundform
of this $m$-dimensional manifold. The latter is thus itself an
$m$-dimensional Riemann space, and the size of any portion of it
may be calculated from formula~\Eq{(14)} in the case $m = n$. In this
way the lengths of segments of lines and the areas of portions of
surfaces may be determined.
\PageSep{95}
\Section{12.}{Continuation. Dynamical View of Metrical Properties}
We shall now revert to the theory of surfaces in Euclidean
space. The \Emph{curvature} of a plane curve may be defined in the
\index{Curvature!Gaussian}%
\index{Gaussian curvature}%
following way as the measure of the rate at which the normals to
the curve diverge. From a fixed point~$O$ we trace out the vector~$Op$,
the ``normal'' to the curve at an \Typo{arbitary}{arbitrary} point~$P$, and make it
of unit length. This gives us a point~$P$, corresponding to~$P$, on the
circle of radius unity. If $P$~traverses a small arc~$\Delta s$ of the curve,
the corresponding point~$p$ will traverse an arc~$\Delta \sigma$ of the circle; $\Delta \sigma$~is
the plane angle which is the sum of the angles that the normals
erected at all points of the arc of the curve make with their respective
neighbours. The limiting value of the quotient~$\dfrac{\Delta \sigma}{\Delta s}$ for an
%[** TN: Moved up three lines]
\Figure{6}
element of arc~$\Delta s$ which contracts to a point~$P$ is the curvature at~$P$.
Gauss defined the curvature of a surface as the measure of the rate
at which its normals diverge in an exactly analogous manner. In
place of the unit circle about~$O$, he uses the unit sphere. Applying
the same method of representation he makes a small portion~$d\omega$ of
this sphere correspond to a small area~$do$ of the surface; $d\omega$~is
equal to the solid angle formed by the normals erected at the
points of~$do$. The ratio~$\dfrac{d\omega}{do}$ for the limiting case when $do$~becomes
vanishingly small is the \emph{Gaussian measure of curvature}. \emph{Gauss
made the important discovery that this curvature is determined by
the inner measure-relations of the surface alone, and that it can be
calculated from the co-efficients of the metrical groundform as a
differential expression of the second order.} The curvature accordingly
remains unaltered if the surface be bent without being distorted by
stretching. By this geometrical means a \Emph{differential invariant
of the quadratic differential forms} of two variables was discovered,
that is to say, a quantity was found, formed of the co-efficients
of the differential form in such a way that its value
was the same for two differential forms that arise from each other
\PageSep{96}
by a transformation (and also for parametric pairs which correspond
to one another in the transformation).
Riemann succeeded in extending the conception of curvature to
quadratic forms of three and more variables. He then found that
it was no longer a scalar but a tensor (we shall discuss this in §\,15
of the present chapter). More precisely it may be stated that
Riemann's space has a definite curvature at every point in the
\index{Space!form of@{(as form of phenomena)}}%
normal direction of every surface. The characteristic of Euclidean
space is that its curvature is nil at every point and in every direction.
Both in the case of Bolyai-Lobatschefsky's geometry and
spherical geometry the curvature has a value~$a$ independent of the
place and of the surface passing through it: this value is positive
in the case of spherical geometry, negative in that of Bolyai-Lobatschefsky.
(It may therefore be put $= ±1$ if a suitable unit
of length be chosen.) If an $n$-dimensional space has a constant
curvature~$a$, then if we choose appropriate co-ordinates~$x_{i}$, its
metrical groundform must be of the form
\[
\frac{\left(1 + a\sum_{i} x_{i}^{2}\right) · \sum_{i} dx_{i}^{2}
- a\left(\sum_{i} x_{i}\, \Typo{dx}{dx_{i}}\right)^{2}}
{\left(1 + a\sum_{i} x_{i}^{2}\right)^{2}}.
\]
It is thus completely defined in a single-valued manner. If space
is everywhere homogeneous in all directions, its curvature must be
constant, and consequently its metrical groundform must be of the
form just given. Such a space is necessarily either Euclidean,
spherical, or Lobatschefskyan. Under these circumstances not only
have the line elements an existence which is independent of place
and direction, but any arbitrary finitely extended figure may be
transferred to any arbitrary place and put in any arbitrary direction
without altering its metrical conditions, i.e.\ its displacements are
congruent. This brings us back to congruent transformations
which we used as a starting-point for our reflections on space in~§\,1.
Of these three possible cases the Euclidean one is characterised
by the circumstance that the group of translations having the
special properties set out in~§\,1 are unique in the group of congruent
transformations. The facts which are summarised in this
paragraph are mentioned briefly in Riemann's essay; they have
been discussed in greater detail by Christoffel, Lipschitz, Helmholtz,
and Sophus Lie (\textit{vide} \FNote{7}).
Space is a form of phenomena, and, by being so, is necessarily
homogeneous. It would appear from this that out of the rich
abundance of possible geometries included in Riemann's conception
\PageSep{97}
only the three special cases mentioned come into consideration
from the outset, and that all the others must be rejected without
further examination as being of no account: \textit{parturiunt montes,
nascetur ridiculus mus!} Riemann held a different opinion, as is
evidenced by the concluding remarks of his essay. Their full
purport was not grasped by his contemporaries, and his words died
away almost unheard (with the exception of a solitary echo in the
writings of W.~K. Clifford). Only now that Einstein has removed
the scales from our eyes by the magic light of his theory of gravitation
do we see what these words actually mean. To make them
quite clear I must begin by remarking that Riemann contrasts
\Emph{discrete} manifolds, i.e.\ those composed of single isolated elements,
with \Emph{continuous} manifolds. The measure of every part of such a
discrete manifold is determined by the \Emph{number} of elements belonging
\index{Manifold!discrete}%
to it. Hence, as Riemann expresses it, a discrete manifold
has the principle of its metrical relations in itself, \textit{a~priori}, as a
consequence of the concept of number. In Riemann's own words:---
``The question of the validity of the hypotheses of geometry in
the infinitely small is bound up with the question of the ground of
the metrical relations of space. In this question, which we may
still regard as belonging to the doctrine of space, is found the
application of the remark made above; that in a discrete manifold,
the principle or character of its metric relations is already given in
the notion of the manifold, whereas in a continuous manifold this
ground has to be found elsewhere, i.e.\ has to come from outside.
Either, therefore, the reality which underlies space must form a
discrete manifold, or we must seek the ground of its metric relations
(measure-conditions) outside it, in binding forces which act upon it. %''
``A decisive answer to these questions can be obtained only by
starting from the conception of phenomena which has hitherto
been justified by experience, to which Newton laid the foundation,
and then making in this conception the successive changes required
by facts which admit of no explanation on the old theory; researches
of this kind, which commence with general \Erratum{motions}{notions},
cannot be other than useful in preventing the work from being
hampered by too narrow views, and in keeping progress in the
knowledge of the inter-connections of things from being checked
by traditional prejudices. %''
``This carries us over into the sphere of another science, that of
physics, into which the character and purpose of the present discussion
will not allow us to enter.''
If we discard the first possibility, ``that the reality which underlies
space forms a discrete manifold''---although we do not by this
\PageSep{98}
in any way mean to deny finally, particularly nowadays in view of
the results of the quantum-theory, that the ultimate solution of the
problem of space may after all be found in just this possibility---we
see that Riemann rejects the opinion that had prevailed up to
his own time, namely, that the metrical structure of space is fixed
and inherently independent of the physical phenomena for which
it serves as a background, and that the real content takes possession
of it as of residential flats. \emph{He asserts, on the contrary, that space
in itself is nothing more than a three-dimensional manifold devoid of
all form; it acquires a definite form only through the advent of the
material content filling it and determining its metric relations.}
There remains the problem of ascertaining the laws in accordance
with which this is brought about. In any case, however, the
metrical groundform will alter in the course of time just as the
disposition of matter in the world changes. We recover the
possibility of displacing a body without altering its metric relations
by making the body carry along with it the ``metrical field'' which
it has produced (and which is represented by the metrical groundform\Add{)};
just as a mass, having assumed a definite shape in equilibrium
under the influence of the field of force which it has itself produced,
would become deformed if one could keep the field of force fixed
while displacing the mass to another position in it; whereas, in
reality, it retains its shape during motion (supposed to be sufficiently
slow), since it carries the field of force, which it has produced,
along with itself. We shall illustrate in greater detail this bold
idea of Riemann concerning the metrical field produced by matter,
and we shall show that if his opinion is correct, any two portions
of space which can be transformed into one another by a continuous
deformation, must be recognised as being congruent in the sense
we have adopted, and that the same material content can fill one
portion of space just as well as the other.
To simplify this examination of the underlying principles we
assume that the material content can be described fully by scalar
phase quantities such as mass-density, density of charge, and so
forth. We fix our attention on a definite moment of time. During
this moment the density~$\rho$ of charge, for example, will, if we choose
a certain co-ordinate system in space, be a definite function
$f(x_{1}\Com x_{2}\Com x_{3})$ of the co-ordinates~$x_{1}$ but will be represented by a different
function $f^{*}(x_{1}^{*}\Com x_{2}^{*}\Com x_{3}^{*})$ if we use another co-ordinate system in~$x_{i}^{*}$.
\emph{A parenthetical note.} Beginners are often confused by failing to
notice that in mathematical literature symbols are used throughout
to designate \Emph{functions}, whereas in physical literature (including
the mathematical treatment of physics) they are used exclusively
\PageSep{99}
to denote ``\Emph{magnitudes}'' (quantities). For example, in \Chg{thermo-dynamics}{thermodynamics}
\index{Magnitudes}%
the energy of a gas is denoted by a definite letter, say~$E$,
irrespective of whether it is a function of the pressure~$p$ and the
temperature~$\theta$ or a function of the volume~$v$ and the temperature~$\theta$.
The mathematician, however, uses two different symbols to express
this:---
\[
E = \phi(p, \theta) = \psi(v, \theta).
\]
The partial derivatives $\dfrac{\dd \phi}{\dd \theta}$, $\dfrac{\dd \psi}{\dd \theta}$, which are totally different in meaning,
consequently occur in physics books under the common expression~$\dfrac{\dd E}{\dd \theta}$.
A suffix must be added (as was done by Boltzmann),
or it must be made clear in the text that in one case~$p$, in the other
case~$v$, is kept constant. The symbolism of the mathematician is
clear without any such addition.\footnote
{This is not to be taken as a criticism of the physicist's nomenclature
which is fully adequate to the purposes of physics, which deals with
\Emph{magnitudes}.}
Although the true state of things is really more complex we
shall assume the most simple system of geometrical optics, the
fundamental law of which states that the ray of light from a point~$M$
emitting light to an observer at~$P$ is a ``geodetic'' line, which is
the shortest of all the lines connecting $M$ with~$P$: we take no
account of the finite velocity with which light is propagated. We
ascribe to the receiving consciousness merely an optical faculty of
perception and simplify this to a ``point-eye'' that immediately\Pagelabel{99}
observes the differences of direction of the impinging rays, these
directions being the values of~$\theta$ given by~\Eq{(15)}; the ``point-eye''
thus obtains a picture of the directions in which the surrounding
objects lie (colour factors are ignored). The Law of Continuity
governs not only the action of physical things on one another but
also psycho-physical interactions. The direction in which we observe
objects is determined not by their places of occupation alone,
but also by the direction of the ray from them that strikes the
retina, that is, by the state of the optical field directly in contact
with that elusive body of reality whose essence it is to have an
objective world presented to it in the form of experiences of consciousness.
To say that a material content~$G$ is the same as the
material content~$G'$ can obviously mean no more than saying that
to every point of view~$P$ with respect to~$G$ there corresponds a
point of view~$P'$ with respect to~$G'$ (and conversely) in such a way
that an observer at~$P'$ in~$G'$ receives the same ``direction-picture''
as an observer in~$G$ receives at~$P$.
\PageSep{100}
Let us take as a basis a definite co-ordinate system~$x_{i}$. The
\index{Field action of electricity!metrical@{(metrical)}}%
scalar phase-quantities, such as density of electrification~$\rho$, are then
represented by definite functions
\[
\rho = f(x_{1}\Com x_{2}\Com x_{3}).
\]
Let the metrical groundform be
\[
\sum_{i,k=1}^{3} g_{ik}\, dx_{i}\, dx_{k}
\]
in which the~$g_{ik}$'s likewise (in ``mathematical'' terminology) denote
definite functions of $x_{1}$,~$x_{2}$,~$x_{3}$. Furthermore, suppose any continuous
transformation of space into itself to be given, by which
a point~$P'$ corresponds to each point~$P$ respectively. Using this
co-ordinate system and the modes of expression
\[
P = (x_{1}\Com x_{2}\Com x_{3}),\qquad
P' = (x_{1}'\Com x_{2}'\Com x_{3}')\Add{,}
\]
suppose the transformation to be represented by
\[
x_{i}' = \phi(x_{1}\Com x_{2}\Com x_{3})\Add{.}
\Tag{(16)}
\]
Suppose this transformation convert the portion~$\vS$ of space into~$\vS'$,
I shall show that if Riemann's view is correct $\vS'$~is congruent with~$\vS$
in the sense defined.
I make use of a second co-ordinate system by taking as co-ordinates
of the point~$P$ the values of~$x_{i}'$ given by~\Eq{(16)}; the expressions~\Eq{(16)}
then become the formulæ of transformation. The
mathematical region in three variables represented by~$\vS$ in the
co-ordinates~$x'$ is identical with that represented by~$\vS'$ in the co-ordinates~$x$.
An arbitrary point~$P$ has the same co-ordinates in~$x'$
as $P'$~has in~$x$. I now imagine space to be filled by matter in some
other way, namely, that represented by the formulæ
\[
\rho = f(x_{1}'\Com x_{2}'\Com x_{3}')
\]
at the point~$P$, with similar formulæ for the other scalar quantities.
If the metric relations of space are taken to be independent of the
contained matter, the metrical groundform will, as in the case of
the first content, be of the form
\[
\sum_{i\Com k} g_{ik}\, dx_{i}\, dx_{k}
= \sum_{i\Com k} g_{ik}'(x_{1}'\Com x_{2}'\Com x_{3}')\, dx_{i}'\, dx_{k}',
\]
the right-hand member of which denotes the expression after
transformation to the new co-ordinate system. If, however, the
metric relations of space are determined by the matter filling it---we
assume, with Riemann, that this is actually so---then, since the
second occupation by matter expresses itself in the co-ordinates~$x'$
\PageSep{101}
in exactly the same way as does the first in~$x$, the metrical groundform
for the second occupation will be
\[
\sum_{i\Com k} g_{ik}(x_{1}'\Com x_{2}'\Com x_{3}')\, dx_{i}'\, dx_{k}'.
\]
In consequence of our underlying principle of geometrical optics
assumed above, the content in the portion~$\vS'$ of space during the
first occupation will present exactly the same appearance to an
observer at~$P'$ as the material content in~$\vS$ during the second
occupation presents to an observer at~$P$. If the older view of
``residential flats'' is correct, this would of course not be the case.
The simple fact that I can squeeze a ball of modelling clay with
my hands into any irregular shape totally different from a sphere
would seem to reduce Riemann's view to an absurdity. This, however,
proves nothing. For if Riemann is right, a deformation of
the inner atomic structure of the clay is entirely different from that
which I can effect with my hands, and a rearrangement of the masses
in the universe, would be necessary to make the distorted ball of
clay appear spherical to an observer from all points of view.
The essential point is that a piece of space has no visual form at
all, but that this form depends on the material content occupying
the world, and, indeed, occupying it in such a way that by means
of an appropriate rearrangement of the mode of occupation I can
give it any visual form. By this I can also metamorphose any
two \Emph{different} pieces of space into the \Emph{same} visual form by choosing
an appropriate disposition of the matter. Einstein helped to
lead Riemann's ideas to victory (although he was not directly
influenced by Riemann). Looking back from the stage to which
Einstein has brought us, we now recognise that these ideas could
give rise to a valid theory only after \Emph{time} had been added as a
fourth dimension to the three-space dimensions in the manner set
forth in the so-called special theory of relativity. As, according to
Riemann, the conception ``congruence'' leads to no metrical system
at all, not even to the general metrical system of Riemann, which is
governed by a quadratic differential form, we see that ``the inner
ground of the metric relations'' must indeed be sought elsewhere.
Einstein affirms that it is to be found in the ``binding forces'' of
\Emph{Gravitation}. In Einstein's theory (Chapter~IV) the co-efficients~$g_{ik}$
of the metrical groundform play the same part as does gravitational
potential in Newton's theory of gravitation. The laws
according to which space-filling matter determines the metrical
structure are the laws of gravitation. The gravitational field affects
light rays and ``rigid'' bodies used as measuring rods in such a
\PageSep{102}
way that when we use these rods and rays in the usual manner to
take measurements of objects, a geometry of measurement is found
to hold which deviates very little from that of Euclid in the regions
accessible to observation. These metric relations are not the outcome
of space being a form of phenomena, but of the physical
behaviour of measuring rods and light rays as determined by the
gravitational field.
After Riemann had made known his discoveries, mathematicians
busied themselves with working out his system of geometrical ideas
formally; chief among these were Christoffel, Ricci, and Levi-Civita
(\textit{vide} \FNote{8}). Riemann, in the last words of the above
quotation, clearly left the real development of his ideas in the
hands of some subsequent scientist whose genius as a physicist
could rise to equal flights with his own as a mathematician. After
a lapse of seventy years this mission has been fulfilled by Einstein.
Inspired by the weighty inferences of Einstein's theory to
examine the mathematical foundations anew the present writer
made the discovery that Riemann's geometry goes only half-way
towards attaining the ideal of a pure infinitesimal geometry. It still
remains to eradicate the last element of geometry ``at a distance,''
a remnant of its Euclidean past. Riemann assumes that it is possible
to compare the lengths of two line elements at \Emph{different} points
of space, too; \Emph{it is not permissible to use comparisons at a
distance in an ``infinitely near'' geometry}. One principle alone
is allowable; by this a division of length is transferable from one
point to that infinitely adjacent to it.
After these introductory remarks we now pass on to the
\index{Affine!manifold}%
systematic development of pure infinitesimal geometry (\textit{vide}
\FNote{9}), which will be traced through three stages; from the
\Emph{continuum}, which eludes closer definition, by way of \Emph{affinely
connected manifolds}, to \Emph{metrical space}. This theory which,
in my opinion, is the climax of a wonderful sequence of logically-connected
ideas, and in which the result of these ideas has found
its ultimate shape, is a true \emph{geometry}, a doctrine of \emph{space itself}
and not merely like Euclid, and almost everything else that has
been done under the name of geometry, a doctrine of the configurations
that are possible in space.
\Section{13.}{Tensors and Tensor-densities in any Arbitrary
Manifold}
\index{Manifold!metrical}%
\Par{An $n$-dimensional Manifold.}---Following the scheme outlined
above we shall make the sole assumption about space that it is
an $n$-dimensional continuum. It may accordingly be referred to
\PageSep{103}
\index{Continuous relationship}%
\index{Displacement current!infinitesimal, of a point}%
\index{Line-element!generally@{(generally)}}%
\index{Relationship!continuous}%
$n$-co-ordinates $x_{1}\Com x_{2}\Com \dots\Com x_{n}$, of which each has a definite numerical
value at each point of the manifold; different value-systems of the
co-ordinates correspond to different points. If $\bar{x}_{1}\Com \bar{x}_{2}\Com \dots \bar{x}_{n}$ is a
second system of co-ordinates, then there are certain relations
\[
x_{i} = f_{i}(\bar{x}_{1}\Com \bar{x}_{2}\Com \dots \bar{x}_{n})
\text{ where }
(i = 1, 2, \dots\Add{,} n)
\Tag{(17)}
\]
between the $x$-co-ordinates and the $\bar{x}$-co-ordinates; these relations
are conveyed by certain functions~$f_{i}$. We do not only assume that
they are continuous, but also that they have continuous derivatives
\[
\alpha_{k}^{i} = \frac{\dd f_{i}}{\dd \bar{x}_{k}}
\]
whose determinant is non-vanishing. The latter condition is
necessary and sufficient to make affine geometry hold in infinitely
small regions, that is, so that reversible linear relations exist
between the differentials of the co-ordinates in both systems, i.e.\
\[
dx_{i} = \sum_{k} \alpha_{k}^{i}\, d\bar{x}_{k}\Add{.}
\Tag{(18)}
\]
We assume the existence and continuity of higher derivatives wherever
we find it necessary to use them in the course of our investigation.
In every case, then, a meaning which is invariant and
independent of the co-ordinate system has been assigned to the
conception of continuous functions of a point which have continuous
first, second, third, or higher derivatives as required; the
co-ordinates themselves are such functions.
\Par{Conception of a Tensor.}---The relative co-ordinates~$dx$ of a
\index{Components, co-variant, and contra-variant!tensor@{of a tensor}!generally@{(\emph{generally})}}%
\index{Components, co-variant, and contra-variant!tensor@{of a tensor}!linear@{(in a linear manifold)}}%
\index{Contra-variant tensors!(generally)}%
\index{Co-variant tensors!(generally)}%
\index{Tensor!general@{(general)}}%
point $P' = (x_{i} + dx_{i})$ infinitely near to the point $P = (x_{i})$ are the
components of a \Emph{line element} at~$P$ or of an \Emph{infinitesimal displacement}~$\Vector{PP'}$
of~$P$. The transformation to another co-ordinate
system is effected for these components by formulæ~\Eq{(18)},
in which $\alpha_{k}^{i}$~denote the values of the respective derivatives at the
point~$P$. The infinitesimal displacements play the same part in the
development of Tensor Calculus as do displacements in Chapter~I\@.
It must, however, be noticed that, here, \Emph{a displacement is essentially
bound to a point}, and that there is no meaning in saying
that the infinitesimal displacements of two different points are the
equal or unequal. It might occur to us to adopt the convention
of calling the infinitesimal displacements of two points equal if
they have the same components; but it is obvious from the fact
that the~$\alpha_{k}^{i}$'s in~\Eq{(18)} are not constants, that if this were the case
for one co-ordinate system it need in no wise be true for another.
Consequently we may only speak of the infinitesimal displacement
\PageSep{104}
\index{Continuous relationship}%
\index{Linear equation!tensor}%
\index{Relationship!continuous}%
of a \Emph{point} and not, as in Chapter~I, of the whole of space; hence
we cannot talk of a vector or tensor simply, but must talk of a
\Emph{vector} or \Emph{tensor} as being \Emph{at a point~$P$}. A tensor at a point~$P$ is
a linear form, in several series of variables, which is dependent on
a co-ordinate system to which the immediate neighbourhood of~$P$
is referred in the following way: the expressions of the linear form
in any two co-ordinate systems $x$~and~$\bar{x}$ pass into one another if
certain of the series of variables (with upper indices) are transformed
co-grediently, the remainder (with lower indices) contra-grediently,
to the differentials~$dx_{i}$, according to the scheme
\[
\xi^{i} = \sum_{k} \alpha_{k}^{i} \bar{\xi}^{k}
\text{ and }
\bar{\xi}_{i} = \sum_{k} \alpha_{i}^{k} \xi_{k}
\text{ respectively\Add{.}}
\Tag{(19)}
\]
By $\alpha_{k}^{i}$ we mean the values of these derivatives \Emph{at the point~$P$}. The
co-efficients of the linear form are called the components of the
tensor in the co-ordinate system under consideration; they are co-variant
in those indices that belong to the variables with an upper
index, contra-variant in the remaining ones. The conception of
tensors is possible owing to the circumstance that the transition from
one co-ordinate system to another expresses itself as a \Emph{linear} transformation
in the differentials. One here uses the exceedingly fruitful
mathematical device of making a problem ``linear'' by reverting to
infinitely small quantities. The whole of \Emph{Tensor Algebra}, by
whose operations only tensors \Emph{at the same point} are associated,
\Emph{can now be taken over from Chapter~I}. Here, again, we shall
call tensors of the first order \Emph{vectors}. There are contra-variant
and co-variant vectors. Whenever the word vector is used without
being defined more exactly we shall understand it as meaning a
contra-variant vector. Infinitesimal quantities of this type are the
line elements in~$P$. Associated with every co-ordinate system there
are $n$~``unit vectors''~$\ve_{i}$ at~$P$, namely, those which have components
\index{Unit vectors}%
\[
\begin{array}{c|ccccc}
\ve_{1} & 1, & 0, & 0, & \dots & 0 \\
\ve_{2} & 0, & 1, & 0, & \dots & 0 \\
\dots & \hdotsfor{5} \\
\ve_{n} & 0, & 0, & 0, & \dots & 1 \\
\end{array}
\]
in the co-ordinate system. Every vector~$\vx$ at~$P$ may be expressed
in linear terms of these unit vectors. For if $\xi^{i}$~are its components,
then
\[
\vx = \xi^{1} \ve_{1} + \xi^{2} \ve_{2} + \dots + \xi^{n} \ve_{n} \text{ holds.}
\]
The unit vectors~$\bar{\ve}_{i}$ of another co-ordinate system~$\bar{x}$ are derived
from the~$\ve_{i}$'s according to the equations
\PageSep{105}
\[
\bar{\ve}_{i} = \sum_{k} \alpha_{i}^{k} \ve_{k}.
\]
The possibility of passing from co-variant to contra-variant components
of a tensor does not, of course, come into question here.
\index{Tensor!field}%
Each two linearly independent line elements having components
$dx_{i}$,~$\delta x_{i}$ map out a \Emph{surface element} whose components are
\[
dx_{i}\, \delta x_{k} - dx_{k}\, \delta x_{i} = \Delta x_{ik}.
\]
Each three such line elements map out a three-dimensional space
element and so forth. Invariant differential forms that assign a
number linearly to each arbitrary line element, surface element,
etc., respectively are \Emph{linear tensors} ($=$~co-variant skew-symmetrical
\index{Linear equation!tensor-density}%
tensors, \textit{vide} §\,7). The above convention about omitting
signs of summation will be retained.
\Par{Conception of a Curve.}---If to every value of a parameter~$s$\Pagelabel{105}
a point $P = P(s)$ is assigned in a continuous manner, then if we
interpret $s$ as time, a ``\Emph{motion}'' is given. In default of a better
\index{Motion!(in mathematical sense)}%
expression we shall apply this name in a purely mathematical
sense, even when we do not interpret~$s$ in this way. If we use a
definite co-ordinate system we may represent the motion in the
form
\[
x_{i} = x_{i}(s)
\Tag{(20)}
\]
by means of $n$~continuous functions~$x_{i}(s)$, which we assume not
only to be continuous, but also continuously differentiable.\footnote
{I.e.\ have continuous differential co-efficients.}
In
passing from the parametric value~$s$ to~$s + ds$, the corresponding
point~$P$ suffers an infinitesimal displacement having components~$dx_{i}$.
If we divide this vector at~$P$ by~$ds$, we get the ``\Emph{velocity},'' a
\index{Velocity}%
vector at~$P$ having components~$\dfrac{dx_{i}}{ds} = u^{i}$. The formulæ~\Eq{(20)} is at
the same time a parametric representation of the \Emph{trajectory} of
the motion. Two motions describe the same \Emph{curve} if, and only
if, the one motion arises from the other when the parameter~$s$ is
subjected to a transformation $s = \omega(\bar{s})$, in which $\omega$~is a continuous
and continuously differentiable uniform function~$\omega$. Not the components
of velocity at a point are determinate for a curve, but only
their ratios (which characterise the \Emph{direction} of the curve).
\Par{Tensor Analysis.}---A \Emph{tensor field} of a certain kind is defined in
a region of space if to every point~$P$ of this region a tensor of this
kind at~$P$ is assigned. Relatively to a co-ordinate system the
components of the tensor field appear as definite functions of the
co-ordinates of the variable ``point of emergence''~$P$: we assume
them to be continuous and to have continuous derivatives. The
\PageSep{106}
Tensor Analysis worked out in Chapter~I, §\,8, cannot, without
alteration, be applied to any arbitrary continuum. For in defining
the general process of differentiation we earlier used arbitrary co-variant
and contra-variant vectors, whose components were \Emph{independent
of the point in question}. This condition is indeed
invariable for linear transformations, but not for any arbitrary
ones since, in these, the~$\alpha_{k}^{i}$'s are not constants. For an arbitrary
manifold we may, therefore, set up only the analysis of \Emph{linear}
tensor fields: this we proceed to show. Here, too, there is
derived from a scalar field~$f$ by means of differentiation, independently
of the co-ordinate system, a linear tensor field of the first
order having components
\[
f_{i} = \frac{\dd f}{\dd x_{i}}\Add{.}
\Tag{(21)}
\]
From a linear tensor field~$f_{i}$ of the first order we get one of the
second order
\[
f_{ik} = \frac{\dd f_{i}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\Add{.}
\Tag{(22)}
\]
From one of the second order,~$\Typo{f^{ik}}{f_{ik}}$, we get a linear tensor field of
the third order
\[
f_{ikl} = \frac{\dd f_{kl}}{\dd x_{i}}
+ \frac{\dd f_{li}}{\dd x_{k}}
+ \frac{\dd f_{ik}}{\dd x_{l}}\Add{,}
\Tag{(23)}
\]
and so forth.
If $\phi$~is a given scalar field in space and if $x_{i}$,~$\bar{x}_{i}$ denote any two
co-ordinate systems, then the scalar field will be expressed in each
in turn as a function of the~$x_{i}$'s or $\bar{x}_{i}$'s respectively, i.e.\
\[
\phi = f(x_{1}\Com x_{2}\Com \dots\Com x_{n})
= \bar{f}(\bar{x}_{1}\Com \bar{x}_{2}\Com \dots\Com \bar{x}_{n}).
\]
If we form the increase of~$\phi$ for an infinitesimal displacement of
\index{Gradient!(generalised)}%
the current point, we get
\[
d\phi = \sum_{i} \frac{\dd f}{\dd x_{i}}\, dx_{i}
= \sum_{i} \frac{\dd \bar{f}}{\dd \bar{x}_{i}}\, d\bar{x}_{i}.
\]
From this we see that the $\dfrac{\dd f}{\dd x_{i}}$'s are components of a co-variant
tensor field of the first order, which is derived from the scalar field~$\phi$
in a manner independent of all co-ordinate systems. We have
here a simple illustration of the conception of vector fields. At
the same time we see that the operation ``grad'' is invariant not
only for linear transformations, but also for any arbitrary transformations
of the co-ordinates whatsoever, and this is what we
enunciated.
\PageSep{107}
To arrive at~\Eq{(22)} we perform the following construction. From
\Pagelabel{107}%
the point $P = P_{00}$ we draw the two line elements with components
$dx_{i}$ and~$\delta x_{i}$, which lead to the two infinitely near points $P_{10}$ and~$P_{01}$.
We displace (by ``variation'') the line element~$dx$ in some way so
that its point of emergence describes the distance $P_{00} P_{01}$; suppose
it to have got to $\Vector{P_{01} P_{11}}$ finally. We shall call this process the displacement~$\delta$.
Let the components~$dx_{i}$ have increased by~$\delta dx_{i}$, so
that
\[
\delta dx_{i}
= \bigl\{x_{i}(P_{11}) - x_{i}(P_{01})\bigr\}
- \bigl\{x_{i}(P_{10}) - x_{i}(P_{00})\bigr\}\Add{.}
\]
We now interchange $d$ and~$\delta$. By an analogous displacement~$d$ of
the line element~$\delta x$ along $P_{00} P_{10}$, by which it finally takes up the
position $\Vector{P_{10} \Typo{P_{11}}{P_{11}'}}$, its components are increased by
\[
d \delta x_{i}
= \bigl\{x_{i}(P_{11}') - x_{i}(P_{10})\bigr\}
- \bigl\{x_{i}(P_{01}) - x_{i}(P_{00})\bigr\}.
\]
Hence it follows that
\[
\delta dx_{i} - d \delta x_{i} = x_{i}(P_{11}) - x_{i}(P_{11}')\Add{.}
\Tag{(24)}
\]
If, and only if, the two points $P_{11}$~and~$P_{11}'$ coincide, i.e.\ if the two
line elements $dx$ and~$\delta x$ sweep out the same infinitesimal ``parallelogram''
during their displacements $\delta$ and~$d$ respectively---that is how
we shall view it---then we shall have
\[
\delta dx_{i} - d \delta x_{i} = 0\Add{.}
\Tag{(25)}
\]
If, now, a co-variant vector field with components~$f_{i}$ is given, then
we form the change in the invariant $df = f_{i}\, dx_{i}$ owing to the displacement~$\delta$
thus:
\[
\delta df = \delta f_{i}\, dx_{i} + f_{i}\, \delta dx_{i}.
\]
Interchanging $d$ and~$\delta$, and then subtracting, we get
\[
\Delta f = (\delta d - d\delta)f
= (\delta f_{i}\, dx_{i} - df_{i}\, \delta x_{i})
+ f_{i}(\delta dx_{i} - d \delta x_{i})
\]
and if both displacements pass over the same infinitesimal parallelogram
we get, in particular,
\[
\Delta f = \delta f_{i}\, dx_{i} - df_{i}\, \delta x_{i}
= \left(\frac{\dd f_{i}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\right) dx_{i}\, \delta x_{k}\Add{.}
\Tag{(26)}
\]
If one is inclined to distrust these perhaps too venturesome
operations with infinitesimal quantities the differentials may be
replaced by differential co-efficients. Since an infinitesimal element
of surface is only a part (or more correctly, the limiting value of the
part) of an arbitrarily small but finitely extended surface, the argument
will run as follows. Let a point~$(s\Com t)$ of our manifold be
assigned to every pair of values of two parameters $s$,~$t$ (in a certain
region encircling $s = 0$, $t = 0$). Let the functions $x_{i} = x_{i}(s\Com t)$, which
represents this ``two-dimensional motion'' (extending over a surface)
in any co-ordinate system~$x_{i}$, have continuous first and second
\PageSep{108}
differential co-efficients. For every point~$(s\Com t)$ there are two velocity
vectors with components $\dfrac{dx_{i}}{ds}$ and~$\dfrac{dx_{i}}{dt}$. We may assign our parameters
so that a prescribed point $P = (0\Com 0)$ corresponds to $s = 0$,
$t = 0$, and that the two velocity vectors at it coincide with two arbitrarily
given vectors $u^{i}$,~$v^{i}$ (for this it is merely necessary to make
the~$x_{i}$'s linear functions of $s$~and~$t$). Let $d$~denote the differentiation~$\dfrac{d}{ds}$,
and $\delta$~denote~$\dfrac{d}{dt}$. Then
\[
df = f_{i}\, \frac{dx_{i}}{ds},\qquad
\delta df
= \frac{\dd f_{i}}{\dd x_{k}}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{dt}
+ f_{i}\, \frac{d^{2} x_{i}}{dt\, ds}.
\]
By interchanging $d$ and~$\delta$, and then subtracting, we get
\[
\Delta f = \delta df - d \delta f
= \left(\frac{\dd f_{i}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\right)
\frac{dx_{i}}{ds}\, \frac{dx_{k}}{dt}\Add{.}
\Tag{(27)}
\]
By setting $s = 0$ and $t = 0$, we get the invariant at the point~$P$
\[
\left(\frac{\dd \Typo{f}{f_{i}}}{\dd x_{k}} - \frac{\dd f_{k}}{\dd x_{i}}\right) u^{i} v^{k}
\]
which depends on two arbitrary vectors $u$,~$v$ at that point. The
connection between this view and that which uses infinitesimals
consists in the fact that the latter is applied in rigorous form to
the infinitesimal parallelograms into which the surface $x_{i} = x_{i}(s\Com t)$
is divided by the co-ordinate lines $s = \text{const.}$ and $t = \text{const.}$
\Emph{Stokes' Theorem} may be recalled in this connection. The
\index{Stokes' Theorem}%
invariant linear differential $f_{i}\, dx_{i}$ is called \Emph{integrable} if its integral
\index{Integrable}%
along every closed curve (its ``curl'') $= 0$. (This is true, as we
know, only for a total differential.) Let any arbitrary surface given
in a parametric form $x_{i} = x_{i}(s\Com t)$ be spread out within the closed
curve, and be divided into infinitesimal parallelograms by the co-ordinate
lines. The curl taken around the perimeter of the whole
surface may then be traced back to the single curls around these
little surface meshes, and their values are given for every mesh by
our expression~\Eq{(27)}, after it has been multiplied by~$ds\, dt$. A differential
division of the curl is produced in this way, and the tensor~\Eq{(22)}
is a measure of the ``intensity of the curl'' at every point.
In the same way we pass on to the next higher stage~\Eq{(23)}. In
place of the infinitesimal parallelogram we now use the three-dimensional
parallelepiped mapped out by the three line elements
$d$,~$\delta$, and~$\dd$. We shall just indicate the steps of the argument
briefly.
\PageSep{109}
\[
\Typo{\delta}{\dd}(f_{ik}\, dx_{i}\, \delta x_{k})
= \frac{\dd f_{ik}}{\dd x_{l}}\, dx_{i}\, \delta x_{k}\, \dd x_{l}
+ f_{ik}(\dd \Typo{dx_{k}}{dx_{i}} · \delta x_{k}
+ \dd \delta x_{k} · dx_{i})\Add{.}
\Tag{(28)}
\]
Since $f_{ki} = -f_{ik}$, the second term on the right is
\[
= f_{ik}(\dd dx_{i} · \delta x_{k}
- \dd \delta x_{i} · dx_{k})\Add{.}
\Tag{(29)}
\]
If we interchange $d$,~$\delta$, and~$\dd$ cyclically, and then sum up, the six
members arising out of~\Eq{(29)} will destroy each other in pairs on
account of the conditions of symmetry~\Eq{(25)}.
\Par{Conception of Tensor-density.}---If $\int \vW\, dx$, in which $dx$~represents
\index{Intensity of field!quantities}%
\index{Linear equation!tensor-density}%
\index{Multiplication!of a tensor-density!by a number}%
\index{Quantities!intensity}%
\index{Quantities!magnitude}%
\index{Sum of!tensor-densities}%
\index{Vector!density@{-density}}%
\index{Tensor!density}%
briefly the element of integration $dx_{1}\Typo{,}{\,} dx_{2} \dots dx_{n}$, is an invariant
integral, then $\vW$~is a quantity dependent on the co-ordinate
system in such a way that, when transformed to another co-ordinate
system, its value become multiplied by the absolute
(numerical) value of the functional determinant. If we regard
this integral as a measure of the quantity of substance occupying
the region of integration, then $\vW$~is its density. We may, therefore,
call a quantity of the kind described a \Emph{scalar-density}.
\index{Scalar-Density}%
This is an important conception, equally as valuable as the conception
of scalars; it cannot be reduced to the latter. In an
analogous sense we may speak of \Emph{tensor-densities} as well as
scalar-densities. A linear form of several series of variables which
is dependent on the co-ordinate system, some of the variables
carrying upper indices, others lower ones, is a \Emph{tensor-density} at
a point~$P$, if, when the expression for this linear form is known
for a given co-ordinate system, its expression for any other arbitrary
co-ordinate system, distinguished by bars, is obtained by multiplying
it with the absolute or numerical value of the functional determinant
\[
\Delta = \text{abs.}\, |\alpha_{i}^{k}|\quad
\text{i.e.\ the absolute value of~$|\alpha_{i}^{k}|$,}
\]
and by transforming the variable according to the old scheme~\Eq{(19)}.
The words, components, co-variant, contra-variant, symmetrical,
skew-symmetrical, field, and so forth, are used exactly as in the
case of tensors. By contrasting tensors and tensor-densities, it
seems to me that we have grasped rigorously the difference between
\Emph{quantity} and \Emph{intensity}, so far as this difference has a
physical meaning: \Emph{tensors are the magnitudes of intensity,
tensor-densities those of quantity}. The same unique part that
co-variant skew-symmetrical tensors play among tensors is taken
among tensor-densities by contra-variant symmetrical tensor-densities,
which we shall term briefly \Emph{linear tensor-densities}.
\Par{Algebra of Tensor-densities.}---As in the realm of tensors so
have here the following operations:---
\PageSep{110}
1. Addition of tensor-densities of the same type; multiplication
\index{Addition of tensors!of tensor-densities}%
\index{Multiplication!of a tensor-density!by a tensor}%
of a tensor-density by a number.
2. Contraction.
3. Multiplication of a tensor by a tensor-density (\Emph{not} multiplication
of two tensor-densities by each other). For, if two scalar-densities,
for example, were to be multiplied together, the result
would not again be a scalar-density but a quantity which, to be
transformed to another co-ordinate system, would have to be multiplied
by the square of the functional determinant. Multiplying a
tensor by a tensor-density, however, always leads to a tensor-density
(whose order is equal to the sum of the orders of both factors).
Thus, for example, if a contra-variant vector with components~$f^{i}$ and
a co-variant tensor-density with components~$\vw_{ik}$ be multiplied
together, we get a mixed tensor-density of the third order with
components~$f^{i} \vw_{kl}$ produced in a manner independent of the co-ordinate
system.
\Emph{The analysis of tensor-densities} can be established only for
\Emph{linear} fields in the case of an arbitrary manifold. It leads to the
following \Emph{processes resembling the operation of divergence}:---
\begin{align*}
\frac{\dd \vw^{i}}{\dd x_{i}} &= \vw\Add{,}
\Tag{(30)} \displaybreak[0]\\
\frac{\dd \vw^{ik}}{\dd x_{k}} &= \vw^{i}\Add{,}
\Tag{(31)} \\
\multispan{2}{\dotfill}.
\end{align*}
As a result of~\Eq{(30)} a linear tensor-density field~$\vw^{i}$ of the first order
gives rise to a scalar-density field~$\vw$, whereas \Eq{(31)}~produces from a
linear field of the second order ($\vw^{ki} = -\vw^{ik}$) a linear field of the
first order, and so forth. These operations are independent of the
co-ordinate system. The divergence~\Eq{(30)} of a field~$\vw^{i}$ of the first
order which has been produced from one,~$\vw^{ik}$, of the second order
by means of~\Eq{(31)} is~$= 0$; an analogous result holds for the higher
orders. To prove that \Eq{(30)}~is invariant, we use the following known
result of the theory of the motion of continuously extended masses.
If $\xi^{i}$~is a given vector field, then
\[
\bar{x}_{i} = x_{i} + \xi^{i} · \delta t
\Tag{(32)}
\]
expresses an \Emph{infinitesimal displacement} of the points of the
\index{Displacement current!vector@{of a vector}}%
\index{Infinitesimal!displacement}%
continuum, by which the point with the co-ordinates~$x_{i}$ is transferred
to the point with the co-ordinates~$\bar{x}_{i}$. Let the constant infinitesimal
factor~$\delta t$ be defined as the element of time during which the
deformation takes place. The determinant of transformation
$A = \left\lvert\dfrac{\dd x^{i}}{\dd x_{k}}\right\rvert$ differs from unity by~$\delta t\, \dfrac{\dd \xi^{i}}{\dd x_{i}}$. The displacement causes
\PageSep{111}
portion~$\vG$ of the continuum, to which, if $x^{i}$'s~are used to denote
its co-ordinates, the mathematical region~$\XX$ in the variables~$x_{i}$ corresponds,
to pass into the region~$\Bar{\vG}$, from which $\vG$~differs by an
infinitesimal amount. If $\vs$~is a scalar-density field, which we
regard as the density of a substance occupying the medium, then
the quantity of substance present in~$\vG$
\begin{align*}
&= \int_{\XX} \vs(x)\, dx
\intertext{whereas that which occupies~$\Bar{\vG}$}
&= \int \vs(\bar{x})\, d\bar{x}
= \int_{\XX} \vs(\bar{x}) A\, dx,
\end{align*}
whereby the values~\Eq{(32)} are to be inserted in the last expression for
the arguments~$\bar{x}_{i}$ of~$\vs$. (I am here displacing the volume with respect
to the substance; instead of this, we can of course make the
substance flow through the volume; $\vs \xi^{i}$~then represents the intensity
of the current.) The increase in the amount of substance that
the region~$\vG$ gains by the displacement is given by the integral
$\vs(\bar{x})A - \vs(x)$ taken with respect to the variables~$x_{i}$ over~$\XX$. We,
however, get for the integrand
\[
\vs(\bar{x}) (A - 1) + \bigl\{\vs(\bar{x}) - \vs(x)\bigr\}
= \delta t\left(\vs\, \frac{\dd \xi^{i}}{\dd x_{i}} + \frac{\dd \vs}{\dd x_{i}}\, \xi^{i}\right)
= \delta t · \frac{\dd(\vs \xi^{i})}{\dd x_{i}}.
\]
Consequently the formula
\[
\frac{\dd(\vs \xi^{i})}{\dd x_{i}} = \vw
\]
establishes an invariant connection between the two scalar-density
fields $\vs$~and~$\vw$ and the contra-variant vector field with the components~$\xi^{i}$.
Now, since every vector-density~$\vw^{i}$ is representable in
the form~$\vs \xi^{i}$---for if in a \Emph{definite} co-ordinate system a scalar-density~$\vs$
and a vector field~$\xi$ be defined by $\vs = 1$, $\xi^{i} = \vw^{i}$, then the equation
$\vw^{i} = \vs \xi^{i}$ holds for \Emph{every} co-ordinate system---the required proof is
complete.
In connection with this discussion we shall enunciate the\Pagelabel{111}
%[** TN: Original entry points to page 110]
\index{Partial integration (principle of)}%
\Emph{Principle of Partial Integration} which will be of frequent use
below. If the functions~$\vw^{i}$ vanish at the boundary of a region~$\vG$,
then the integral
\[
\int_{\vG} \frac{\dd \vw^{i}}{\dd x_{i}}\, dx = 0.
\]
For this integral, multiplied by~$\delta t$, signifies the change that the
``volume'' $\Dint dx$ of this region suffers through an infinitesimal deformation
whose components $= \delta t · \vw^{i}$.
\PageSep{112}
The invariance of the process of divergence~\Eq{(30)} enables us
easily to advance to further stages, the next being~\Eq{(31)}. We enlist
the help of a co-variant vector field~$f_{i}$, which has been derived
from a potential~$f$; i.e.\
\[
f_{i} = \frac{\dd f}{\dd x_{i}}.
\]
We then form the linear tensor-density~$\Typo{\vw_{ik}}{\vw^{ik}} f_{i}$ of the first order
and also its divergence
\[
\frac{\dd(\vw^{ik} f_{i})}{\dd x_{k}}
= f_{i}\, \frac{\dd \vw^{ik}}{\dd x_{k}}.
\]
The observation that the~$f_{i}$'s may assume any arbitrarily assigned
values at a point~$P$ concludes the proof. In a similar way we
proceed to the third and higher orders.
\Section{14.}{Affinely Related Manifolds}
\Par{The Conception of Affine Relationship.}---We shall call a point~$P$
\index{Affine!geometry!(infinitesimal)}%
\index{Geodetic calibration!co-ordinate system}%
\index{Relationship!affine}%
of a manifold affinely related to its neighbourhood if we are given
\index{Manifold!affinely connected}%
the vector~$P'$ into which every vector at~$P$ is transformed by a
parallel displacement from $P$ to~$P'$; $P'$~is here an arbitrary point
infinitely near~$P$ (\textit{vide} \FNote{10}). No more and no less is required of
this conception than that it is endowed with all the properties that
were ascribed to it in the affine geometry of Chapter~I\@. That is,
we postulate: \emph{There is a co-ordinate system (for the immediate
neighbourhood of~$P$) such that, in it, the components of any vector at~$P$
are not altered by an infinitesimal parallel displacement.} This
postulate characterises parallel displacements as being such that
they may rightly be regarded as leaving vectors \Emph{unchanged}. Such
co-ordinate systems are called \Emph{geodetic} at~$P$. What is the effect
of this in an arbitrary co-ordinate system~$x_{i}$? Let us suppose that,
in it, the point~$P$ has the co-ordinate~$x_{i}^{\Typo{\circ}{0}}$, $P'$~the co-ordinates $x_{i}^{\Typo{\circ}{0}} + dx_{i}$;
let $\xi^{i}$~be the components of an arbitrary vector at~$P$, $\xi^{i} + d\xi^{i}$
the components of the vector resulting from it by parallel displacement
towards~$P'$. Firstly, since the parallel displacement from $P$
to~$P'$ causes all the vectors at~$P$ to be mapped out linearly or
affinely by all the vectors at~$P'$, $d\xi^{i}$~must be linearly dependent on~$\xi^{i}$,
i.e.\
\[
d\xi^{i} = -d\gamma_{r}^{i} \xi^{r}\Add{.}
\Tag{(33)}
\]
Secondly, as a consequence of the postulate with which we started,
the~$d\gamma_{r}^{i}$'s must be linear forms of the differentials~$dx_{i}$, i.e.\
\[
d\gamma_{r}^{i} = \Gamma_{rs}^{i}\, dx_{s}
\Tag{(33')}
\]
in which the number co-efficients~$\Gamma$, the ``components of the affine
relationship,'' satisfy the condition of symmetry
\[
\Gamma_{sr}^{i} = \Gamma_{rs}^{i}\Add{.}
\Tag{(33'')}
\]
\PageSep{113}
To prove this, let $\bar{x}_{i}$ be a geodetic co-ordinate system at~$P$; the
formulæ of transformation \Eq{(17)}~and \Eq{(18)} then hold. It follows
from the geodetic character of the co-ordinate system~$\bar{x}_{i}$ that, for a
parallel displacement,
\index{Parallel!displacement!infinitesimal@{(infinitesimal, of a contra-variant vector)}}%
\[
d\xi^{i} = d(\alpha_{r}^{i} \bar{\xi}^{r}) = d\alpha_{r}^{i} \bar{\xi}^{r}.
\]
If we regard the~$\xi^{i}$'s as components~$\delta x_{i}$ of a line element at~$P$ we
must have
\[
-d\gamma_{r}^{i}\, \delta x_{r}
= \frac{\dd^{2} f_{i}}{\dd \bar{x}_{r}\, \dd \bar{x}_{s}}\,
\delta\bar{x}_{r}\, d\bar{x}_{s}
\]
(in the case of the second derivatives we must of course insert
their values at~$P$). The statement contained in our enunciation
follows directly from this. Moreover, the symmetrical bilinear form
\[
-\Gamma_{rs}^{i}\, \delta\bar{x}_{r}\, d\bar{x}_{s}
\quad\text{is derived from}\quad
\frac{\dd^{2} f_{i}}{\dd \bar{x}_{r}\, \dd \bar{x}_{s}}\,
\delta\bar{x}_{r}\, d\bar{x}_{s}
\Tag{(34)}
\]
by transformation according to~\Eq{(18)}. This exhausts all the aspects
of the question. Now, if $\Gamma_{rs}^{i}$~are arbitrarily given numbers that
satisfy the condition of symmetry~\Eq{(33'')}, and if we define the
affine relationship by \Eq{(33)}~and~\Eq{(33')}, the transformation formulæ
lead to
\[
x_{i} - x_{i}^{0}
= \bar{x}_{i} - \tfrac{1}{2}\Gamma_{rs}^{i} \bar{x}_{r} \bar{x}_{s},
\]
that is, to a geodetic co-ordinate system~$\bar{x}_{i}$ at~$P$, since the equations~\Eq{(34)}
are fulfilled for them at~$P$. In fact this transformation at~$P$
gives us
\[
\bar{x}_{i} = 0,\qquad
d\bar{x}_{i} = dx_{i}\quad (\alpha_{k}^{i} = \delta_{k}^{i}),\qquad
\frac{\dd^{2} \Typo{f}{f_{i}}}{\dd \bar{x}_{r}\, \dd \bar{x}_{s}}
= -\Gamma_{rs}^{i}.
\]
The formulæ according to which the components~$\Gamma_{rs}^{i}$ of the
affine relationship are transformed in passing from one co-ordinate
system to another may easily be obtained from the above
discussion; we do not, however, require them for subsequent
work. The $\Gamma$'s are certainly \Emph{not} components of a tensor (contra-variant
in~$i$, co-variant in $r$~and~$s$) at the point~$P$; they have this
character with regard to \emph{linear} transformations, but lose it when
subjected to \emph{arbitrary} transformations. For they all vanish in a
geodetic co-ordinate system. Yet every virtual change of the
affine relationship~$[\Gamma_{rs}^{i}]$, whether it be finite or ``infinitesimal,'' is
a tensor. For
\[
[d\xi^{i}] = [\Gamma_{rs}^{i}]\xi^{r}\, dx_{s}
\]
is the difference of the two vectors that arise as a result of the two
parallel displacements of the vector~$\xi$ from $P$ to~$P'$.
The meaning of the \Emph{parallel displacement of a co-variant
\PageSep{114}
vector~$\xi_{i}$} at the point~$P$ to the infinitely near point~$P'$ is defined
uniquely by the postulate that the invariant product~$\xi_{i} \eta^{i}$ of the
vector~$\xi_{i}$ and any arbitrary contra-variant vector~$\eta^{i}$ remain unchanged
after the simultaneous parallel displacements, i.e.\
\[
d(\xi_{i} \eta^{i})
= (d\xi_{i} · \eta^{i}) + (\xi_{r}\, d\eta^{r})
= (d\xi_{i} - d\gamma_{i}^{r}\, \xi_{r}) \eta^{i} = 0\Add{,}
\]
whence
\[
d\xi_{i} = \sum_{r} d\gamma_{i}^{r}\, \xi_{r}\Add{.}
\Tag{(35)}
\]
We shall call a contra-variant \Emph{vector field~$\xi^{i}$} \emph{stationary} at the point~$P$,
\index{Stationary!field}%
\index{Stationary!vectors}%
if the vectors at the points~$P'$ infinitely near~$P$ arise from the
vector at~$P$ by parallel displacement, that is, if the total differential
equations\Pagelabel{114}
\[
d\xi^{i} + d\gamma_{r}^{i}\, \xi^{r} = 0\quad
%[** TN: Large parentheses in the original]
(\text{or } \frac{\dd \xi^{i}}{\dd x_{s}} + \Gamma_{rs}^{i} \xi^{r} = 0)
\]
are satisfied at~$P$. A vector field can obviously always be found
such that it has arbitrary given components at a point~$P$ (this remark
will be used in a construction which is to be carried out in
the sequel). The same conception may be set up for a co-variant
vector field.
From now onwards we shall occupy ourselves with \Emph{affine
manifolds; they are such that every point of them is
affinely related to its neighbourhood}. For a definite co-ordinate
system the components~$\Gamma_{rs}^{i}$ of the affine relationship
\index{Relationship!of a manifold as a whole (conditions of)}%
are continuous functions of the co-ordinates~$x_{i}$. By selecting the
appropriate co-ordinate system the $\Gamma_{rs}^{i}$'s may, of course, be made to
vanish at a single point~$P$, but it is, in general, not possible to
achieve this simultaneously for all points of the manifold. There
is no difference in the nature of any of the affine relationships
holding between the various points of the manifold and their immediate
neighbourhood. The manifold is homogeneous in this
sense. There are not various types of manifolds capable of being
distinguished by the nature of the affine relationships governing
each kind. The postulate with which we set out admits of
only one definite kind of affine relationship.
\Par{Geodetic Lines.}---If a point which is in motion carries a
\index{Geodetic calibration!line (general)}%
\index{Line, straight!geodetic}%
vector (which is arbitrarily variable) with it, we get for every value
of the time parameter~$s$ not only a point
\[
P = (s):\ x = x_{i}(s)
\]
of the manifold, but also a vector at this point with components
$v^{i} = v^{i}(s)$ dependent on~$s$. The vector remains stationary at the
moment~$s$ if
\PageSep{115}
\[
\frac{dv^{i}}{ds} + \Gamma_{\alpha\beta}^{i}\, v^{\alpha}\, \frac{dx_{\beta}}{ds} = 0\Add{.}
\Tag{(36)}
\]
(This will relieve the minds of those who disapprove of operations
with differentials; they have here been converted into
differential co-efficients.) In the case of a vector being carried along
according to any arbitrary rule, the left-hand side~$V^{i}$ of~\Eq{(36)} consists
of the components of a vector in~$(s)$ connected invariantly with the
motion and indicating how much the vector~$v^{i}$ changes per unit
of time at this point. For in passing from the point $P = (s)$ to
$P' = (s + ds)$, the vector~$v^{i}$ at~$P$ becomes the vector
\[
v^{i} + \frac{dv^{i}}{ds}\, ds
\]
at~$P'$. If, however, we displace~$v^{i}$ from $P$ to~$P'$ leaving it unchanged,
we there get
\[
v^{i} + \delta v^{i} = v^{i} - \Gamma_{\alpha\beta}^{i}\, v^{\alpha}\, dx_{\beta}.
\]
Accordingly, the difference between these two vectors at~$P'$, the
change in~$v$ during the time~$ds$ has components
\[
\frac{dv^{i}}{ds}\, ds - \delta v^{i} = V^{i}\, ds.
\]
In analytical language the invariant character of the vector~$V$ may
\index{Parallel!displacement!co-variant vector}%
\index{Translation of a point!(in the kinematical sense)}%
be recognised most readily as follows. Let us take an arbitrary
auxiliary co-variant vector $\xi_{i} = (s)$ at~$P$, and let us form the change
in the invariant~$\xi_{i} v^{i}$ in its passage from~$(s)$ to~$(s + ds)$, whereby the
vector~$\xi_{i}$ is taken along unchanged. We get
\[
\frac{d(\xi_{i} v^{i})}{ds} = \xi_{i} V^{i}.
\]
If $V$~vanishes for every value of~$s$, the vector~$v$ glides with the
point~$P$ along the trajectory during the motion \emph{without becoming
changed}.
Every motion is accompanied by the vector $u^{i} = \dfrac{dx_{i}}{ds}$ of its
velocity; for this particular case, $V$~is the vector
\[
U^{i} = \frac{du^{i}}{ds} + \Gamma_{\alpha\beta}^{i}\, u^{\alpha} u^{\beta}
= \frac{d^{2} x}{ds^{2}}
+ \Gamma_{\alpha\beta}^{i}\, \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds}:
\]
namely, the \Emph{acceleration}, which is a measure of the change of
\index{Acceleration}%
velocity per unit of time. A motion, in the course of which the
velocity remains unchanged throughout, is called a \Emph{translation}.
The trajectory of a translation, being a curve which preserves its
direction unchanged, is a \Emph{straight} or \Emph{geodetic line}. According
\PageSep{116}
to the translational view (cf.\ Chapter~I, §\,1) this is the inherent
property of the straight line.
\Emph{The analysis of tensors and tensor-densities} may be developed
for an affine manifold just as simply and completely as
for the linear geometry of Chapter~I\@. For example, if $f_{i}^{k}$~are the
components (co-variant in~$i$, contra-variant in~$k$) of a tensor field of
the second order, we take two auxiliary arbitrary vectors at the
point~$P$, of which the one,~$\xi$, is contra-variant and the other,~$\eta$, is
co-variant, and form the invariant
\[
f_{i}^{k} \xi^{i} \eta_{k}
\]
and its change for an infinitesimal displacement~$d$ of the current
point~$P$, by which $\xi$~and~$\eta$ are displaced parallel to themselves.
Now
\[
d(f_{i}^{k} \xi^{i} \eta_{k})
= \frac{\dd f_{i}^{k}}{\dd x_{l}}\, \xi^{i} \eta_{k}\, dx_{l}
- f_{r}^{k} \eta_{k}\, d\gamma_{i}^{r}\, \xi^{i}
+ f_{i}^{r} \xi^{i}\, d\gamma_{r}^{k}\, \eta_{k},
\]
hence
\[
f_{il}^{k} = \frac{\dd f_{i}^{k}}{\dd x_{l}}
- \Gamma_{il}^{r} f_{r}^{k}
+ \Gamma_{rl}^{k} f_{i}^{r}
\]
are the components of a tensor field of the third order, co-variant
in~$i\Com l$ and contra-variant in~$k$: this tensor field is derived from the
given one of the second order by a process independent of the co-ordinate
system. The additional terms, which the components of
the affine relationship contain, are characteristic quantities in
which, following Einstein, we shall later recognise the influence of
the gravitational field. The method outlined enables us to differentiate
a tensor in every conceivable case.
Just as the operation ``grad'' plays the fundamental part in
tensor analysis and all other operations are derivable from it, so the
operation ``div'' defined by~\Eq{(30)} is the basis of the analysis of
tensor-densities. The latter leads to processes of a similar character
for tensor-densities of any order. For instance, if we wish
to find an expression for the divergence of a mixed tensor-density~$\vw_{i}^{k}$
of the second order, we make use of an auxiliary stationary
vector field~$\xi^{i} \vw_{i}^{k}$ at~$P$ and find the divergence of the tensor-density~$\xi^{i} \vw_{i}^{k}$:
\[
\frac{\dd(\Typo{\xi_{i}}{\xi^{i}} \vw_{i}^{k})}{\dd x_{k}}
= \frac{\dd \xi^{r}}{\dd x_{k}} \vw_{r}^{k} + \xi^{i} \frac{\dd \vw_{i}^{k}}{\dd x_{k}}
= \xi^{i} \left(-\Gamma_{ik}^{r}\, \vw_{r}^{k} + \frac{\dd \vw_{i}^{k}}{\dd x_{k}}\right).
\]
This quantity is a scalar-density, and since the components of a
% [** TN: Extra word "the" in the original]
vector field which is stationary at~$P$ may assume any values at\Typo{ the}{}
this point~$(P)$, namely,
\PageSep{117}
\[
\frac{\dd \vw_{i}^{k}}{\dd x_{k}} - \Gamma_{is}^{r} \vw_{r}^{s}\Add{,}
\Tag{(37)}
\]
it is a co-variant tensor-density of the first order which has been
derived from~$\vw_{i}^{k}$ in a manner independent of every co-ordinate
system.
Moreover, not only can we reduce a tensor-density to one of the
next lower order by carrying out the process of \Emph{divergence}, but we
can also transpose a tensor-density into one of the next higher order
by \Emph{differentiation}. Let $\vs$ denote a scalar-density, and let us again
use a stationary vector field~$\xi^{i}$ at~$P$: we then form the divergence
\index{Vector!transference}%
of current-density,~$\vs \xi^{i}$:
\[
\frac{\dd(\vs \xi^{i})}{\dd x_{i}}
= \frac{\dd \vs}{\dd x_{i}}\, \xi^{i} + \vs\, \frac{\dd \xi^{i}}{\dd x_{i}}
= \left(\frac{\dd \vs}{\dd x_{i}} - \Gamma_{ir}^{r} \vs\right) \xi^{i}\Add{.}
\]
We thus get
\[
\frac{\dd \vs}{\dd x_{i}} - \Gamma_{ir}^{r} \vs
\]
as the components of a co-variant vector-density. To extend
differentiation beyond scalar tensor-densities to any tensor-densities
whatsoever, for example, to the mixed tensor-density~$\vw_{i}^{k}$ of the
second order, we again proceed, as has been done repeatedly above,
to make use of two stationary vector fields at~$P$, namely, $\xi^{i}$~and~$\eta^{i}$,
the latter being co-variant and the former contra-variant. We
differentiate the scalar-density $\vw_{i}^{k} \xi^{i}\eta_{k}$. If the tensor-density that
has been derived by differentiation be contracted with respect to
the symbol of differentiation and one of the contra-variant indices,
the divergence is again obtained.
\Section{15.}{Curvature}
If $P$~and~$P^{*}$ are two points connected by a curve, and if a vector
is given at~$P$, then this vector may be moved parallel to itself along
the curve from $P$ to~$P^{*}$. Equations~\Eq{(36)}, giving the unknown
components~$v^{i}$ of the vector which is being subjected to a continuous
parallel displacement, have, for given initial values of~$v^{i}$, one and
only one solution. The \Emph{vector transference} that comes about in
this way is in general \Emph{non-integrable}, that is, the vector which we
get at~$P^{*}$ is dependent on the path of the displacement along
which the transference is effected. Only in the particular case, in
which integrability occurs, is it allowable to speak of the \emph{same}
vector at two different points $P$~and~$P^{*}$; this comprises those
vectors that are generated from one another by parallel displacement.
Let such a manifold be called \Emph{Euclidean-affine}. If we
\PageSep{118}
\index{Equality!of vectors}%
subject all points of such a manifold to an infinitesimal displacement,
which is in each case representable by an ``\Emph{equal}'' infinitesimal
vector, then the space is said to have undergone an infinitesimal
\Emph{total translation}. With the help of this conception, and following
the line of reasoning of Chapter~I. (without entering on a rigorous
proof), we may construct ``linear'' co-ordinate systems which are
characterised by the fact that, in them, the same vectors have the
same components at different points of the systems. In a linear
co-ordinate system the components of the affine relationship vanish
identically. Any two such systems are connected by \Emph{linear}
formulæ of transformation. The manifold is then an affine space
in the sense of Chapter~I.: \emph{The integrability of the vector transference
is the infinitesimal geometrical property which distinguishes
``linear'' spaces among affinely related spaces.}
We must now turn our attention to the \Emph{general case}; it must
not be expected in this that a vector that has been taken round a
closed curve by parallel displacement finally returns to its initial
position. Just as in the proof of Stokes's Theorem, so here we
stretch a surface over the closed curve and divide it into infinitely
small parallelograms by parametric lines. The change in any
arbitrary vector after it has traversed the periphery of the surface
is reduced to the change effected after it has traversed each of the
infinitesimal parallelograms marked out by two line elements $dx_{i}$
and~$\delta x_{i}$ at a point~$P$. This change has now to be determined. We
shall adopt the convention that the amount $\Delta \vx = (\Delta \xi^{i})$, by which
a vector $\vx = \xi^{i}$ increases, is derived from~$\vx$ by a linear transformation,
% [** TN: Sans-serif "F" in the original]
a matrix~$\Delta \vF$, i.e.\
\[
\Delta \vx = \Delta \vF(\vx);\qquad
\Delta \xi^{\alpha} = \Delta F_{\beta}^{\alpha} · \xi^{\beta}\Add{.}
\Tag{(38)}
\]
If $\Delta \vF = 0$, then the manifold is ``\Emph{plane}'' at the point~$P$ in the
surface direction assumed by the surface element; if this is true
for all elements of a finitely extended portion of surface, then every
vector that is subjected to parallel displacement along the edge of
the surface returns finally to its initial position. $\Delta \vF$~is linearly
dependent on the element of surface:
\[
\Delta \vF = \vF_{ik}\, dx_{i}\, \delta x_{k}
= \tfrac{1}{2} \vF_{ik} \Delta x_{ik}\qquad
(\Delta x_{ik} = dx_{i}\, \delta x_{k} - dx_{k}\, \delta x_{i},
\]
and
\[
\vF_{ki} = -\vF_{ik})\Add{.}
\Tag{(39)}
\]
The differential form that occurs here characterises the \Emph{curvature},
\index{Curvature!generally@{(generally)}}%
\index{Curvature!vector}%
that is, the deviation of the manifold from plane-ness at the point~$P$
for all possible directions of the surface; since its co-efficients are
not numbers, but matrices, we might well speak of a ``linear
matrix-tensor of the second order,'' and this would undoubtedly
best characterise the quantitative nature of curvature. If, however,
\PageSep{119}
we revert from the matrices back to their components---supposing
$F_{\beta ik}^{\alpha}$~to be the components of~$\vF_{ik}$ or else the co-efficients
of the form
\[
\Delta F_{\beta}^{\alpha} = F_{\beta ik}^{\alpha}\, dx_{i}\, \delta x_{k}
\Tag{(40)}
\]
---then we arrive at the formula
\[
\Delta \vx\, F_{\beta ik}^{\alpha} \ve_{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}\Add{.}
\Tag{(41)}
\]
From this we see that the~$F_{\beta ik}^{\alpha}$'s are the components of a tensor of the
fourth order which is contra-variant in~$\alpha$ and co-variant in $\beta$,~$i$ and~$k$.
Expressed in terms of the components~$\Gamma_{rs}^{i}$ of the affine relationship,
it is
\[
F_{\beta ik}^{\alpha}
= \left(\frac{\dd \Gamma_{\beta k}^{\alpha}}{\dd x_{i}}
- \frac{\dd \Gamma_{\beta i}^{\alpha}}{\dd x_{k}}\right)
+ (\Gamma_{ri}^{\alpha} \Gamma_{\beta k}^{r}
- \Gamma_{rk}^{\alpha} \Gamma_{\beta i}^{r})\Add{.}
\Tag{(42)}
\]
According to this they fulfil the conditions of ``skew'' and
``cyclical'' symmetry, namely:---
\[
F_{\beta ki}^{\alpha} = -F_{\beta ik}^{\alpha};\qquad
F_{\beta ik}^{\alpha} + f_{ik\beta}^{\alpha} + F_{k\beta i}^{\alpha} = 0\Add{.}
\Tag{(43)}
\]
The vanishing of the curvature is the invariant differential law
which distinguishes Euclidean spaces among affine spaces in terms
\index{Euclidean!manifolds, Chapter I (from the point of view of infinitesimal geometry)}%
of general infinitesimal geometry.
To prove the statements above enunciated we use the same
process of sweeping twice over an infinitesimal parallelogram as
we used on \Pageref[p.]{107} to derive the curl tensor; we use the same notation
as on that occasion. Let a vector $\vx = \vx(P_{00})$ with components~$\xi^{i}$
be given at the point~$P_{00}$. The vector~$\vx(P_{10})$ that is derived
from~$\vx(P_{00})$ by parallel displacement along the line element~$dx$ is
attached to the end point~$P_{10}$ of the same line element. If the
%[** TN: "then" set on same line as displayed equation in the original]
components of~$\vx(P_{10})$ are $\xi^{i} + d\xi^{i}$ then
\[
d\xi^{\alpha} = -d\gamma_{\beta}^{\alpha}\, \xi^{\beta}
= -\Gamma_{\beta i}^{\alpha}\, \xi^{\beta}\, dx_{i}.
\]
Throughout the displacement~$\delta$ to which the line element~$dx$ is to
be subjected (and which need by no means be a parallel displacement)
let the vector at the end point be bound always by the
specified condition to the vector at the initial point. The $d\xi^{\alpha}$'s are
then increased, owing to the displacement, by an amount
\[
\delta d\xi^{\alpha}
= -\delta\Gamma_{\beta i}^{\alpha}\, dx_{i}\, \xi^{\beta}
- \Gamma_{\beta i}^{\alpha}\, \delta dx_{i}\, \xi^{\beta}
- d\gamma_{r}^{\alpha}\, \delta \xi^{r}.
\]
If, in particular, the vector at the initial point of the line element
remains parallel to itself during the displacement, then $\delta \xi^{r}$~must be
replaced in this formula by~$-\delta\gamma_{\beta}^{r}\, \xi^{\beta}$. In the final position $\Vector{P_{01} P_{11}}$
of the line element we then get, at the point~$P_{01}$, the vector~$\vx(P_{01})$,
which is derived from~$\vx(P_{00})$ by parallel displacement along~$\Vector{P_{00}P_{01}}$;
\PageSep{120}
at~$P_{11}$ we get the vector~$\vx(P_{11})$, into which $\vx(P_{01})$~is converted by
parallel displacement along~$\Typo{P_{01} P_{11}}{\Vector{P_{01} P_{11}}}$, and we have
\[
\delta d\xi^{\alpha}
= \bigl\{\xi^{\alpha}(P_{11}) - \xi^{\alpha}(P_{01})\bigr\}
- \bigl\{\xi^{\alpha}(P_{10}) - \xi^{\alpha}(P_{00})\bigr\}.
\]
If the vector that is derived from~$\vx(P_{10})$ by parallel displacement
along $\Vector{P_{10} P_{11}}$ is denoted by~$\vx_{*}P_{11}$, then, by interchanging $d$~and~$\delta$,
we get an analogous expression for
\[
d\delta \xi^{\alpha}
= \bigl\{\xi_{*}^{\alpha}(P_{11}) - \xi^{\alpha}(P_{10})\bigr\}
- \bigl\{\xi^{\alpha}(P_{01}) - \xi^{\alpha}(P_{00})\bigr\}.
\]
By subtraction we get
\begin{align*}
\Delta \xi^{\alpha}
&= \delta d\xi^{\alpha} - d\delta \xi^{\alpha} \\
&= \left\{
\begin{aligned}
&-\delta \Gamma_{\beta i}^{\alpha}\, dx_{i}
+ d\gamma_{r}^{\alpha}\, \delta\gamma_{\beta}^{r}
- \Gamma_{\beta i}^{\alpha}\, \delta dx_{i} \\
&+ d\Gamma_{\beta k}^{\alpha}\, \delta x_{k}
- d\gamma_{r}^{\alpha}\, d\gamma_{\beta}^{r}
+ \Gamma_{\beta i}^{\alpha}\, d\delta x_{i}
\end{aligned}
\right\} \xi^{\beta}.
\end{align*}
Since $\delta dx_{i} = d\delta x_{i}$ the two last terms on the right destroy one another,
and we are left with
\[
\Delta \xi^{\alpha} = \Delta F_{\beta}^{\alpha} · \xi^{\beta}
\]
in which the~$\Delta \xi^{\alpha}$'s are the components of a vector~$\Delta \vx$ at~$P_{11}$, which
is the difference of the two vectors $\vx$~and~$\vx_{*}$ \Emph{at the same point},
i.e.\
\[
-\Delta \xi^{\alpha} = \xi^{\alpha}(P_{11}) - \xi_{*}^{\alpha}(P_{11}).
\]
Since, when we proceed to the limit, $P_{11}$~coincides with $P = P_{00}$,
this proves the statements enunciated above.
%[** TN: [sic] "become"]
The foregoing argument, based on infinitesimals, become rigorous
as soon as we interpret $d$~and~$\delta$ in terms of the differentiations
$\dfrac{d}{ds}$~and~$\dfrac{d}{dt}$, as was done earlier. To trace the various stages of the
vector~$\vx$ during the sequence of infinitesimal displacements, we
may well adopt the following plan. Let us ascribe to every pair
of values $s$,~$t$, not only a point $P = (s\Com t)$, but also a co-variant vector
at~$P$ with components~$f_{i}(s\Com t)$. If $\xi^{i}$~is an arbitrary vector at~$P$,
then $d(f_{i} \xi^{i})$~signifies the value that $\dfrac{d(f_{i} \xi^{i})}{ds}$ assumes if $\xi^{i}$~is taken
along unchanged from the point~$(s\Com t)$ to the point $(s + ds, t)$. And
$d(f_{i} \xi^{i})$~is itself again an expression of the form~$f_{i} \xi^{i}$ excepting that
instead of~$f_{i}$ there are now other functions~$f_{i}'$ of $s$~and~$t$. We may,
therefore, again subject it to the same process, or to the analogous
one~$\delta$. If we do the latter, and repeat the whole operation in the
reverse order, and then subtract, we get
\[
\delta d(f_{i} \xi^{i})
= \delta df_{i}\Chg{ · }{\,} \xi^{i}
+ df_{i}\, \delta \xi^{i}
+ \delta \Typo{f}{f_{i}}\, d\xi^{i}
+ f_{i}\, \delta d\xi^{i},
\]
and then, since
\[
\delta df_{i} = \frac{d^{2} \Typo{f}{f_{i}}}{dt\, ds}
= \frac{d^{2} f_{i}}{ds\, dt}
= d\delta f_{i},
\]
\PageSep{121}
we have
\[
\Delta(f_{i} \xi^{i})
= (\delta d - d\delta)(f_{i} \xi^{i})
= f_{i}\, \Delta \xi^{i}.
\]
In the last expression $\Delta \xi^{i}$~is precisely the expression found above.
The invariant obtained is, for the point $P = (0\Com 0)$,
\[
F_{\beta ik}^{\alpha} f_{\alpha} \xi^{\beta} u^{i} v^{k}.
\]
It depends on an arbitrary co-variant vector with components~$f_{i}$ at
this point, and on three contra-variant vectors $\xi$,~$u$,~$v$; the $F_{\beta ik}^{\alpha}$'s
are accordingly the components of a tensor of the fourth order.
\Section{16.}{Metrical Space}
\Par{The Conception of Metrical Manifolds.}---A manifold \Emph{has a
\index{Distance (generally)}%
\index{Manifold!metrical}%
\index{Measure-index of a distance}%
\index{Metrics or metrical structure!(general)}%
\index{Perpendicularity}%
\index{Right angle}%
measure-determination at the point~$P$}, if the line elements at~$P$
may be compared with respect to length; we herein assume that
the Pythagorean law (of Euclidean geometry) is valid for infinitesimal
regions. \emph{Every vector~$\vx$ then defines a distance at~$P$;
and there is a non-degenerate quadratic form~$\vx^{2}$, such that $\vx$~and~$vy$
define the same distance if, and only if, $\vx^{2} = \vy^{2}$.} This postulate
determines the quadratic form fully, if a factor of proportionality
differing from zero be prefixed. The fixing of the latter serves to
\Emph{calibrate} the manifold at the point~$P$. We shall then call~$\vx^{2}$ the
measure of the vector~$\vx$, or since it depends only on the distance
defined by~$\vx$, we may call it the \Emph{measure~$l$ of this distance}.
Unequal distances have different measures; the distances at a
point~$P$ therefore constitute a one-dimensional totality. If we replace
this calibration by another, the new measure~$\bar{l}$ is derived
\index{Calibration}%
from the old one~$l$ by multiplying it by a constant factor $\lambda \neq 0$,
independent of the distance; that is, $\bar{l} = \lambda l$. The relations between
the measures of the distances are independent of the calibration.
So we see that just as the characterisation of a vector at~$P$
by a system of numbers (its components) depends on the choice
of the co-ordinate system, so the fixing of a distance by a number
depends on the calibration; and just as the components of a vector
undergo a homogeneous linear transformation in passing to another
co-ordinate system, so also the measure of an arbitrary distance
when the calibration is altered. We shall call two vectors $\vx$ and~$\vy$
(at~$P$), for which the symmetrical bilinear form~$\vx · \vy$ corresponding
to~$\vx^{2}$ vanishes, \Emph{perpendicular} to one another; this reciprocal relation\Pagelabel{121}
is not affected by the calibration factor. The fact that the
form~$\vx^{2}$ is definite is of no account in our subsequent mathematical
propositions, but, nevertheless, we wish to keep this case uppermost
in our minds in the sequel. If this form has $p$~positive and
$q$~negative dimensions ($p + q = n$), we say that the manifold is
$(p + q)$-dimensional at the point in question. If $p \neq q$ we
\PageSep{122}
fix the sign of the metrical fundamental form~$\vx^{2}$ once and for all
by the postulate that $p > q$; the calibration ratio~$\lambda$ is then always
positive. After choosing a definite co-ordinate system and a certain
calibration factor, suppose that, for every vector~$\vx$ with components~$\xi^{i}$,
we have
\[
\vx^{2} = \sum_{i\Com k} g_{ik} \xi^{i} \xi^{k}\qquad
(g_{ki} = g_{ik})\Add{.}
\Tag{(44)}
\]
\Emph{We now assume that our manifold has a measure-determination
at every point.} Let us calibrate it everywhere, and
insert in the manifold a system of $n$~co-ordinates~$x_{i}$---we must do
this so as to be able to express in numbers all quantities that
occur---then the~$g_{ik}$'s in~\Eq{(44)} are perfectly definite functions of the
co-ordinates~$x_{i}$; we assume that these functions are continuous
and differentiable. Since the determinant of the~$g_{ik}$'s vanishes at
no point, the integral numbers $p$ and~$q$ will remain the same in the
whole domain of the manifold; we assume that $p > q$.
For a manifold to be a metrical space, it is not sufficient for it
to have a measure-determination at every point; in addition, every
point must be \Emph{metrically related} to the domain surrounding it.
The conception of metrical relationship is analogous to that of
affine relationship; just as the latter treats of \Emph{vectors}, so the
former deals with distances. A point is thus metrically related to
the domain in its immediate neighbourhood, if the distance is
known to which every distance at~$P$ gives rise when it passes by a
congruent displacement from~$P$ to any point~$P'$ infinitely near~$P$.
The immediate vicinity of~$P$ may be calibrated in such a way that
the measure of any distance at~$P$ has undergone no change after
congruent displacements to infinitely near points. Such a calibration
is called \emph{geodetic} at~$P$. If, however, the manifold is
calibrated in any way, and if $l$~is the measure of any arbitrary
distance at~$P$, and $l + dl$~the measure of the distance at~$P'$ resulting
from a congruent displacement to the infinitely near point~$P'$,
there is necessarily an equation
\[
dl = -l\, d\phi
\Tag{(45)}
\]
in which the infinitesimal factor~$d\phi$ is independent of the displaced
distance, for the displacement effects a representation of the distances
at~$P$ similar to that at~$P'$. In~\Eq{(45)}, $d\phi$~corresponds to the~$d\gamma_{r}^{i}$'s
in the formula for vector displacements~\Eq{(33)}. If the calibration
is altered at~$P$ and its neighbouring points according to the
formula $\bar{l} = l\lambda$ (the calibration ratio~$\lambda$ is a positive function of the
position), we get in place of~\Eq{(45)}
\PageSep{123}
\[
d\bar{l} = -\bar{l}\, d\bar{\phi}
\text{ in which }
d\bar{\phi} = d\phi - \frac{d\lambda}{\lambda}\Add{.}
\Tag{(46)}
\]
The necessary and sufficient condition that an appropriate value of~$\lambda$
make $d\bar{\phi}$~vanish identically at~$P$ with respect to the infinitesimal
displacement $\Vector{PP'} = (dx_{i})$ is clearly that $d\phi$~must be a differential
form, that is,
\[
d\phi = \phi_{i}\, dx_{i}\Add{.}
\Tag{(45')}
\]
The inferences that may be drawn from the postulate enunciated
at the outset are exhausted in \Eq{(45)}~and~\Eq{(45')}. (In short, the~$\phi_{i}$'s
are definite numbers at the point~$P$. If $P$~has co-ordinates $x_{i} = \Typo{o}{0}$,
we need only assume $\log \lambda$ equal to the linear function $\sum \phi_{i} x_{i}$ to
get $d\phi = \Typo{o}{0}$ there.) All points of the manifold are identical as
regards the measure-determinations governing each and as regards
their metrical relationship with their neighbouring points. Yet,
according as $n$~is even or odd, there are respectively $\dfrac{n}{2} + 1$ or $\dfrac{n + 1}{2}$
different types of metrical manifolds which are distinguishable from
one another by the inertial index of the metrical groundform. One
kind, with which we shall occupy ourselves particularly, is given
by the case in which $p = n$, $q = \Typo{o}{0}$ (or $p = \Typo{o}{0}$, $q = n$); other cases
are $p = n - 1$, $q = 1$ (or $p = 1$, $q = n - 1$), or $p = n - 2$, $q = 2$
(or $p = 2$, $q = n - 2$), and so forth.
We may summarise our results thus. \emph{The metrical character
of a manifold is characterised relatively to a system of reference $(=
\text{co-ordinate system} + \text{calibration})$ by two fundamental forms,
namely, a quadratic differential form $Q = \sum_{i\Com k} g_{ik}\, dx_{i}\, dx_{k}$ and a linear
one $d\phi = \sum_{i} \phi_{i}\, dx_{i}$. They remain invariant during transformations
to new co-ordinate systems. If the calibration is changed, the first
form receives a factor~$\lambda$, which is a positive function of position with
continuous derivatives, whereas the second function becomes diminished
by the differential of~$\log \lambda$.} Accordingly all quantities
or relations that represent metrical conditions analytically must
contain the functions~$g_{ik} \phi_{i}$ in such a way that invariance holds
(1)~for any transformation of co-ordinate (\emph{co-ordinate invariance}),
(2)~for the substitution which replaces $g_{ik}$~and $\phi_{i}$ respectively by
\[
\lambda · g_{ik}\quad\text{and}\quad
\phi_{i} - \frac{1}{\lambda} · \frac{\dd\lambda}{\dd x_{i}}
\]
\PageSep{124}
no matter, in~\Eq{(2)}, what function of the co-ordinates $\lambda$~may be.
(This may be termed \emph{calibration invariance}.)
In the same way as in §\,15, in which we determined the change
\index{Axioms!of metrical geometry!(infinitesimal)}%
\index{Normal calibration of Riemann's space}%
in a vector which, remaining parallel to itself, traverses the periphery
of an infinitesimal parallelogram bounded by $dx_{i}$,~$\delta x_{i}$, so here
we calculate the change~$\Delta l$ in the measure~$l$ of a distance subjected
to an analogous process. Making use of $dl = -l\, d\phi$ we get
\[
\delta dl = -\delta l\, d\phi - l\, \delta d\phi
= l\, \delta\phi\, d\phi - l\, \delta d\phi,
\]
%[** TN: "i.e." and "where" on same line as next equation in the original]
i.e.\
\[
\Delta l = \delta dl - d\delta l
= -l\, \Delta\phi
\]
where
\[
\Delta\phi = (\delta d - d\delta)\phi
= f_{ik}\, dx_{i}\, \delta x_{k}\quad\text{and}\quad
f_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}\Add{.}
\Tag{(47)}
\]
Hence we may call the linear tensor of the second order with components~$f_{ik}$
the \emph{distance curvature} of metrical space as an analogy
\index{Curvature!distance}%
to the \emph{vector curvature} of affine space, which was derived in~§\,15.
Equation~\Eq{(46)} confirms analytically that the distance curvature is
independent of the calibration; it satisfies the equations of invariance
\[
\frac{\dd f_{kl}}{\dd x_{i}} +
\frac{\dd f_{li}}{\dd x_{k}} +
\frac{\dd f_{ik}}{\dd x_{l}} = 0.
\]
\emph{Its vanishing is the necessary and sufficient condition that every
distance may be transferred from its initial position, in a manner
independent of the path, to all points of the space.} This is the only
case that Riemann considered. If metrical space is a \Emph{Riemann
space}, there is meaning in speaking of the \Emph{same} distance at different
points of space; the manifold may then be calibrated (\emph{normal
calibration}) so that $d\phi$~vanishes identically. (Indeed, it follows
from $f_{ik} = 0$, that $d\phi$~is a total differential, namely, the differential
of a function~$\log \lambda$; by re-calibrating in the calibration ratio~$\lambda$, $d\phi$~may
then be made equal to zero everywhere.) In normal calibration
the metrical groundform~$Q$ of Riemann's space is determined
except for an arbitrary \Emph{constant} factor, which may be fixed by
choosing once and for all a unit distance (no matter at which
point; the normal meter may be transported to any place).
\Par{The Affine Relationship of a Metrical Space.}---We now
arrive at a fact, which may almost be called the \Emph{key-note of
infinitesimal geometry}, inasmuch as it leads the logic of
geometry to a wonderfully harmonious conclusion. In a metrical
space the conception of infinitesimal parallel displacements may
be given in only one way if, in addition to our previous postulate,\Pagelabel{124}
it is also to satisfy the almost self-evident one: \emph{parallel displacement
of a vector must leave unchanged the distance which it determines.
Thus, the principle of transference of distances or lengths
\PageSep{125}
%[** TN: Next line unitalicized in the original]
which is the basis of metrical geometry, carries with it a
principle of transference of direction}; in other words, \Emph{an affine
\index{Affine!relationship of a metrical space}%
relationship is inherent in metrical space}.\Pagelabel{125}
\Proof.---We take a definite system of reference. In the case
of all quantities~$\Typo{\alpha}{a}^{i}$ which carry an upper index~$i$ (not necessarily
excluding others) we shall define the lowering of the index by
equations
\[
\Typo{\alpha}{a}_{i} = \sum_{j} g_{ij} a^{j} %[** TN: RHS OK in the original!]
\]
and the reverse process of raising the index by the corresponding
inverse equations. If the vector~$\xi^{i}$ at the point $P = (x_{i})$ is to be
transformed into the vector $\xi^{i} + d\xi^{i}$ at $P' \Typo{(= x_{i} + dx_{i})}{= (x_{i} + dx_{i})}$ by the
parallel displacement to~$P'$ which we are about to explain, then
\[
d\xi^{i} = -d\gamma_{k}^{i}\, \xi^{k},\qquad
d\gamma_{k}^{i} = \Gamma_{kr}^{i}\, dx_{r},
\]
and the equation
\[
dl = -l\, d\phi
\]
must hold for the measure
\[
l = g_{ik} \xi^{i} \xi^{k}
\]
according to the postulate enunciated, and this gives
\[
2\xi_{i}\, d\xi^{i} + \Typo{\xi}{\xi^{i}}\xi^{k}\, dg_{ik}
= -(g_{ik} \xi^{i} \xi^{k})\, d\phi.
\]
The first term on the left
\[
= -2\xi_{i} \xi^{k}\, d\gamma_{k}^{i}
= -2\xi^{i} \xi^{k}\, d\gamma_{ik}
= -\xi^{i} \xi^{k} (d\gamma_{ik} + d\gamma_{ki}).
\]
Hence we get
\[
d\gamma_{ik} + d\gamma_{ki} = dg_{ik} + g_{ik}\, d\phi\Add{,}
\]
or
\[
\Gamma_{i,kr} + \Gamma_{k,ir} = \frac{\dd g_{ik}}{\dd x_{r}} + g_{ik} \phi_{r}\Add{.}
\Tag{(48)}
\]
By interchanging the indices $i\Com k\Com r$ cyclically, then adding the last
two and subtracting the first from the resultant sum, we get, bearing
in mind that the~$\Gamma$'s must be symmetrical in their last two
indices,
\[
\Gamma_{r,ik}
= \tfrac{1}{2} \left(
\frac{\dd g_{ir}}{\dd x_{k}}
+ \frac{\dd g_{kr}}{\dd x_{i}}
- \frac{\dd g_{ik}}{\dd x_{r}}\right)
+ \tfrac{1}{2}(g_{ir} \phi_{k} + g_{kr} \phi_{i} - g_{ik} \phi_{r})\Add{.}
\Tag{(49)}
\]
From this the~$\Gamma_{ik}^{r}$ are determined according to the equation
\[
\Gamma_{r,ik} = g_{rs} \Gamma_{ik}^{s}\quad
\text{ or, explicitly,}\quad
\Gamma_{ik}^{r} = g^{rs} \Gamma_{s,ik}\Add{.}
\Tag{(50)}
\]
These components of the affine relationship fulfil all the postulates
that have been enunciated. It is in the nature of metrical space to
be furnished with this affine relationship; in virtue of it the whole
analysis of tensors and tensor-densities with all the conceptions
\PageSep{126}
\index{Direction-curvature}%
worked out above, such as geodetic line, curvature, etc., may be
\index{Curvature!direction}%
applied to metrical space. If the curvature vanishes identically,
the space is metrical and Euclidean in the sense of Chapter~I\@.
In the case of \Emph{vector curvature} we have still to derive an important
\index{Vector!curvature}%
decomposition into components, by means of which we
prove that distance curvature is an inherent constituent of the
former. This is quite to be expected since vector transference is
automatically accompanied by distance transference. If we use the
symbol $\Delta = \delta d - d\delta$ relating to parallel displacement as before,
then the measure~$l$ of a vector~$\xi^{i}$ satisfies
\[
\Delta l = -l\, \Delta\phi,\qquad
\Delta \xi_{i} \xi^{i} = -(\xi_{i} \xi^{i})\, \Delta\phi\Add{.}
\Tag{(47)}
\]
Just as we found for the case in which $f_{i}$~are any functions of
position that
\[
\Delta(f_{i} \xi^{i}) = f_{i}\, \Delta \xi^{i}
\]
so we see that
\[
\Delta(\xi_{i} \xi^{i}) = \Delta(g_{ik} \xi^{i} \xi^{k})
= g_{ik}\, \Delta \xi^{i} · \xi^{k}
+ g_{ik} \xi^{i} · \Delta \xi^{k}
= 2\xi_{i}\, \Delta \xi^{i}\Add{,}
\]
and equation~\Eq{(47)} then leads to the following result. If for the
vector $\vx = (\xi^{i})$ we set
\[
\Delta \vx = *\Delta \vx - \vx · \tfrac{1}{2} \Delta \phi,
\]
then $\Delta \vx$ appears split up into a component at right angles to~$\vx$ and
another parallel to~$\vx$, namely, $*\Delta \vx$~and $-\vx · \frac{1}{2} \Delta \phi$ respectively. This
is accompanied by an analogous resolution of the curvature tensor,
i.e.\
\[
F_{\beta ik}^{\alpha}
= *F_{\beta ik}^{\alpha} - \tfrac{1}{2} \delta_{\beta}^{\alpha} \Typo{f^{ik}}{f_{ik}}\Add{.}
\Tag{(51)}
\]
The first component~$*F$ will be called ``\Emph{direction curvature}''; it
is defined by
\[
*\Delta \vx = *F_{\beta ik}^{\alpha} \ve_{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}.
\]
The perpendicularity of~$*\Delta \vx$ to~$\vx$ is expressed by the formula
\[
*F_{\beta ik}^{\alpha} \xi_{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}
= *F_{\alpha\beta ik} \xi^{\alpha} \xi^{\beta}\, dx_{i}\, \delta x_{k}
= 0.
\]
The system of numbers~$*F_{\alpha\beta ik}$ is skew-symmetrical not only with
respect to $i$~and~$k$ but also with respect to the index pair $\alpha$~and~$\beta$.
In consequence we have also, in particular,
\[
*F_{\alpha ik}^{\alpha} = 0.
\]
\Par{Corollaries.}---If the co-ordinate system and calibration around
a point~$P$ is chosen so that they are geodetic at~$P$, then we have,
at~$P$, $\phi_{i} = 0$, $\Gamma_{ik}^{r} = 0$, or, according to \Eq{(48)}~and~\Eq{(49)}, the equivalent
\[
\phi_{i} = 0,\qquad
\frac{\dd g_{ik}}{\dd x_{r}} = 0.
\]
\PageSep{127}
\index{Calibration!(geodetic)}%
\index{Geodetic calibration}%
\index{Geodetic calibration!null-line}%
\index{Geodetic calibration!systems of reference}%
\index{Null-lines, geodetic}%
The linear form~$d\phi$ vanishes at~$P$ and the co-efficients of the
quadratic groundform become stationary; in other words, those
conditions come about at~$P$, which are obtained in Euclidean space
simultaneously for all points by a single system of reference. This
results in the following explicit definition of the parallel displacement
of a vector in metrical space. A geodetic system of reference
at~$P$ may be recognised by the property that the~$\phi_{i}$'s at~$P$ vanish
relatively to it and the~$g_{ik}$'s assume stationary values. A vector is
displaced from~$P$ parallel to itself to the infinitely near point~$P'$ by
leaving its components in \Emph{a system of reference belonging to~$P$}
unaltered. (There are always geodetic systems of reference; the
\index{Systems of reference!geodetic}%
choice of them does not affect the conception of parallel displacements.)
Since, in a \Emph{translation} $x_{i} = x_{i}(s)$, the velocity vector $u_{i} = \dfrac{dx_{i}}{ds}$
moves so that it remains parallel to itself, it satisfies
\[
\frac{d(u_{i} u^{i})}{ds} + (u_{i} u^{i})(\phi_{i} u^{i}) = 0\quad
\text{in metrical geometry}\Add{.}
\Tag{(52)}
\]
If at a certain moment the~$u^{i}$'s have such values that $u_{i} u^{i} = 0$ (a
case that may occur if the quadratic groundform~$Q$ is indefinite),
then this equation persists throughout the whole translation: we
shall call the trajectory of such a translation a \Emph{geodetic null-line}.
An easy calculation shows that the geodetic null-lines do not alter
if the metric relationship of the manifold is changed in any way, as
long as the measure-determination is kept fixed at every point.\Pagelabel{127}
\Par{Tensor Calculus.}---It is an essential characteristic of a tensor
\index{Weight of tensors and tensor-densities}%
that its components depend only on the co-ordinate system and not
on the calibration. In a generalised sense we shall, however, also
call a linear form which depends on the co-ordinate system and the
\Emph{calibration} a tensor, if it is transformed in the usual way when
the co-ordinate system is changed, but becomes multiplied by the
factor~$\lambda^{e}$ (where $\lambda = \text{the calibration ratio}$) when the calibration is
changed; we say that it is of \Emph{weight~$e$}. Thus the~$g_{ik}$'s are components
of a symmetrical co-variant tensor of the second order and
of weight~$1$. Whenever tensors are mentioned without their weight
being specified, we shall take this to mean that those of weight~$0$
are being considered. The relations which were discussed in tensor
analysis are relations, which are independent of calibration and
co-ordinate system, between tensors and tensor-densities \Emph{in this
special sense}. We regard the extended conception of a tensor,
and also the analogous one of tensor-density of weight~$e$, merely as
an auxiliary conception, which is introduced to simplify calculations.
They are convenient for two reasons: (1)~They make it possible to
\PageSep{128}
``juggle with indices'' in this extended region. By lowering a contra-variant
index in the components of a tensor of weight~$e$ we get the
components of a tensor of weight~$e + 1$, the components being co-variant
with respect to this index. The process may also be carried
out in the reverse direction. (2)~Let $g$ denote the determinant of
the~$g_{ik}$'s, furnished with a plus or minus sign according as the
number~$g$ of the negative dimensions is even or uneven, and let $\sqrt{g}$~be
the positive root of this positive number~$g$. Then, \Emph{by multiplying
any tensor by~$\sqrt{g}$ we get a tensor-density whose weight
is $\dfrac{n}{2}$~more than that of the tensor}; from a tensor of weight~$-\dfrac{n}{2}$
we get, in particular, a tensor-density in the true sense. The
proof is based on the evident fact that $\sqrt{g}$~is itself a scalar-density
of weight~$\dfrac{n}{2}$. We shall always indicate when a quantity is multiplied
by~$\sqrt{g}$ by changing the ordinary letter which designates the
quantity into the corresponding one printed in Clarendon type.
Since, in Riemann's geometry, the quadratic groundform~$Q$ is fully
\index{Geodetic calibration!line (general)!(in Riemann's space}%
determined by normal calibration (we need not consider the arbitrary
\Emph{constant} factor), the difference in the weights of tensors disappears
here: since, in this case, every quantity that may be
represented by a tensor may also be represented by the tensor-density
that is derived from it by multiplying it by~$\sqrt{g}$, the difference
between tensors and tensor-densities (as well as between
co-variant and contra-variant) is effaced. This makes it clear why
for a long time tensor-densities did not come into their right as
compared with tensors. The main use of tensor calculus in
geometry is an \Emph{internal} one, that is, to construct fields that are
derived invariantly from the metrical structures. We shall give
two examples that are of importance for later work. Let the
metrical manifold be $(3 + 1)$-dimensional, so that $-g$~will be
the determinant of the~$g_{ik}$'s. In this space, as in every other, the
distance curvature with components~$f_{ik}$ is a true linear tensor
field of the second order. From it is derived the contra-variant
tensor~$f^{ik}$ of weight~$-2$, which, on account of its weight differing
from zero, is of no actual importance; multiplication by~$\sqrt{g}$ leads
to~$\vf^{ik}$, a true linear tensor-density of the second order.
\[
\vl = \tfrac{1}{4} f_{ik} \vf^{ik}
\Tag{(53)}
\]
is the simplest scalar-density that can be formed; consequently
$\Dint \vl\, dx$ is the simplest invariant integral associated with the metrical
basis of a $(3 + 1)$-dimensional manifold. On the other hand, the
\PageSep{129}
integral $\Dint \sqrt{g}\, dx$, which occurs in Riemann's geometry as ``volume,''
is meaningless in general geometry. We can derive the intensity
of current (vector-density) from~$\vf^{ik}$ by means of the operation
divergence thus:
\[
\frac{\dd \vf^{ik}}{\dd x_{k}} = \vs^{i}.
\]
In physics, however, we use the tensor calculus not to describe the
metrical condition but to describe fields expressing physical states
in metrical space---as, for example, the electromagnetic field---and
to set up the laws that hold in them. Now, we shall find at the
close of our investigations that this distinction between physics and
geometry is false, and that physics does not extend beyond geometry.
The world is a $(3 + 1)$-dimensional metrical manifold, and all
physical phenomena that occur in it are only modes of expression
of the metrical field. In particular, the affine relationship of the
world is nothing more than the gravitational field, but its metrical
character is an expression of the state of the ``æther'' that fills the
world; even matter itself is reduced to this kind of geometry and
loses its character as a permanent substance. Clifford's prediction,
in an article of the \Title{Fortnightly Review} of~1875, becomes confirmed
here with remarkable accuracy; in this he says that ``the
theory of space curvature hints at a possibility of describing matter
and motion in terms of extension only''.
These are, however, as yet dreams of the future. For the
present, we shall maintain our view that physical states are foreign
states in space. Now that the principles of infinitesimal geometry
have been worked out to their conclusion, we shall set out, in the
next paragraph, a number of observations about the special case of
Riemann's space and shall give a number of formulæ which will
be of use later.
\Section{17.}{Observations about Riemann's Geometry as a Special
Case}
General tensor analysis is of great utility even for Euclidean
geometry whenever one is obliged to make calculations, not in a
Cartesian or affine co-ordinate system, but in a curvilinear co-ordinate
system, as often happens in mathematical physics. To
illustrate this application of the tensor calculus we shall here
write out the fundamental equations of the electrostatic and the
magnetic field due to stationary currents in terms of general curvilinear
co-ordinates.
Firstly, let $E_{i}$ be the components of the electric intensity of field
\PageSep{130}
\index{Maxwell's!application of stationary case to Riemann's space}%
in a Cartesian co-ordinate system. By transforming the quadratic
and the linear differential forms
\[
ds^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}\qquad
E_{1}\, dx_{1} + E_{2}\, dx_{2} + E_{3}\, dx_{3}
\]
respectively, into terms of arbitrary curvilinear co-ordinates (again
denoted by~$x_{i}$), each form being independent of the Cartesian co-ordinate
system, suppose we get
\[
ds^{2} = g_{ik}\, dx_{i}\, dx_{k}\quad\text{and}\quad E_{i}\, dx_{i}.
\]
Then the $E_{i}$'s are in every co-ordinate system the components of
the same co-variant vector field. From them we form a vector-density
with components
\[
\vE^{i} = \sqrt{g} · g^{ik} E_{k}\qquad
(g = |g_{ik}|).
\]
We transform the potential~$-\phi$ as a scalar into terms of the new
co-ordinates, but we define the density~$\rho$ of electricity as being the
electric charge given by $\Dint \rho\, dx_{1}\, dx_{2}\, dx_{3}$ contained in any portion of
space; $\rho$~is not then a scalar but a scalar density. The laws are
expressed by
\[
\left.
\begin{gathered}
E_{i} = \frac{\dd \phi}{\dd x_{i}}\qquad
\frac{\dd E_{i}}{\dd x_{k}} - \frac{\dd E_{k}}{\dd x_{i}} = 0 \\
\frac{\dd \vE^{i}}{\dd x_{i}} = \rho \\
\vS_{i}^{k} = E_{i} \vE^{k} - \tfrac{1}{2}\delta_{i}^{k} \vS,
\end{gathered}
\right\}
\Tag{(54)}
\]
in which $\vS$, $= E_{i} \vE^{i}$, are the components of a mixed tensor-density
of the second order, namely, the potential difference. The proof is
sufficiently indicated by the remark that these equations, in the
form we have written them, are absolutely invariant in character,
but pass into the fundamental equations, which were set up earlier,
for a Cartesian co-ordinate system.
The magnetic field produced by stationary currents was characterised
in Cartesian co-ordinate systems by an invariant skew-symmetrical
bilinear form~$H_{ik}\, dx_{i}\, \delta x_{k}$. By transforming the latter
into terms of arbitrary curvilinear co-ordinates, we get~$H_{ik}$, the
components of a linear tensor of the second order, namely, of the
\emph{magnetic field}, these components being co-variant with respect to
arbitrary transformations of the co-ordinates. Similarly, we may
deduce the components~$\phi_{i}$ of the vector potential as components of
a co-variant vector field in any curvilinear co-ordinate system. We
now introduce a linear tensor-density of the second order by means
of the equations
\[
\vH^{ik} = \sqrt{g} · g^{i\alpha} g^{k\beta} H_{\alpha\beta}.
\]
\PageSep{131}
The laws are then expressed by
\[
\left.
\begin{gathered}
H_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
\quad\text{or}\quad
\frac{\dd H_{kl}}{\dd x_{i}} +
\frac{\dd H_{li}}{\dd x_{k}} +
\frac{\dd H_{ik}}{\dd x_{l}} = 0 \\
\text{respectively\Add{,}} \\
\frac{\dd \vH^{ik}}{\dd x_{k}} = \vs^{i}\Add{,} \\
\vS_{i}^{k} = H_{ir} \vH^{kr} - \tfrac{1}{2} \delta_{i}^{k} \vS\Add{,}\qquad
\vS = \tfrac{1}{2} H_{ik} \vH^{ik}\Add{.}
\end{gathered}
\right\}
\Tag{(55)}
\]
The~$\vs^{i}$'s are the components of a vector-density, the electric \emph{intensity
of current}; the potential differences~$\vS_{i}^{k}$ have the same invariant
\index{Current!electric}%
\index{Electrical!current}%
character as in the electric field. These formulæ may be specialised
for the case of, for example, spherical and cylindrical co-ordinates.
No further calculations are required to do this, if we
have an expression for~$ds^{2}$, the distance between two adjacent
points, expressed in these co-ordinates; this expression is easily
obtained from considerations of infinitesimal geometry.
It is a matter of greater fundamental importance that \Eq{(54)}~and
\Eq{(55)} furnish us with the underlying laws of stationary electromagnetic
fields if unforeseen reasons should compel us to give up
the use of Euclidean geometry for physical space and replace it by
\Emph{Riemann's geometry} with a new groundform. For even in the
case of such generalised geometric conditions our equations, in
virtue of their invariant character, represent statements that are
independent of all co-ordinate systems, and that express formal
relationships between charge, current, and field. In no wise can
it be doubted that they are the direct transcription of the laws of
the stationary electric field that hold in Euclidean space; it is
indeed astonishing how simply and naturally this transcription is
effected by means of the tensor calculus. The question whether
space is Euclidean or not is quite irrelevant for the laws of the
electromagnetic field. The property of being Euclidean is expressed
in a universally invariant form by differential equations
of the second order in the~$g_{ik}$'s (denoting the vanishing of the
curvature) but only the~$g_{ik}$'s and their first derivatives appear in
these laws. It must be emphasised that a transcription of such
a simple kind is possible only for laws dealing with \Emph{action at
infinitesimal distances}. To derive the laws of action at a
distance corresponding to Coulomb's, and Biot and Savart's Law
from these laws of contiguous action is a purely mathematical
problem that amounts in essence to the following. In place of the
usual potential equation $\Delta \phi = 0$ we get as its invariant generalisation
(\textit{vide}~\Eq{(54)}) in Riemann's geometry the equation
\PageSep{132}
\[
\frac{\dd}{\dd x_{i}}\left(\sqrt{g} · g^{ik} \frac{\dd \phi}{\dd x_{k}}\right) = 0
\]
that is, a linear differential equation of the second order whose
co-efficients are, however, no longer constants. From this we are
to get the ``standard solution,'' tending to infinity, at any arbitrary
given point; this solution corresponds to the ``standard solution''~$\dfrac{1}{r}$
of the potential equation. It presents a difficult mathematical
problem that is treated in the theory of linear partial differential
equations of the second order. The same problem is presented
when we are limited to Euclidean space if, instead of investigating
events in empty space, we have to consider those taking place in a
non-homogeneous medium (for example, in a medium whose dielectric
constant varies at different places with the time). Conditions
are not so favourable for transcribing electromagnetic laws,
if real space should become disclosed as a metrical space of a still
more general character than Riemann assumed. In that case it
would be just as inadmissible to assume the possibility of a calibration
that is independent of position in the case of currents and
charges as in the case of distances. Nothing is gained by pursuing
this idea. The true solution of the problem lies, as was indicated
in the concluding words of the previous paragraph, in quite another
direction.
Let us rather add a few observations about \Emph{Riemann's space
\index{Riemann's!curvature}%
\index{Riemann's!space}%
as a special case}. Let the unit measure ($1$~centimetre) be chosen
once and for all; it must, of course, be the same at all points. The
metrical structure of the Riemann space is then described by an
invariant quadratic differential form $g_{ik}\, dx_{i}\, dx_{k}$ or, what amounts
to the same thing, by a co-variant symmetrical tensor field of the
second order. The quantities~$\phi_{i}$, that are now equal to zero, must
be struck out everywhere in the formulæ of general metrical
geometry. Thus, the components of the affine relationship,
which here bear the name ``Christoffel three-indices symbols'' and
\index{Christoffel's $3$-indices symbols}%
are usually denoted by $\dChr{ik}{r}$, are determined from
\[
\Chrsq{ik}{r}
= \tfrac{1}{2}\left(\frac{\dd g_{ir}}{\dd x_{k}}
+ \frac{\dd g_{kr}}{\dd x_{i}}
- \frac{\dd g_{ik}}{\dd x_{r}}\right),
\qquad
\Chr{ik}{r} = g^{rs} \Chrsq{ik}{s}\Add{.}
\Tag{(56)}
\]
(We give way to the usual nomenclature---although it disagrees
flagrantly with our own convention regarding rules about the
position of indices---so as to conform to the usage of the text-books.)
\PageSep{133}
\begin{Remark}
The following formulæ are now tabulated for future reference:---
\begin{gather*}
\frac{1}{\sqrt{g}}\, \frac{\dd \sqrt{g}}{\dd x_{i}} - \Chr{ir}{r} = 0\Add{,}
\Tag{(57)}\displaybreak[0] \\
\frac{1}{\sqrt{g}}\, \frac{\Typo{(\dd \sqrt{g} · g^{ik})}{\dd (\sqrt{g} · g^{ik})}}{\dd x_{k}} + \Chr{rs}{i} g^{rs} = 0\Add{,}
\Tag{(57')}\displaybreak[0] \\
\frac{1}{\sqrt{g}}\, \frac{\dd (\sqrt{g} · g^{ik})}{\dd x_{l}}
+ \Chr{lr}{i} g^{rk} + \Chr{lr}{k} g^{ri} - \Chr{lr}{r} g^{ik} = 0\Add{.}
\Tag{(57'')}
\end{gather*}
These equations hold because $\sqrt{g}$~is a scalar and $\sqrt{g} · g^{ik}$~is a tensor-density;
hence, according to the rules given by the analysis of tensor-densities, the left-hand
members of these equations, multiplied by~$\sqrt{g}$, are likewise tensor-densities.
If, however, we use a co-ordinate system $\left(\dfrac{\dd g^{ik}}{\dd x_{r}}\right) = 0$, which is geodetic at~$P$, then
all terms vanish. Hence, in virtue of the invariant nature of these equations,
they also hold in every other co-ordinate system. Moreover,
\[
\frac{dg}{g} = g^{ik}\, dg_{ik},\qquad
\frac{d\sqrt{g}}{\sqrt{g}} = \tfrac{1}{2} g^{ik}\, dg_{ik}\Add{.}
\Tag{(58)}
\]
For the total differential of a determinant with $n^{2}$ (independent and variable)
elements~$g_{ik}$ is equal to~$G^{ik}\, dg_{ik}$, where $G^{ik}$~denotes the minor of~$g_{ik}$. If $\vt^{ik}$ ($= \vt^{ki}$)\Typo{.}{}
is any symmetrical system of numbers, then we always have
\[
\vt^{ik}\, dg_{ik} = -\vt_{ik}\, dg^{ik}\Add{.}
\Tag{(59)}
\]
From
\[
g_{ij} g^{jk} = \delta_{i}^{k}
\]
it follows that
\[
g_{ij}\, dg^{jk} = -g^{jk}\, dg_{ij}.
\]
If these equations are multiplied by~$\vt_{k}^{i}$ (this symbol cannot be misinterpreted
% [** TN: Displayed in the original]
since $\vt_{k}^{i} = g_{kl} \vt^{il} = g_{kl} \vt^{li} = \vt_{k}^{i}$)
the required result follows. In particular, in place of~\Eq{(58)} we may also write
\[
\frac{dg}{g} = -g_{ik}\, dg^{ik}\Add{.}
\Tag{(58')}
\]
The co-variant \Emph{components $R_{\alpha\beta ik}$ of curvature} in Riemann's space,
which we denote by~$R$ instead of~$F$, satisfy the conditions of symmetry
\begin{gather*}
R_{\alpha\beta ki} = -R_{\alpha\beta ik},\qquad
R_{\beta\alpha ki} = -R_{\alpha\beta ik}, \\
R_{\alpha\beta ki} + R_{\alpha ik \beta} + R_{\alpha k \beta i} = 0,
\end{gather*}
(for the ``distance curvature'' vanishes). It is easy to show that, from them, it
follows that (\textit{vide} \FNote{11})
\[
R_{ik \alpha\beta} = R_{\alpha\beta ik}.
\]
As the result of an observation on \Pageref{57}, it follows that all those conditions taken
together enable us to characterise the curvature tensor completely by means of a
quadratic form that is dependent on an arbitrary element of surface, namely,
\[
\tfrac{1}{4} R_{\alpha\beta ik}\, \Delta x_{\alpha\beta}\, \Delta x_{ik}\qquad
(\Delta x_{ik} = dx_{i}\, \delta x_{k} - dx_{k}\, \delta x_{i}).
\]
If this quadratic form is divided by the square of the magnitude of the surface
element, the quotient depends only on the ratio of the~$\Delta x_{ik}$'s, i.e.\ on the position
\PageSep{134}
of the surface element; Riemann calls this number the curvature of the space
\index{Curvature!scalar of}%
at the point~$P$ in the surface direction in question. In two-dimensional
Riemann space (on a surface) there is only one surface direction and the
tensor degenerates into a scalar (Gaussian curvature). In Einstein's theory of
gravitation the contracted tensor of the second order
\[
R_{i\alpha k}^{\alpha} = R_{ik}
\]
which is symmetrical in Riemann's space, becomes of importance: its
components are
\[
R_{ik} = \frac{\dd}{\dd x_{r}} \Chr{ik}{r} - \frac{\dd}{\dd x_{k}} \Chr{ir}{r}
+ \Chr{ik}{r} \Chr{rs}{s} - \Chr{ir}{s} \Chr{ks}{r}\Add{.}
\Tag{(60)}
\]
Only in the case of the second term on the right, the symmetry with respect to
$i$~and~$k$ is not immediately evident; according to~\Eq{(57)}, however, it is equal to
\[
\tfrac{1}{2}\, \frac{\dd^{2} (\log g)}{\dd x_{i}\, \dd x_{k}}.
\]
Finally, by applying contraction once more we may form the \Emph{scalar of
curvature}
\[
R = g^{ik} R_{ik}.
\]
In general metrical space the analogously formed scalar of curvature~$F$ is
expressed in the following way (as is easily shown) by the Riemann expression~$R$,
which is dependent only on the~$g_{ik}$'s and which has no distinct meaning in
that space:---
\[
F = R - (n - 1) \frac{1}{\sqrt{g}}\, \frac{\dd (\sqrt{g} \phi^{i})}{\dd x_{i}}
- \frac{(n - 1)(n - 2)}{4} (\phi_{i} \phi^{i})\Add{.}
\Tag{(61)}
\]
$F$~is a scalar of weight~$-1$. Hence, in a region in which $F \neq 0$ we may define a
unit of length by means of the equation $F = \text{constant}$. This is a remarkable result
inasmuch as it contradicts in a certain sense the original view concerning the
transference of lengths in general metrical space, according to which a direct
comparison of lengths at a distance is not possible; it must be noticed, however,
that the unit of length which arises in this way is dependent on the conditions
of curvature of the manifold. (The existence of a unique uniform calibration of
this kind is no more extraordinary than the possibility of introducing into
Riemann's space certain unique co-ordinate systems arising out of the metrical
structure.) The ``volume'' that is measured by using this unit of length is
represented by the invariant integral
\[
\int \sqrt{g · F^{n}}\, dx\Add{.}
\Tag{(62)}
\]
\end{Remark}
For two vectors $\xi^{i}$,~$\eta^{i}$ that undergo parallel displacement we have,
in metrical space,
\[
d(\xi_{i} \eta^{i}) + (\xi_{i} \eta^{i})\, d\phi = 0.
\]
In Riemann's space, the second term is absent. From this it
follows that in Riemann's space the parallel displacement of a
contra-variant vector~$\xi$ is expressed in exactly the same way in
terms of the quantities $\xi_{i} = g_{ik} \xi^{k}$ as the parallel displacement of a
co-variant vector is expressed in terms of its components~$\xi_{i}$:
\[
d\xi_{i} - \Chr{i\alpha}{\beta} dx_{\alpha}\, \xi_{\beta} = 0
\quad\text{or}\quad
d\xi_{i} - \Chrsq{i\alpha}{\beta} dx_{\alpha}\, \xi^{\beta} = 0.
\]
\PageSep{135}
Accordingly, for a translation we have
\[
\frac{du_{i}}{ds}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, u^{\alpha} u^{\beta} = 0\qquad
\left(u^{i} = \frac{dx_{i}}{ds},\ u_{i} = g_{ik} u^{k}\right)
\Tag{(63)}
\]
for, by equation~\Eq{(48)},
\[
\Chrsq{i\alpha}{\beta} + \Chrsq{i\beta}{\alpha}
= \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
\]
and hence for any symmetrical system of numbers~$\vt^{\alpha\beta}$:---
\[
\tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} · \vt_{\alpha\beta}
= \Chrsq{i\alpha}{\beta} \vt^{\alpha\beta}
= \Chr{i\alpha}{\beta} \vt_{\beta}^{\alpha}\Add{.}
\Tag{(64)}
\]
Since the numerical value of the velocity vector remains unchanged
during translations, we get
\[
g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds} = u_{i} u^{i} = \text{const.}
\Tag{(65)}
\]
If, for the sake of simplicity, we assume the metrical groundform
to be definitely positive, then every curve $x_{i} = x_{i}(s)$ [$a \leq s \leq b$] has a
\Emph{length}, which is independent of the mode of parametric representation.
This length is
\[
\int_{a}^{b} \sqrt{Q}\, ds\qquad
\left(Q = g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds}\right).
\]
If we use the length of arc itself as the parameter, $Q$~becomes equal
to~$1$. Equation~\Eq{(65)} states that a body in translation traverses its
path, the geodetic line, with constant speed, namely, that the time-parameter
is proportional to~$s$, the length of arc. In Riemann's
space the geodetic line possesses not only the differential property
of preserving its direction unaltered, but also \Emph{the integral property
that every portion of it is the shortest line connecting its
initial and its final point}. This statement must not, however,
be taken literally, but must be understood in the same sense as
the statement in mechanics that, in a position of equilibrium, the
potential energy is a minimum, or when it is said of a function
$f(x, y)$ in two variables that it has a minimum at points where its
differential
\[
df = \frac{\dd f}{\dd x}\, dx + \frac{\dd f}{\dd y}\, dy
\]
vanishes identically in $dx$ and~$dy$; whereas the true expression is
that it assumes a ``stationary'' value at that point, which may be
a minimum, a maximum, or a ``point of inflexion''. The geodetic
line is not necessarily a curve of least length but is a curve of
stationary length. On the surface of a sphere, for instance, the
\PageSep{136}
great circles are geodetic lines. If we take any two points, $A$ and~$B$,
on such a great circle, the shorter of the two arcs~$AB$ is indeed
the shortest line connecting $A$ and~$B$, but the other arc~$AB$ is also
a geodetic line connecting $A$ and~$B$; it is not of least but of
stationary length. We shall seize this opportunity of expressing
in a rigorous form the principle of infinitesimal variation.
Let any arbitrary curve be represented parametrically by
\[
x_{i} = x_{i}(s),\qquad
(a \leq s \leq b).
\]
We shall call it the ``initial'' curve. To compare it with
neighbouring curves we consider an arbitrary family of curves
involving one parameter:
\[
x_{i} = x_{i}(s; \Typo{e}{\epsilon}),\qquad
(a \leq s \leq b).
\]
The parameter~$\epsilon$ varies within an interval about $\epsilon = 0$; $x_{i}(s; \epsilon)$~are
to denote functions that resolve into~$x_{i}(s)$ when $\epsilon = 0$. Since all
curves of the family are to connect the same initial point with the
same final point, $x_{i}(a; \epsilon)$ and $x_{i}(b; \epsilon)$ are independent of~$\epsilon$. The
length of such a curve is given by
\[
L(\epsilon) = \int_{a}^{b} \sqrt{Q}\, ds\Add{.}
\]
Further, we assume that $s$~denotes the length of an arc of the
initial curve, so that $Q = 1$ for $\epsilon = 0$. Let the direction components
$\dfrac{dx_{i}}{ds}$ of the initial curve $\epsilon = 0$ be denoted by~$u^{i}$. We also set
\[
\epsilon · \left(\frac{dx_{i}}{d\epsilon}\right)_{\epsilon=0}
= \xi^{i}(s) = \delta x_{i}.
\]
These are the components of the ``infinitesimal'' displacement
which makes the initial curve change into the neighbouring curve
due to the ``variation'' corresponding to an infinitely small value
of~$\epsilon$; they vanish at the ends.
\[
\epsilon\left(\frac{dL}{d\epsilon}\right)_{\epsilon=0} = \delta L
\]
is the corresponding variation in the length. $\delta L = 0$ is the condition
that the initial curve has a stationary length as compared
with the other members of the family. If we use the symbol~$\delta Q$
in the same sense, we get
\[
\delta L = \int_{a}^{b} \frac{\delta Q}{2\sqrt{Q}}\, ds
= \tfrac{1}{2} \int_{a}^{b} \delta Q\, ds
\Tag{(66)}
\]
since $Q = 1$ in the case of the initial curve. Now
\[
\frac{dQ}{d\epsilon}
= \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, \frac{dx_{i}}{d\epsilon}\,
\frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds}
+ 2g_{ik}\, \frac{dx_{k}}{ds}\, \frac{d^{2}x_{i}}{d\epsilon\, ds}
\]
\PageSep{137}
and hence (if we interchange ``variation'' and ``differentiation,''
that is the differentiations with respect to $\epsilon$~and~$s$) we get
\[
\delta Q
= \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, u^{\alpha} u^{\beta} \xi^{i}
+ 2 g_{ik} u^{k}\, \frac{d\xi^{i}}{ds}.
\]
If we substitute this in~\Eq{(66)} and rewrite the second term by applying
partial integration, and note that the~$\xi^{i}$'s vanish at the ends
of the interval of integration, then
\[
\delta L = \int_{a}^{b} \left(\tfrac{1}{2} \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, u^{\alpha} u^{\beta} - \frac{du_{i}}{ds}\right) \xi^{i}\, ds.
\]
Hence the condition $\delta L = 0$ is fulfilled for any family of curves if,
and only if, \Eq{(63)}~holds. Indeed, if, for a value $s = s_{0}$ between $a$
and~$b$, one of these expressions, for example the first, namely, $i = 1$,
differed from zero (were greater than zero), say, it would be possible
to mark off a little interval around~$s_{0}$ so small that, within it, the
above expression would be always $> 0$. If we choose a non-negative
function for~$\xi^{1}$ such that it vanishes for points beyond this
interval, all remaining~$\xi^{i}$'s, however, being $= 0$, we find the equation
$\delta L = 0$ contradicted.
Moreover, it is evident from this proof that, of all the motions
that lead from the same initial point to the same final point within
the same interval of time $a \leq s \leq b$, a \Emph{translation} is distinguished
by the property that $\int_{a}^{b} Q\, ds$ has a stationary value.
Although the author has aimed at lucidity of expression many
a reader will have viewed with abhorrence the flood of formulæ
and indices that encumber the fundamental ideas of
infinitesimal geometry. It is certainly regrettable that we have to
enter into the purely formal aspect in such detail and to give it so
much space but, nevertheless, it cannot be avoided. Just as anyone
who wishes to give expressions to his thoughts with ease must
spend laborious hours learning language and writing, so here too
the only way that we can lessen the burden of formulæ is to
master the technique of tensor analysis to such a degree that we
can turn to the real problems that concern us without feeling any
encumbrance, our object being to get an insight into the nature of
space, time, and matter so far as they participate in the structure
of the external world. Whoever sets out in quest of this goal must
possess a perfect mathematical equipment from the outset. Before
\PageSep{138}
we pass on after these wearisome preparations and enter into the
sphere of physical knowledge along the route illumined by the
genius of Einstein, we shall seek to obtain a clearer and deeper
vision of metrical space. Our goal is to grasp the inner necessity
and uniqueness of its metrical structure as expressed in Pythagoras'
Law.
\Section{18.}{Metrical Space from the Point of View of the Theory
of Groups}
\index{Euclidean!group of rotations}%
\index{Groups!of rotations}%
\index{Rotations, group of}%
Whereas the character of affine relationship presents no further
difficulties---the postulate on \Pageref{124} to which we subjected the
conception of parallel displacement, and which characterises it as a
kind of \Emph{unaltered} transference, defines its character uniquely---we
have not yet gained a view of metrical structure that takes us
beyond experience. It was long accepted as a fact that a metrical
character could be described by means of a quadratic differential
form, but this fact was not clearly understood. Riemann many
years ago pointed out that the metrical groundform might, with
equal right essentially, be a homogeneous function of the fourth
order in the differentials, or even a function built up in some other
way, and that it need not even depend rationally on the differentials.
But we dare not stop even at this point. The underlying general
feature that determines the metrical structure at a point~$P$ is the
\Emph{group of rotations}. The metrical constitution of the manifold at
the point~$P$ is known if, among the linear transformations of the
vector body (i.e.\ the totality of vectors), those are known that are
\Emph{congruent} transformations of themselves. There are just as many
different kinds of measure-determinations as there are essentially
different groups of linear transformations (whereby essentially
different groups are such as are distinguished not merely by the
choice of co-ordinate system). In the case of \Emph{Pythagorean
metrical space}, which we have alone investigated hitherto, the
group of rotations consists of all linear transformations that convert
the quadratic groundform into itself. But the group of rotations
need not have an invariant at all in itself (that is, a function which
is dependent on a single arbitrary vector and which remains unaltered
after any rotations).
Let us reflect upon the natural requirements that may be imposed
on the conception of rotation. At a single point, as long as
the manifold has not yet a measure-determination, only the $n$-dimensional
parallelepipeds can be compared with one another in
respect to size. If $\va_{i}$ ($i = 1, 2, \dots\Add{,} n$) are arbitrary vectors
\PageSep{139}
that are defined in terms of the initial unit vectors~$\ve_{i}$ according to
the equations
\[
\va_{i} = \Typo{a}{\alpha}_{i}^{k} \ve_{k}
\]
then the determinant of the~$\Typo{a}{\alpha}_{i}^{k}$'s which, following Grassmann, we
may conveniently denote by
\[
\Det{\va}{\ve}
\]
is, according to definition, the volume of the parallelopiped mapped
out by the $n$~vectors~$\va_{i}$. If we choose another system of unit
vectors~$\bar{\ve}_{i}$ all the volumes become multiplied by a common constant
factor, as we see from the ``multiplication theorem of determinants,''
namely
\[
%[** TN: Superscript typo in the original fixed by macro]
\Det{\va}{\ve} = \Det{\va}{\bar{\ve}}\, \Det{\bar{\ve}}{\ve}.
\]
The volumes are thus determined uniquely and independently of
the co-ordinate system once the unit measure has been chosen.
\emph{Since a rotation is ``not to alter'' the vector body, it must obviously
be a transformation that leaves the infinitesimal elements of volume
unaffected.} Let the rotation that transforms the vector $\vx = (\xi^{i})$
into $\bar{\vx} = (\bar{\xi}^{i})$ be represented by the equations
\[
\bar{\ve}_{i} = \Typo{a}{\alpha}_{i}^{k} \ve_{k}\quad\text{or}\quad
\xi^{i} = \Typo{a}{\alpha}_{k}^{i} \bar{\xi}^{k}.
\]
The determinant of the rotation matrix~$(\Typo{a}{\alpha}_{k}^{i})$ then becomes equal to~$1$.
This being the postulate that applies to a \Emph{single} rotation,
we must demand of the rotations as a whole that they \Emph{form a
group} in the sense of the definition given on \Pageref{9}. Moreover,
this group has to be a \Emph{continuous} one, that is the rotations are to
be elements of a one-dimensional continuous manifold.
If a linear vector transformation be given by its matrix $A = (\Typo{a}{\alpha}_{k}^{i})$
in passing from one co-ordinate system~$(\ve_{i})$ to another~$(\bar{\ve}_{i})$
according to the equations
\[
U : \bar{\ve}_{i} = u_{i}^{k} \ve_{k}\Add{,}
\Tag{(67)}
\]
then $A$~becomes changed into~$UAU^{-1}$ (where $U^{-1}$~denotes the inverse
of~$U$; $UU^{-1}$~and $U^{-1}U$ are equal to identity~$E$). Hence
every group that is derived from a given matrix group~$\vG$ by applying
the operation $UGU^{-1}$ on every matrix~$G$ of~$\vG$ ($U$~being the
same for all~$G$'s) may be transformed into the given matrix group
by an appropriate change of co-ordinate system. Such a group
$U\vG U^{-1}$ will be said to be of the same kind as~$\vG$ (or to differ from~$\vG$
only in orientation). If $\vG$~is the group of rotation matrices at~$P$
and if $U\vG U^{-1}$~is identical with~$\vG$ (this does not mean that $G$~must
\PageSep{140}
again pass into~$G$ as a result of the operation~$UGU^{-1}$, but all that
is required is that $G$~and $UGU^{-1}$ belong to~$\vG$ simultaneously) then
the expressions for the metrical structures of two co-ordinate
systems~\Eq{(67)}, that are transformed into one another by~$U$, are
similar; $U$~is a representation of the vector body on itself, such
that it leaves all the metrical relations unaltered. This is the
conception of \Emph{similar representation}. $\vG$~is included in the
group~$\vG^{*}$ of similar representations as a sub-group.
From the metrical structure at a single point we now pass on
\index{Congruent!transference}%
\index{Groundform, metrical!general@{(in general)}}%
\index{Metrical groundform}%
\index{Similar representation or transformation}%
\index{Transference, congruent}%
\index{Transformation or representation!similar}%
to ``\Emph{metrical relationship}''. The metrical relationship between
the point~$P_{0}$ and its immediate neighbourhood is given if a linear
representation at $P_{0} = x_{i}^{0}$ of the vector body on itself at an infinitely
near point $P = (x_{i}^{0} + dx_{i})$ is a \Emph{congruent transference}. Together
with~$A$ every representation (or transformation) $AG_{0}$, in which $A$~is
followed by a rotation~$G_{0}$ at~$P_{0}$, is likewise a congruent transference;
thus, from one congruent transference~$A$ of the vector body
from $P_{0}$ to~$P$, we get all possible ones by making $G_{0}$ traverse the
group of rotations belonging to~$P_{0}$. If we consider the vector body
belonging to the centre~$P_{0}$ for two positions congruent to one
another, they will resolve into two congruent positions at~$P$ if
subjected to the same congruent transference~$A$; for this reason,
the group of rotations~$\vG$ at~$P$ is equal to~$A\vG_{0} A^{-1}$. The metrical
relationship thus tells us that the group of rotations at~$P$ differs
from that at~$P_{0}$ only in orientation. If we pass continuously from
the point~$P_{0}$ to any point of the manifold, we see that the groups
of rotation are of a similar kind at all points of the manifold; thus
there is homogeneity in this respect.
The only congruent transferences that we take into consideration
are those in which the vector components~$\xi^{i}$ undergo changes~$d\xi^{i}$
that are infinitesimal and of the same order as the displacement of
the centre~$P_{0}$,
\[
d\xi^{i} = d\lambda_{k}^{i} · \xi^{k}.
\]
If $L$ and~$M$ are two such transferences from $P_{0}$ to~$P$, with co-efficients
$d\lambda_{k}^{i}$ and $d\mu_{k}^{i}$ respectively, then the rotation~$ML^{-1}$ is
likewise infinitesimal: it is represented by the formula
\[
d\xi^{i} = d\alpha_{k}^{i} · \xi^{k}
\quad\text{where}\quad
d\alpha_{k}^{i} = d\mu_{k}^{i} - d\lambda_{k}^{i}\Add{.}
\Tag{(68)}
\]
The following will also be true. If an infinitesimal congruent
transference consisting in the displacement~$(dx_{i})$ of the centre~$P_{0}$ is
succeeded by one in which the centre is displaced by~$(\delta x_{i})$, we get
a congruent transference that is effected by the resultant displacement
$dx_{i} + \delta x_{i}$ of the centre (plus an error which is infinitesimal
compared with the magnitude of the displacements). Hence, if
\PageSep{141}
for the transition from $P_{0} = (x_{1}^{0}, x_{2}^{0}, \dots\Add{,} x_{n}^{0})$ to the point
$(x_{1}^{0} + \epsilon, x_{2}^{0}, \dots\Add{,} x_{n}^{0})$, this being an infinitesimal change~$\epsilon$ in the
direction of the first co-ordinate axis,
\[
d\xi^{i} = \epsilon · \Lambda_{k}^{i} \xi^{k}
\]
is a congruent transference, and if $\Lambda_{k2}^{i}, \dots\Add{,} \Lambda_{kn}^{i}$ have a corresponding
meaning for the displacements of~$P_{0}$ in the direction of
% [** TN: Ordinals]
the~2nd up to the $n$th~co-ordinate in turn; then the equation
\[
d\xi^{i} = \Lambda_{kr}^{i}\, dx_{r} · \xi^{k}
\Tag{(69)}
\]
gives a congruent transference for an arbitrary displacement having
components~$dx_{i}$.
Among the various kinds of metrical spaces we shall now
designate by simple intrinsic relations the category to which,
according to Pythagoras' and Riemann's ideas, real space belongs.
The group of rotations that does not vary with position exhibits
a property that belongs to space as a form of phenomena; it
characterises the metrical nature of space. The metrical relationship,\footnote
{Although, as will be shown later, it is everywhere of the same kind.}
from point to point, however, is \emph{not} determined by the
nature of space, nor by the mutual orientation of the groups of
rotation at the various points of the manifold. The metrical
relationship is dependent rather on the disposition of the material
content, and is thus in itself free and capable of any ``virtual''
changes. We shall formulate the fact that it is subject to no
limitation as our first axiom.
\Subsection{I\@. The Nature of Space Imposes no Restriction on the
Metrical Relationship}
It is \Emph{possible} to find a metrical relationship in space between
the point~$P_{0}$ and the points in its neighbourhood such that the
formula~\Eq{(69)} represents a system of congruent transferences to
these neighbouring points \Emph{for arbitrarily given numbers~$\Lambda_{kr}^{i}$}.
Corresponding to every co-ordinate system~$x_{i}$ at~$P_{0}$ there is a
possible conception of parallel displacement, namely, the displacement
of the vectors from~$P_{0}$ to the infinitely near points without
the components undergoing a change in this co-ordinate system.
Such a system of parallel displacements of the vector body from~$P_{0}$
to all the infinitely near points is expressed, as we know, in terms
of a definite co-ordinate system, selected once and for all by the
formula
% [** TN: Reformatted from the original; original code commented out]
\iffalse
\[
d\xi^{i} = -d\gamma^{i} · \xi^{k}
\quad\text{in which the differential forms}\quad
d\gamma_{k}^{i} = \Gamma_{kr}^{i}\, dx_{r}
\]
\fi
\[
d\xi^{i} = -d\gamma^{i} · \xi^{k}
\]
in which the differential forms $d\gamma_{k}^{i} = \Gamma_{kr}^{i}\, dx_{r}$
\PageSep{142}
satisfy the condition of symmetry
\[
\Gamma_{kr}^{i} = \Gamma_{rk}^{i}\Add{.}
\Tag{(70)}
\]
And, indeed, a possible conception of parallel displacement corresponds
to every system of symmetrical co-efficients~$\Gamma$. For a
given metrical relationship the further restriction that the ``parallel
\index{Relationship!metrical}%
displacements'' shall simultaneously be congruent transferences
must be imposed. The second postulate is the one enunciated
above as the fundamental theorem of infinitesimal geometry; for
\index{Geometry!infinitesimal}%
\index{Infinitesimal!geometry}%
\index{Infinitesimal!operation of a group}%
a given metrical relationship there is always a \Emph{single} system of
parallel displacements among the transferences of the vector body.
We treated affine relationship in §\,15 only provisionally as a
\index{Components, co-variant, and contra-variant!affine@{of the affine relationship}}%
rudimentary characteristic of space; the truth is, however, that
parallel displacements, in virtue of their inherent properties, must
be excluded from congruent transferences, and that the conception
of parallel displacement is determined by the metrical relationship.
This postulate may be enunciated thus:---
\Subsection{II\@. The Affine Relationship is Uniquely Determined by the
Metrical Relationship}
Before we can formulate it analytically we must deal with
infinitesimal rotations. A continuous group~$\vG$ of $r$~members is
a continuous $r$-dimensional manifold of matrices. If $s_{1}\Com s_{2}\Com \dots\Add{,} s_{r}$
are co-ordinates in this manifold, then, corresponding to every
value system of the co-ordinates there is a matrix $A(s_{1}\Com s_{2}\Com \dots\Add{,} s_{r})$
of the group which depends on the value-system continuously.
There is a definite value-system---we may assume for it that $s_{1} = 0$---to
which \Emph{identity},~$E$, corresponds. The matrices of the group
that are infinitely near~$E$ differ from~$E$ by
\[
\Alpha_{1}\, ds_{1} + \Alpha_{2}\, ds_{2} + \dots \Add{+} \Alpha_{r}\, ds_{r},
\]
in which $\Alpha_{i} = \left(\dfrac{\dd A}{\dd s_{i}}\right)_{0}$. We call a matrix~$\Alpha$ an infinitesimal
operation of the group if the group contains a transformation
(independent of~$\epsilon$) that coincides with~$E$ and~$\epsilon \Alpha$ to within an
error that converges more rapidly towards zero than~$\epsilon$, for decreasing
small values of~$\epsilon$. The infinitesimal operations of the
group form the linear family
\[
\vg:\ \lambda_{1} \Alpha_{1} + \lambda_{2} \Alpha_{2} + \dots + \lambda_{r} \Alpha_{r}
\quad(\text{$\lambda$ being arbitrary numbers})
\Tag{(71)}
\]
$\vg$~is exactly $r$-dimensional and the~$\Alpha$'s are linearly independent of
one another. For if $\Alpha$~is an arbitrary matrix of the group, the
group property expresses the transformations of the group which
are infinitely near~$A$ in the formula $A(E + \epsilon \Alpha)$, in which $\epsilon$~is an
\PageSep{143}
infinitesimal factor and $\Alpha$~traverses the group~$\vg$. If $\vg$ were of
less dimensions than~$r$, the same would hold at each point of
the manifold; for all values of~$s_{i}$ there would be linear relations
between the derivatives~$\dfrac{\dd A}{\dd s_{i}}$, and $A$~would in reality depend on less
than $r$ parameters. The infinitesimal operations generate and
determine the whole group. If we carry out the infinitesimal
transformation $E + \dfrac{1}{n} \Alpha$ ($n$~being an infinitely great number)
$n$-times successively, we get a matrix (of the group) that is finite
and different from~$E$, namely,
\[
A = \lim_{n \to \infty} \left(E + \frac{1}{n} \Alpha\right)^{n}
= E + \frac{\Alpha}{1!} + \frac{\Alpha^{2}}{2!} + \frac{\Alpha^{3}}{3!} + \dots;
\]
and thus we get every matrix of the group (or at least every one
that may be reached continuously in the group, by starting from
identity) if we make $\Alpha$ traverse the whole family~$\vg$. Not every
arbitrarily given linear family\Eq{(71)} gives a group in this way, but
only those in which the~$\Alpha$'s satisfy a certain condition of integrability.
The latter is obtained by a method quite analogous to that by which,
for example, the condition of integrability is obtained for parallel
displacement in Euclidean space. If we pass from \Emph{Identity},
$E(s_{i} = 0)$, by an infinitesimal change~$ds_{i}$ of the parameters, to the
neighbouring matrix $A_{d} = E + dA$, and thence by a second infinitesimal
change~$\delta s_{i}$, from $A_{\delta}$ to $A_{\delta} A_{d}$ and then reverse these two
operations whilst preserving the same order, we get $A_{\delta}^{-1} A_{d}^{-1} A_{\delta} A_{d}$,
a matrix (of the group) differing by an infinitely small amount
from~$E$. Let $d$~be the change in the direction of the first co-ordinate,
and $\delta$~that in the direction of the second, then we are
dealing with the matrix
\[
A_{st} = A_{t}^{-1} A_{s}^{-1} A_{t} A_{s}
\]
formed from
\[
\Typo{\mathrm{A}}{A_{s}} = A(s, 0, 0, \dots\Add{,} 0)
\quad\text{and}\quad
A_{t} = A(0, t, 0, \dots\Add{,} 0).
\]
Now, $A_{s0} = A_{0t} = E$, hence
\[
\lim_{s \to 0, t \to 0} \frac{A_{st} - E}{s · t}
= \left(\frac{\dd^{2} A_{st}}{\dd s\, \dd t}\right)_{\Subs{s \to 0}{t \to 0}}.
\]
Since $A_{st}$~belongs to the group, this limit is an infinitesimal operation
of the group. We find, however, that
\[
\frac{\dd A_{st}}{\dd t} = -\Alpha_{2} + A_{s}^{-1} \Alpha_{2} A_{s}
\quad\text{for}\quad t = 0;
\]
leading to
\[
\frac{\dd^{2} A_{st}}{\dd s\, \dd t}
= -\Alpha_{1} \Alpha_{2} + \Alpha_{2} \Alpha_{1}
\quad\text{for}\quad
t \to 0, s \to 0.
\]
\PageSep{144}
{\Loosen Accordingly $\Alpha_{1} \Alpha_{2} - \Alpha_{2} \Alpha_{1}$, or, more generally, $\Alpha_{i} \Alpha_{k} - \Alpha_{k} \Alpha_{i}$ must
be an infinitesimal operation of the group: or, what amounts to
\index{Infinitesimal!group}%
the same thing, if $\Alpha$~and $\Beta$ are two infinitesimal operations of the
group, then $\Alpha\Beta - \Beta\Alpha$ must also always be one. Sophus Lie, to
whom we are indebted for the fundamental conceptions and facts
of the theory of continuous transformation groups (\textit{vide} \FNote{12}),
\index{Groups!infinitesimal}%
has shown that this condition of integrability is not only necessary
but also sufficient. Hence we may define an \emph{$r$-dimensional linear
family of matrices as an infinitesimal group having $r$~members if,
whenever any two matrices $\Alpha$ and $\Beta$ belong to the family, $\Alpha\Beta - \Beta\Alpha$
also belongs to the family}. By introducing the infinitesimal operations
of the group, the problem of continuous transformation groups
becomes a linear question.}
If all the transformations of the group leave the elements of
volume unaltered, the ``traces'' of the infinitesimal operations $= 0$.
For the development of the determinant of $E + \epsilon \Alpha$ in powers of~$\epsilon$
begins with the members $1 + \epsilon · \trace(\Alpha)$. $U$~is a similar transformation,
if, for every~$G$ of the group of rotations, $UGU^{-1}$ or,
what comes to the same thing, $UGU^{-1}G^{-1}$, belongs to the group
of rotations~$\vG$. Accordingly, $\Alpha_{0}^{*}$~is an infinitesimal operation of the
group of similar transformations if, and only if, $\Alpha_{0}^{*}\Alpha - \Alpha \Alpha_{0}^{*}$ also
belongs to~$\vg$, no matter which of the matrices~$\Alpha$ of the group of
infinitesimal rotations is used.
The infinitesimal Euclidean rotations
\[
d\xi^{i} = v_{k}^{i} \xi^{k},
\]
that is, the infinitesimal linear transformations that leave the unit
quadratic form
\[
Q_{0} = (\xi^{1})^{2} + (\xi^{2})^{2} + \dots + (\xi^{n})^{2}
\]
invariant, were determined on \Pageref{47}. The condition which
characterises them, namely,
\[
\tfrac{1}{2}dQ_{0} = \xi^{i}\, d\xi^{i} = 0,
\quad\text{implies that}\quad
v_{i}^{k} = -v_{k}^{i}.
\]
Thus it is seen that we are dealing with the infinitesimal group~$\delta$
of all skew-symmetrical matrices; it obviously has $\dfrac{n(n - 1)}{2}$
members. It may be left to the reader to verify by direct calculation
that it possesses the group property. If $Q$~is any quadratic
form that remains invariant during the infinitesimal Euclidean
rotations, i.e.\ $dQ = 0$, then $Q$~necessarily coincides with~$Q_{0}$ except
for a constant factor. Indeed, if
\[
Q = \Typo{\alpha}{a}_{ik} \xi^{i} \xi^{k}\qquad
(\Typo{\alpha}{a}_{ki} = \Typo{\alpha}{a}_{ik})
\]
then for all skew-symmetrical number systems~$v_{k}^{i}$ the equation
\[
\Typo{\alpha}{a}_{rk} v_{i}^{k} + \Typo{\alpha}{a}_{ri} v_{k}^{r} = 0
\Tag{(72)}
\]
\PageSep{145}
must hold. If we assume $k = i$ and notice that the numbers
$v_{i}^{1}, v_{i}^{2}, \dots\Add{,} v_{i}^{n}$ may be chosen arbitrarily for each particular~$i$,
excepting the case $v_{i}^{i} = 0$, we get $\Typo{\alpha}{a}_{ri} = 0$ for $r \neq i$. If we write~$\Typo{\alpha}{a}_{ii}$
for~$\Typo{\alpha}{a}_{i}$, equation~\Eq{(72)} becomes
\[
v_{i}^{k}(\Typo{\alpha}{a}_{i} - \Typo{\alpha}{a}_{k}) = 0
\]
from which we immediately deduce that all~$\Typo{\alpha}{a}_{i}$'s are equal. The
corresponding group~$\delta^{*}$ of similar transformations is derived from~$\delta$
by ``associating'' the single matrix~$E$; this here signifies $d\xi^{i} = \epsilon \xi^{i}$.
For if the matrix $C = (c_{i}^{k})$ belongs to~$\delta^{*}$, that is, if for every skew-symmetrical~$v_{i}^{k}$,
$c_{r}^{i} v_{k}^{r} - v_{r}^{i} c_{k}^{r}$ is also a skew-symmetrical number
system, then the quantities $c_{k}^{i} + c_{i}^{k} = \Typo{\alpha}{a}_{ik}$ satisfy equation~\Eq{(72)};
whence it follows that $\Typo{\alpha}{a}_{ik} = 2\Typo{\alpha}{a} · \delta_{i}^{k}$; that is, $C$~is equal to \emph{$aE$~plus}
a skew-symmetrical matrix.
More generally, let $\delta_{Q}$ denote the infinitesimal group of linear
transformations that transform an arbitrary non-degenerate quadratic
form~$Q$ into itself. $\delta_{Q}$~and $\delta_{Q'}$ are distinguished only by their
orientation, if $Q'$~is generated from~$Q$ by a linear transformation.
Hence there are only a finite number of different kinds of infinitesimal
groups~$\delta_{Q}$ that differ from one another in the inertial index
attached to the form~$Q$. But even these differences are eliminated
if, instead of confining ourselves to the realm of real quantities, we
use that of complex members; in that case, every~$\delta_{Q}$ is of the same
type as~$\delta$.
These preliminary remarks enable us to formulate analytically
the two postulates \Inum{I}~and~\Inum{II}\@. Let $\vg$~be the group of infinitesimal
rotations at~$P$. We take $\Lambda_{kr}^{i}$ to denote every system of $n^{3}$~numbers,
$\Alpha_{kr}^{i}$~to denote every system that is composed of matrices $(\Alpha_{k1}^{i}), (\Alpha_{k2}^{i}), \dots\Add{,} (\Alpha_{kn}^{i})$
belonging to~$\vg$ and $\Gamma_{kr}^{i}$~to denote an arbitrary
system of numbers that satisfies the condition of symmetry~\Eq{(70)}.
If the group of infinitesimal rotations has $N$~members, these
member systems form linear manifolds of $n^{3}$,~$nN$ and $n · \dfrac{n(n + 1)}{2}$
dimensions respectively. Since, according to~\Inum{I}, if the metrical
relationship runs through all possible values, any arbitrary number
systems $\Lambda_{k1}^{i}, \Lambda_{k2}^{i}, \dots\Add{,} \Lambda_{kn}^{i}$ may occur as the co-efficients of $n$~infinitesimal
congruent transferences in the $n$~co-ordinate directions
(cf.~\Eq{(69)}), then, by~\Inum{II} (cf.~\Eq{(68)}) each~$\Lambda$ must be capable of resolution
in one and only one way according to the formula
\[
\Lambda_{kr}^{i} = \Alpha_{kr}^{i} - \Gamma_{kr}^{i}.
\]
\PageSep{146}
This entails two results
1.\qquad $n^{3} = nN + n · \dfrac{n(n + 1)}{2}$\quad or\quad $N = \dfrac{n(n - 1)}{2}$;
2. $\Alpha_{kr}^{i} - \Gamma_{kr}^{i}$ is never equal to zero, unless all the $\Alpha$'s and~$\Gamma$'s
vanish; or, a non-vanishing system~$\Alpha$ can never fulfil the condition
of symmetry, $\Alpha_{kr}^{i} = \Alpha_{rk}^{i}$. To enable us to formulate this condition
invariantly let us define a symmetrical double matrix (an infinitesimal
\index{Infinitesimal!rotations}%
\index{Rotations, group of}%
\index{Trace of a matrix}%
double rotation) belonging to~$\vg$ as a law expressed by
\[
\zeta^{i} = \Alpha_{rs}^{i} \xi^{r} \eta^{s}\qquad
(\Alpha_{rs}^{i} = \Alpha_{sr}^{i}),
\]
which produces from two arbitrary vectors, $\xi$~and~$\eta$, a vector~$\zeta$
as a bilinear symmetrical form, provided that for every fixed vector~$\eta$,
the transition $\xi \to \zeta$ (and hence also for every fixed vector~$\xi$ the
transition $\eta \to \zeta$) is an operation of~$\vg$. We may then summarise
our results thus:---
{\itshape The group of infinitesimal rotations has the following properties
according to our axioms:
\Inum{(\ia)} The trace of every matrix $= 0$;
\Inum{(\ib)} No symmetrical double matrix belongs to~$\vg$ except zero;
\Inum{(\ic)} The dimensional number of~$\vg$ is the highest that is still in
agreement with postulate~\Inum{(\ib)}, namely, $N = \dfrac{n(n - 1)}{2}$.}
These properties retain their meaning for complex quantities as
well as for real ones. We shall just verify that they are true of the
infinitesimal Euclidean group of rotations~$\delta$, that is, that $n^{3}$~numbers~$v_{kl}^{i}$
cannot simultaneously satisfy the conditions of symmetry
\[
v_{lk}^{i} = v_{kl}^{i},\qquad
v_{il}^{k} = -v_{kl}^{i},
\]
without all of them vanishing. This is evident from the calculation
which was undertaken on \Pageref{125} to determine the affine
relationship. For if we write down the three equations that we
get from $v_{kl}^{i} + v_{il}^{k} = 0$ by interchanging the indices $i\Com k\Com l$ cyclically,
and then subtract the second from the sum of the first and the
third, we get, as a result of the first condition of symmetry, $v_{kl}^{i} = 0$.
It seems highly probable to the author that $\delta$~is the only infinitesimal
group that satisfies the postulates \Inum{\Chg{\ia}{(\ia)}}, \Inum{\Chg{\ib}{(\ib)}}, and~\Inum{\Chg{\ic}{(\ic)}}; or, more
exactly, in the case of complex quantities every such infinitesimal
group may be made to coincide with~$\delta$ by choosing the appropriate
co-ordinate system. If this is true, then the group of infinitesimal
rotations must be identical with a certain group~$\delta_{Q}$, in which $Q$~is
a non-degenerate quadratic form. $Q$~itself is determined by~$\vg$
except for a constant of proportionality. It is real if $\vg$~is real.
\PageSep{147}
For if we split~$Q$ (in which the variables are taken as real) into a
real and an imaginary part $Q_{1} + iQ_{2}$, then $\vg$~leaves both these forms
$Q_{1}$~and $Q_{2}$ invariant. Hence we must have
\[
Q_{1} = c_{1}Q\qquad
Q_{2} = c_{2}Q.
\]
One of these two constants is certainly different from zero, since
$c_{1} + ic_{2} = 1$, and hence $Q$~must be a real form excepting for a
constant factor. This would link up with the line of argument
followed in the preceding paragraph and would complete the
Analysis of Space; we should then be able to claim to have made
intelligible the nature of space and the source of the validity of
Pythagoras' Theorem, by having explored the ultimate grounds
accessible to mathematical reasoning (\textit{vide} \FNote{13}). If the
supposed mathematical proposition is not true, definite characteristics
and essentials of space will yet have escaped us. The
author has proved that the proposition holds actually for the
lowest dimensional numbers $n = 2$ and $n = 3$. It would lead too
far to present these purely mathematical considerations here.
In conclusion, it will be advisable to call attention to two points.
Firstly, axiom~\Inum{I} is in no wise contradicted by the result of axiom~\Inum{II}
which states that not only the metrical structure, but also the
metrical relationship is of the same kind at every point, namely, of
the simplest type imaginable. For every point there is a geodetic
co-ordinate system such that the shifting of all vectors at that point,
which leaves its components unaltered, to a neighbouring point is
always a congruent transference. Secondly, the possibility of grasping
the unique significance of the metrical structure of Pythagorean
space in the way here outlined depends solely on the circumstance
that the quantitative metrical conditions admit of considerable virtual
changes. This possibility stands or falls with the dynamical view
of Riemann. It is this view, the truth of which can scarcely be
doubted after the success that has attended Einstein's Theory of
Gravitation (Chapter~IV), that opens up the road leading to the
discovery of the ``Rationality of Space''.
The investigations about space that have been conducted in
Chapter~II seemed to the author to offer a good example of the
kind of analysis of the modes of existence (\textit{Wesensanalyse}) which is
the object of Husserl's phenomenological philosophy, an example
that is typical of cases in which we are concerned with non-immanent
modes. The historical development of the problem of
space teaches how difficult it is for us human beings entangled
in external reality to reach a definite conclusion. A prolonged
phase of mathematical development, the great expansion of geometry
dating from Euclid to Riemann, the discovery of the physical
\PageSep{148}
facts of nature and their underlying laws from the time of Galilei,
together with the incessant impulses imparted by new empirical
data, finally the genius of individual great minds---Newton, Gauss,
Riemann, Einstein---all these factors were necessary to set us free
from the external, accidental, non-essential characteristics which
would otherwise have held us captive. Certainly, once the true
point of view has been adopted reason becomes flooded with light,
and it recognises and appreciates what is of itself intelligible to it.
Nevertheless, although reason was, so to speak, always conscious of
this point of view in the whole development of the problem, it had
not the power to penetrate into it with one flash. This reproach
must be directed at the impatience of those philosophers who
believe it possible to describe adequately the mode of existence on
the basis of a single act of typical presentation (\textit{exemplarischer
Vergegenwärtigung}): in principle they are right: yet from the point
of view of human nature, how utterly they are wrong! The problem
of space is at the same time a very instructive example of that
question of phenomenology that seems to the author to be of
greatest consequence, namely, in how far the delimitation of the
essentialities perceptible in consciousness expresses the structure
peculiar to the realm of presented objects, and in how far mere
convention participates in this delimitation.
\PageSep{149}
\Chapter{III}
{Relativity of Space and Time}
\index{Galilei's Principle of Relativity and Newton's Law of Inertia}%
\index{Relativity!principle of!Galilei's}%
\index{World ($=$ space-time)!-line}%
\index{World ($=$ space-time)!-point}%
\Section{19.}{Galilei's Principle of Relativity}
\First{We} have already discussed in the introduction how it is
possible to measure time by means of a clock and how,
after an arbitrary initial point of time and a time-unit has
been chosen, it is possible to characterise every point of time by a
number~$t$. But the \Emph{union of space and time} gives rise to difficult
further problems that are treated in the theory of relativity.
The solution of these problems, which is one of the greatest feats in
the history of the human intellect, is associated above all with the
names of \Emph{Copernicus} and \Emph{Einstein} (\textit{vide} \FNote{1}).
By means of a clock we fix directly the time-conditions of
%[** TN: Original entry points to page 148]
\index{Now@{\emph{Now}}}%
only such events as occur just at the locality at which the clock
happens to be situated. Inasmuch as I, as an unenlightened being,
fix, without hesitation, the things that I see into the moment of
their perception, I extend my time over the whole world. I believe
that there is an objective meaning in saying of an event which is
happening somewhere that it is happening ``now'' (at the moment at
which I pronounce the word!); and that there is an objective meaning
in asking which of two events that have happened at different
places has occurred earlier or later than the other. \Emph{We shall for
the present accept the point of view implied in these assumptions.}
Every space-time event that is strictly localised, such as
the flash of a spark that is instantaneously extinguished, occurs at
a definite space-time-point or \Emph{world-point}, ``here-now''. As a
result of the point of view enunciated above, to every world-point
there corresponds a definite time-co-ordinate~$t$.
We are next concerned with fixing the position of such a point-event
in space. For example, we ascribe to two point-masses a
distance separating them at a definite moment. We assume that
the world-points corresponding to a definite moment~$t$ form a three-dimensional
point-manifold for which Euclidean geometry holds.
(In the present chapter we adopt the view of space set forth in
\PageSep{150}
Chapter~I\@.) We choose a definite unit of length and a rectangular
co-ordinate system at the moment~$t$ (such as the corner of a room).
Every world-point whose time-co-ordinate is~$t$ then has three
definite space-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$.
Let us now fix our attention on another moment~$t'$. We assume
that there is a definite objective meaning in stating that measurements
are carried out at the moment~$t'$ with the same unit length
as that used at the moment~$t$ (by means of a ``rigid'' measuring
staff that exists both at the time~$t$ and at the time~$t'$). In addition
to the unit of time we shall adopt a unit of length fixed once and
for all (centimetre, second). We are then still free to choose the
position of the Cartesian co-ordinate system independently of the
choice of time~$t$. Only when we believe that there is objective
meaning in stating that two point-events happening at arbitrary
\Figure{7}
moments take place at the \Emph{same} point of space, and in saying that
a body is \Emph{at rest}, are we able to fix the position of the co-ordinate
\index{Rest}%
system for all times on the basis of the position chosen arbitrarily at
a certain moment, without having to specify additional ``individual
objects''; that is, we accept the postulate that the co-ordinate
system remains permanently at rest. After choosing an initial
point in the time-scale and a definite co-ordinate system at this
initial moment we then get four definite co-ordinates for every
world-point. To be able to represent conditions graphically we
suppress one space-co-ordinate, assuming space to be only two-dimensional,
a Euclidean plane.
We construct a graphical picture by representing in a space
carrying the rectangular set of axes $(x_{1}, x_{2}, t)$ the world-point by a
``picture''-point with co-ordinates $(x_{1}, x_{2}, t)$. We can then trace
\PageSep{151}
out graphically the ``time-table'' of all moving point-masses; the
motion of each is represented by a ``world-line,'' whose direction
has always a positive component in the direction of the $t$-axis. The
world-lines of point-masses that are at rest are parallels to the
$t$-axis. The world-line of a point-mass which is in uniform translation
is a straight line. On a section $t = \text{constant}$ we may read off
the position of all the point-masses at the same time~$t$. If we
choose an initial point in the time-scale and also some other Cartesian
co-ordinate system, and if $(x_{1}, x_{2}, t)$, $(x_{1}', x_{2}', t')$ are the co-ordinates
of an arbitrary world-point in the first and second
co-ordinate system respectively, the transformation formulæ
\[
\left.
\begin{alignedat}{3}
x_{1} &= \alpha_{11} x_{1}' &&{}+{} &\alpha_{12} x_{2}' &+ \alpha_{1} \\
x_{2} &= \alpha_{21} x_{1}' &&{}+{} &\alpha_{22} x_{2}' &+ \alpha_{2} \\
t &= && & t' &+ a
\end{alignedat}
\right\}
\Chg{\textTag{I}}{\textTag{(I)}}
\]
hold; in them, the $\Typo{\alpha}{\alpha_{i}}$'s and the~$a$ denote constants, the $\alpha_{ik}$'s, in
particular, are the co-efficients of an orthogonal transformation. The
world-co-ordinates are thus fixed \Emph{except for an arbitrary transformation
of this kind} in an objective manner without individual
objects or events being specified. In this we have not yet taken
into consideration the arbitrary choice of both units of measure.
If the initial point remains unchanged both in space and in time,
%[** TN: For rest of paragraph, "x"s upright in the original]
so that $\alpha_{1} = \alpha_{2} = a = 0$, then $(x_{1}', x_{2}', t')$ are the co-ordinates with
respect to a rectilinear system of axes whose $t'$~axis coincides with
the $t$-axis, whereas the axes $x_{1}'$,~$x_{2}'$ are derived from $x_{1}$,~$x_{2}$ by a
rotation in their plane $t = 0$.
A moment's reflection suffices to show that one of the assumptions
adopted is not true, namely, the one which states that the
conception of rest has an objective content.\footnote
{Even Aristotle was clear on this point, for he denotes ``place'' (\textgreek{t'opos}) as
the relation of a body to the bodies in its neighbourhood.}
When I arrange to
meet some one at the same place to-morrow as that at which we
met to-day, this means in the same material surroundings, at the
same building in the same street (which, according to Copernicus,
may be in a totally different part of stellar space to-morrow). All
this acquires meaning as a result of the fortunate circumstance
that at birth we are introduced into an essentially stable world, in
which changes occur in conjunction with a comparatively much
more comprehensive set of permanent factors that preserve their
constitution (which is partly perceived directly and partly deduced)
unchanged or almost unchanged. The houses stand still; ships
travel at so and so many knots: these things are always understood
in ordinary life as referring to the firm ground on which we
\PageSep{152}
stand. \Emph{Only the motions of bodies (point-masses) relative to
one another have an objective meaning}, that is, the distances
and angles that are determined from simultaneous positions of the
point-masses and their functional relation to the time-co-ordinate.
The connection between the co-ordinates of the same world-point
expressed in two different systems of this kind is given by formulæ\Pagelabel{152}
\[
\left.
\begin{alignedat}{3}
x_{1} &= \alpha_{11}(t') x_{1}' &&{}+{} &\alpha_{12}(t') x_{2}' &+ \alpha_{1}(t') \\
x_{2} &= \alpha_{21}(t') x_{1}' &&{}+{} &\alpha_{22}(t') x_{2}' &+ \alpha_{2}(t') \\
t &= t' + a
\end{alignedat}
\right\}
\Chg{\textTag{II}}{\textTag{(II)}}
\]
in which the $\alpha_{i}$'s and $\alpha_{ik}$'s may be any continuous functions of~$t'$,
and the~$\alpha_{ik}$'s are the co-efficients of an orthogonal transformation for
all values of~$t'$. If we map out the \Erratum{curves}{surfaces} $t' = \text{const.}$, as also $x_{1}' = \text{const.}$
and $x_{2}' = \text{const.}$ by our graphical method, then the surfaces
of the first family are again planes that coincide with the planes
$t = \text{const.}$; on the other hand, the other two families of \Erratum{curves}{surfaces} are
curved surfaces. The transformation formulæ are no longer linear.
Under these circumstances we achieve an important aim, when
investigating the motion of systems of point-masses, such as
planets, by choosing the co-ordinate system so that the functions
$x_{1}(t)$,~$x_{2}(t)$ that express how the space-co-ordinates of the point-masses
depend on the time become as simple as possible or at
least satisfy laws of the greatest possible simplicity. This is the
substance of the discovery of Copernicus that was afterwards
elaborated to such an extraordinary degree by Kepler, namely, that
there is in fact a co-ordinate system for which the laws of planetary
motion assume a much simpler and more expressive form than if
they are referred to a motionless earth. The work of Copernicus
produced a revolution in the philosophic ideas about the world inasmuch
a\Emph{s he shattered the belief in the absolute importance
of the earth}. His reflections as well as those of Kepler are purely
\Emph{kinematical} in character. Newton crowned their work by discovering
the true ground of the kinematical laws of Kepler to lie in
the fundamental \Emph{dynamical} law of mechanics and in the law of
attraction. Every one knows how brilliantly the mechanics of
Newton has been confirmed both for celestial as well as for earthly
phenomena. As we are convinced that it is valid universally and
not only for planetary systems, and as its laws are by no means
invariant with respect to the transformations~\Chg{\textEq{II}}{\textEq{(II)}}, it enables us to
fix the co-ordinate system in a manner independent of all individual
specification and much more definitely than is possible on the
kinematical view to which the principle of relativity~\textEq{(II)} leads.
\index{Relativity!of motion}%
\Par{Galilei's Principle of Inertia} (Newton's First Law of
\index{Inertia!principle of (Galilei's and Newton's)}%
\PageSep{153}
Motion) forms the foundation of mechanics. It states that a point-mass
which is subject to no forces from without executes a uniform
translation. Its world-line is consequently a straight line, and the
space-co-ordinates $x_{1}$,~$x_{2}$ of the point-mass are linear functions of
the time~$t$. If this principle holds for the two co-ordinate systems
connected by~\textEq{(II)}, then $x_{1}$~and~$x_{2}$ must become linear functions of~$t'$,
when linear functions of~$t'$ are substituted for $x_{1}'$~and~$x_{2}'$. It
straightway follows from this that the~$\alpha_{ik}$'s must be constants, and
that $\alpha_{1}$~and~$\alpha_{2}$ must be linear functions of~$t$; that is, the one Cartesian
co-ordinate system (in space) must be moving uniformly in
a straight line relatively to the other co-ordinate system. Conversely,
it is easily shown that if $\vC_{1}$,~$\vC_{2}$ are two \Emph{such} co-ordinate
systems, then if the principle of inertia and Newtonian mechanics
holds for~$\vC$ it will also hold for~$\vC'$. Thus, in mechanics, any two
``allowable'' co-ordinate systems are connected by formulæ
\[
\left.
\begin{alignedat}{4}
x_{1} &= \alpha_{11} x_{1}' &&+ \alpha_{12} x_{2}' &{}+{} && \gamma_{1} t' &+ \alpha_{1} \\
x_{2} &= \alpha_{21} x_{1}' &&+ \alpha_{22} x_{2}' &{}+{} && \gamma_{2} t' &+ \alpha_{2} \\
t &= && && & t' &+ a
\end{alignedat}
\right\}
\Chg{\textTag{III}}{\textTag{(III)}}
\]
in which the~$\alpha_{ik}$'s are constant co-efficients of an orthogonal transformation,
and $a$,~$\alpha_{i}$ and~$\gamma_{i}$ are arbitrary constants. Every transformation
of this kind represents a transition from one allowable
co-ordinate system to another. (This is the \Emph{Principle of Relativity
of Galilei and Newton}.) The essential feature of this
transition is that, if we disregard the naturally arbitrary directions
of the axis in space and the arbitrary initial point, there is invariance
with respect to the transformations
\[
x_{1} = x_{1}' + \gamma_{1} t',\qquad
x_{2} = x_{2}' + \gamma_{2} t',\qquad
t = t'\Add{.}
\Tag{(1)}
\]
In our graphical representation (\textit{vide} \Fig{7}) $x_{1}'$,~$x_{2}'$,~$t'$ would be
the co-ordinates taken with respect to a rectilinear set of axes in
which the $x_{1}'$-,~$x_{2}'$-axes coincide with the $x_{1}$-,~$x_{2}$-axes, whereas the
new $t'$-axis has some new direction. The following considerations
show that the laws of Newtonian mechanics are not altered in passing
from one co-ordinate system~$\vC$ to another~$\vC'$. According to the
law of attraction the gravitational force with which one point-mass
acts on another at a certain moment is a vector, in space, which is
independent of the co-ordinate system (as is also the vector that
connects the simultaneous positions of both point-masses with one
another). Every force, no matter what its physical origin, must
be the same kind of magnitude; this is entailed in the assumptions
of Newtonian mechanics, which demands a physics that satisfies
this assumption in order to be able to give a content to its conception
of force. We may prove, for example, in the theory of
\PageSep{154}
elasticity that the stresses (as a consequence of their relationship
to deformation quantities) are of the required kind.
Mass is a scalar that is independent of the co-ordinate system.
Finally, on account of the transformation formulæ that result from~\Eq{(1)}
for the motion of a point-mass,
\[
\frac{dx_{1}}{dt} = \frac{dx_{1}'}{dt'} + \gamma_{1},\
\frac{dx_{2}}{dt} = \frac{dx_{2}'}{dt'} + \gamma_{2};\quad
\frac{d^{2}x_{1}}{dt^{2}} = \frac{d^{2}x_{1}'}{dt'^{2}},\
\frac{d^{2}x_{2}}{dt^{2}} = \frac{d^{2}x_{2}'}{dt'^{2}}
\]
not the velocity, but the acceleration is a vector (in space) independent
of the co-ordinate system. Accordingly, the fundamental
law: \Emph{mass} times \Emph{acceleration} = \Emph{force}, has the required
invariant property.
According to Newtonian mechanics the centre of inertia of
every isolated mass-system not subject to external forces moves in
a straight line. If we regard the sun and his planets as such a
system, there is no meaning in asking whether the centre of inertia
of the solar system is at rest or is moving with uniform translation.
The fact that astronomers, nevertheless, assert that the sun is
moving towards a point in the constellation of Hercules, is based
on the statistical observation that the stars in that region seem on
the average to diverge from a certain centre---just as a cluster of
trees appears to diverge as we approach them. If it is certain that
the stars are on the average at rest, that is, that the centre of
inertia of the stellar firmament is at rest, the statement about the
sun's motion follows. It is thus merely an assertion about the
relative motion of the centre of inertia and of that of the stellar
firmament.
To grasp the true meaning of the principle of relativity, one
must get accustomed to thinking not in ``space,'' nor in ``time,''
but ``in the world,'' that is in \Emph{space-time}. Only the coincidence
(or the immediate succession) of two events in space-time has a
meaning that is directly evident, it is just the fact that in these
cases space and time cannot be dissociated from one another
absolutely that is asserted by the principle of relativity. Following
the mechanistic view, according to which all physical happening
can be traced back to mechanics, we shall assume that not only
mechanics but the whole of the physical uniformity of Nature is
subject to the principle of relativity laid down by Galilei and
Newton, which states \emph{that it is impossible to single out from the
systems of reference that are equivalent for mechanics and of which
each two are correlated by the formula of transformation~\Chg{\Eq{III}}{\textEq{(III)}} special
systems without specifying} \Emph{individual objects}. These formulæ
condition \Emph{the geometry of the four-dimensional world} in exactly
\PageSep{155}
\index{World ($=$ space-time)!-vectors}%
the same way as the group of transformation substitutions connecting
two Cartesian co-ordinate systems condition the Euclidean
geometry of three-dimensional space. A relation between world-points
has an objective meaning if, and only if, it is defined by such
arithmetical relations between the co-ordinates of the points as are
invariant with respect to the transformations~\textEq{(III)}. Space is said
to be \Emph{homogeneous} at all points and homogeneous in all directions
at every point. These assertions are, however, only parts of the
\Emph{complete statement of homogeneity} that all Cartesian co-ordinate
\index{Homogeneity!of the world}%
systems are equivalent. In the same way the principle
of relativity determines exactly the sense in which the \emph{world}
($=$~space-time as the ``form'' of phenomena, not its ``accidental''
non-homogeneous material content) is homogeneous.
It is indeed remarkable that two mechanical events that are
fully alike kinematically, may be different dynamically, as a comparison
of the dynamical principle of relativity~\textEq{(III)} with the much
more general kinematical principle of relativity~\textEq{(II)} teaches us. A
rotating spherical mass of fluid existing all alone, or a rotating fly-wheel,
cannot in itself be distinguished from a spherical fluid mass
or a fly-wheel at rest; in spite of this the ``rotating'' sphere becomes
flattened, whereas the one at rest does not change its shape, and
stresses are called up in the rotating fly-wheel that cause it to
burst asunder, if the rate of rotation be sufficiently great, whereas
\index{Rotation!general@{(general)}}%
\index{Rotation!relativity of}%
no such effect occurs in the case of a fly-wheel which is at rest.
The cause of this varying behaviour can be found only in the
``metrical structure of the world,'' that reveals itself in the centrifugal
forces as an active agent. This sheds light on the idea quoted
from Riemann above; if there corresponds to metrical structure (in
this case that of the world and not the fundamental metrical tensor
of space) something just as real, which acts on matter by means of
forces, as the something which corresponds to Maxwell's stress
tensor, then we must assume that, conversely, matter also reacts on
this real something. We shall revert to this idea again later in
Chapter~IV\@.
For the present we shall call attention only to the linear
character of the transformation formulæ~\textEq{(III)}; this signifies that
\Emph{the world is a four-dimensional affine space}. To give a
systematic account of its geometry we accordingly use \Emph{world-vectors}
or displacements in addition to world-points. A displacement
of the world is a transformation that assigns to every world-point~$P$
a world-point~$P'$, and is characterised by being expressible in
an allowable co-ordinate system by means of equations of the form
\[
x_{i}' = x_{i} + \Typo{a}{\alpha}_{i}\qquad
(i = 0, 1, 2, 3)
\]
\PageSep{156}
in which the~$x_{i}$'s denote the four space-time-co-ordinates of~$P$
($t$~being represented by~$x_{\Typo{o}{0}}$), and the~$x_{i}'$'s are those of~$P'$ in this co-ordinate
system, whereas the~$\Typo{a}{\alpha}_{i}$'s are constants. This conception
is independent of the allowable co-ordinate system selected. The
displacement that transforms $P$ into~$P'$ (or transfers $P$ to~$P'$) is
denoted by~$\Vector{PP'}$. The world-points and displacements satisfy all
the axioms of the affine geometry whose dimensional number is
$n = 4$. Galilei's Principle of Inertia (Newton's First Law of
Motion) is an affine law; it states what motions realise the
straight lines of our four-dimensional affine space (``world''),
namely, those executed by point-masses moving under no forces.
From the \Emph{affine} point of view we pass on to the \Emph{metrical} one.
\index{Metrics or metrical structure}%
From the graphical picture, which gave us an affine view of the
world (one co-ordinate being suppressed), we can read off its
essential metrical structure; this is quite different from that of
Euclidean space. The world is ``stratified''; the planes, $t = \text{const.}$,
in it have an absolute meaning. After a unit of time has been
chosen, each two world-points $A$~and~$B$ have a definite time-difference,
the time-component of the vector $\Vector{AB} = \vx$; as is
generally the case with vector-components in an affine co-ordinate
system, the time-component is a linear form~$t(\vx)$ of the arbitrary
vector~$\vx$. The vector~$\vx$ points into the past or the future according
as $t(\vx)$~is negative or positive. Of two world-points $A$ and~$B$, $A$~is
earlier than, simultaneous with, or later than~$B$, according as
\[
t(\Vector{AB}) > 0,\ = 0,\ \text{or}\ < 0.
\]
Euclidean geometry, however, holds in each ``stratum''; it is
based on a definite quadratic form, which is in this case defined
only for those world-vectors~$\vx$ that lie in one and the same
stratum, that is, that satisfy the equation $t(\vx) = 0$ (for there is
sense only in speaking of the distance between \Emph{simultaneous}
positions of two point-masses). Whereas, then, the \Emph{metrical
structure} of Euclidean geometry is based on a definitely positive
quadratic form, that \Emph{of Galilean geometry is based on}
1. \emph{A linear form $t(\vx)$ of the arbitrary vector~$\vx$} (the ``duration''
of the displacement~$\vx$).
{\Loosen 2. \emph{A definitely positive quadratic form~$(\vx\Com \vx)$} (the square of the
``length'' of~$\vx$), \emph{which is defined only for the three-dimensional
linear manifold of all the vectors~$\vx$ that satisfy the equation
$t(\vx) = 0$}.}
We cannot do without a definite space of reference, if we wish to
form a picture of physical conditions. Such a space depends on the
\PageSep{157}
choice of an arbitrary displacement~$\ve$ in the world (within which
the time-axis falls in the picture), and is then defined by the convention
that all world-points that lie on a straight line of direction~$\ve$,
meet at the \Emph{same point of space}. In geometrical language, we
are merely dealing with the process of \Emph{parallel projection}. To
\index{Parallel!projection}%
\index{Projection}%
arrive at an appropriate formulation we shall begin with some
geometrical considerations that relate to an arbitrary $n$-dimensional
affine space. To enable us to form a picture of the processes we
shall confine ourselves to the case $n = 3$. Let us take a family of
straight lines in space all drawn parallel to the vector~$\ve$ ($\neq \Typo{0}{\0}$). If we
look into space along these rays, all the space-points that lie behind
one another in the direction of such a straight line would coincide;
it is in no wise necessary to specify a plane on to which the points are
projected. Hence our definition assumes the following form.
Let~$\ve$, a vector differing from~$\Typo{0}{\0}$, be given. If $A$~and~$A'$ are two
points such that $\Vector{AA'}$~is a multiple of~$\ve$, we shall say that they pass
into one and the same point~$\vA$ of the \Emph{minor space} defined by~$\ve$.
\index{Minor space}%
We may represent~$\vA$ by the straight line parallel to~$\ve$, on which all
these coincident points $A$,~$A'$\Add{,}~\dots\ in the minor space lie. Since every
displacement~$\vx$ of the space transforms a straight line parallel to~$\ve$
again into one parallel to~$\ve$, $\vx$~brings about a definite displacement~$\vx$
of the minor space; but each two displacements $\vx$~and~$\vx'$ become
coincident in the minor space, if their difference is a multiple of~$\ve$.
We shall denote the transition to the minor space, ``the projection
in the direction of~$\ve$,'' by printing the symbols for points and displacements
in heavy oblique type. Projection converts
\[
\text{$\lambda \vx$, $\vx + \vy$, and $\Vector{AB}$ into $\lambda x$, $x + y$, $\Vector{\sfA\sfB}$}
\]
that is, the projection has a true affine character; this means that
in the minor space affine geometry holds, of which the dimensions
are less by one than those of the original ``complete'' space.
If the space is \Emph{metrical} in the Euclidean sense, that is, if it is
based on a non-degenerate quadratic form which is its metrical
groundform, $Q(\vx) = (\vx\Com \vx)$,---to simplify the picture of the process we
shall keep the case for which $Q$~is definitely positive in view, but
the line of proof is applicable generally,---then we shall obviously
ascribe to the two points of the minor space, which two straight
lines parallel to~$\ve$ appear to be, when we look into the space in the
direction of~$\ve$, a distance equal to the perpendicular distance
between the two straight lines. Let us formulate this analytically.
The assumption is that $(\ve\Com \ve) = e \neq \Typo{0}{\0}$. Every displacement~$\vx$ may
be split up uniquely into two summands
\[
\vx = \xi \ve + \vx^{*}\Add{,}
\Tag{(2)}
\]
\PageSep{158}
of which the first is proportional to~$\ve$ and the second is perpendicular
to it, viz.:\Add{---}
\[
(\vx^{*}\Com \ve) = 0,\qquad
\xi = \frac{1}{e}(\vx\Com \ve)\Add{.}
\Tag{(3)}
\]
We shall call~$\xi$ the \Emph{height} of the displacement~$\vx$ (it is the difference
\index{Height of displacement}%
of height between $A$~and~$B$, if $\vx = \Vector{AB}$). We have
\[
(\vx\Com \vx) = e\xi^{2} + (\vx^{*}\Com \vx^{*})\Add{.}
\Tag{(4)}
\]
$\vx$~is characterised fully, if its height~$\xi$ and the displacement~$\sfx$ of
the minor space produced by~$\vx$ are given; we write
\index{Space!projection@{(as projection of the world)}}%
\[
\vx = \xi \mid \sfx\Add{.}
\]
The ``complete'' space is ``split up'' into height and minor space,
\index{Resolution of tensors into space and time of vectors}%
the ``position-difference''~$\vx$ of two points in the complete space is
split up into the difference of height~$\xi$, and the difference of position~$\sfx$
in the minor space. There is a meaning not only in saying that
two points in space coincide, but also in saying that two points in
the minor space coincide or have the same height, respectively.
Every displacement~$\sfx$ of the minor space is produced by one \Emph{and
only one} displacement~$\vx^{*}$ of the complete space, this displacement
being orthogonal to~$\ve$. The relation between $\vx^{*}$ and~$\sfx$ is singly
reversible and affine. The defining equation
\[
(\sfx\Com \sfx) = (\vx^{*}\Com \vx^{*})
\]
endows the minor space with a metrical structure that is based on
the quadratic groundform~$(\sfx\Com \sfx)$. This converts~\Eq{(4)} into the fundamental
equation of Pythagoras
\[
(\vx\Com \vx) = e\xi^{2} + (\sfx\Com \sfx)
\Tag{(5)}
\]
which, for two displacements, may be generalised in the form
\[
(\vx\Com \vy) = e\xi\eta + (\sfx\Com \sfy)\Add{.}
\Tag{(5')}
\]
Its symbolic form is clear.
These considerations, in so far as they concern affine space, may
be applied directly. The complete space is the four-dimensional
world: $\ve$~is any vector pointing in the direction of the future: the
minor space is what we generally call \Emph{space}. Each two world-points
that lie on a world-line parallel to~$\ve$ project into the same
space-point. This space-point may be represented graphically by
the straight line parallel to~$\ve$ and may be indicated permanently
by a point-mass at rest, that is, one whose world-line is just that
straight line. The metrical structure, however, is, according to the
Galilean principle of relativity, of a kind different from that we
assumed just above. This necessitates the following modifications.
Every world-displacement~$\vx$ has a definite duration $t(\vx) = t$ (this
\PageSep{159}
takes the place of ``height'' in our geometrical argument) and
produces a displacement~$\sfx$ in the minor space; it splits up according
to the formula
\[
\vx = t \mid \sfx
\]
{\Loosen corresponding to the resolution into space and time. In particular
every space-displacement~$\sfx$ may be produced by one and only one
world-displacement~$\vx^{*}$, which satisfies the equation $t(\vx^{*}) = 0$. The
quadratic form $(\vx^{*}\Com \vx^{*})$ as defined for such vectors~$\vx^{*}$, impresses on
space its Euclidean metrical structure}
\[
(\sfx\Com \sfx) = (\vx^{*}\Com \vx^{*})\Add{.}
\]
The space is dependent on the direction of projection. In actual
cases the direction of projection may be fixed by any point-mass
moving with uniform translation (or by the centre of mass of a
closed isolated mass-system).
We have set forth these details with pedantic accuracy so as to
be armed at least with a set of mathematical conceptions which
have been sifted into a form that makes them immediately applicable
to Einstein's principle of relativity for which our powers of intuition
are much more inadequate than for that of Galilei.
To return to the realm of physics. The discovery \Emph{that light is
propagated with a finite velocity} gave the death-blow to the
natural view that things exist simultaneously with their perception.
As we possess no means of transmitting time-signals more rapid
than light itself (or wireless telegraphy) it is of course impossible to
measure the velocity of light by measuring the time that elapses
whilst a light-signal emitted from a station~$A$ travels to a station~$B$.
In 1675 \Chg{Roemer}{Römer} calculated this velocity from the apparent irregularity
of the time of revolution of Jupiter's moons, which took
place in a period which lasted exactly one year: he argued that it
would be absurd to assume a mutual action between the earth and
Jupiter's satellites such that the period of the earth's revolution
caused a disturbance of so considerable an amount in the satellites.
Fizeau confirmed the discovery by measurements carried out on
the earth's surface. His method is based on the simple idea of
making the transmitting station~$A$ and the receiving station~$B$
coincide by reflecting the ray, when it reaches~$B$, back to~$A$.
According to these measurements we have to assume that the
centre of the disturbances is propagated in concentric spheres with
a constant velocity~$c$. In our graphical picture (one space-co-ordinate
again being suppressed) the propagation of a light-signal
emitted at the world-point~$O$ is represented by the circular cone
depicted, which has the equation
\[
c^{2} t^{2} - (x_{1}^{2} + x_{2}^{2}) = 0\Add{.}
\Tag{(6)}
\]
\PageSep{160}
Every plane given by $t = \text{const.}$ cuts the cone in a circle composed
of those points which the light-signal has reached at the moment~$t$.
The equation~\Eq{(6)} is satisfied by all and only by all those world-points
reached by the light-signal (provided that $t > 0$). The
question again arises on what space of reference this description of
the event is based. The \Emph{aberration of the stars} shows that,
\index{Aberration}%
relatively to this reference space, the earth moves in agreement
with Newton's theory, that is, that it is identical with an allowable
reference space as defined by Newtonian mechanics. The propagation
in concentric spheres is, however, certainly not invariant
with respect to the Galilei transformations~\textEq{(III)}; for a $t'$-axis that
is drawn obliquely intersects the planes $t = \text{const.}$ at points that
are excentric to the circles of propagation. Nevertheless, this
cannot be regarded as an objection to Galilei's principle of relativity,
if, accepting the ideas that have long held sway in physics, we
\index{Aether@{Æther}!(as a substance)}%
assume that light is transmitted by a material medium, the \Emph{æther},
whose particles are movable with regard to one another. The
conditions that obtain in the case of light are exactly similar to
those that bring about concentric circles of waves on a surface of
water on to which a stone has been dropped. The latter phenomenon
certainly does not justify the conclusion that the equations
of hydrodynamics are contrary to Galilei's principle of relativity.
For the medium itself, the water or the æther respectively, whose
particles are at rest with respect to one another, if we neglect the
relatively small oscillations, furnishes us with the same system of
reference as that to which the statement concerning the concentric
transmission is referred.
To bring us into closer touch with this question we shall here
insert an account of optics in the theoretical guise that it has preserved
since the time of Maxwell under the name of the theory of
moving electromagnetic fields.
\Section{20.}{The Electrodynamics of Moving Fields
Lorentz's Theorem of Relativity}
In passing from stationary electromagnetic fields to moving
electromagnetic fields (that is, to those that vary with the time) we
have learned the following:---
1. The so-called electric current is actually composed of moving
\index{Current!conduction}%
electricity: a charged coil of wire in rotation produces a magnetic
field according to the law of Biot and Savart. If $\rho$~is the density
of charge, $\vv$~the velocity, then clearly the density~$\vs$ of this convection
current $= \rho\vv$; yet, if the Biot-Savart Law is to remain
valid in the old form, $\vs$~must be measured in other units. Thus
\PageSep{161}
we must set $\vs = \dfrac{\rho\vv}{c}$, in which $c$~is a universal constant having the
dimensions of a velocity. The experiment carried out by Weber
and Kohlrausch, repeated later by Rowland and Eichenwald, gave
a value of~$c$ that was coincident with that obtained for the velocity
of light, within the limits of errors of observation (\textit{vide} \FNote{2}).
We call $\dfrac{\rho}{c} = \rho'$ the electromagnetic measure of the charge-density
\index{Measure!electrostatic and electromagnetic}%
and, so as to make the density of electric force $= \rho' \vE'$ in electromagnetic
units, too, we call $\vE' = c\vE$ the electromagnetic measure
\index{Electrical!intensity of field}%
\index{Electromagnetic field!and electrostatic units}%
\index{Intensity of field}%
of the field-intensity.
2. A moving magnetic field induces a current in a homogeneous
\index{Induction, magnetic!law of}%
wire. It may be determined from the physical law $\vs = \sigma\vE$ and
\Emph{Faraday's Law of Induction}; the latter asserts that the induced
\index{Faraday's Law of Induction}%
electromotive force is equal to the time-decrement of the magnetic
flux through the conductor; hence we have
\[
\int \vE'\, d\vr = - \frac{d}{dt} \int B_{n}\, do\Add{.}
\Tag{(7)}
\]
On the left there is the line-integral along a closed curve, on the
right the surface-integral of the normal components of the magnetic
induction~$\vB$, taken over a surface which fills the curve. The flux
of induction through the conducting curve is uniquely determined
because
\[
\div \vB = 0\Add{;}
\Tag{(8')}
\]
that is, there is no real magnetism. By Stokes' Theorem we get
from~\Eq{(7)} the differential law
\[
\curl \vE + \frac{1}{c}\, \frac{\dd \vB}{\dd t} = \Typo{0}{\0}\Add{.}
\Tag{(8)}
\]
The equation $\curl \vE = \Typo{0}{\0}$, which holds for statistical cases, is hence
increased by the term $\dfrac{1}{c}\, \dfrac{\dd \vB}{\dd t}$ on the left, which is a derivative of
the time. All our electro-technical sciences are based on it; thus
the necessity for introducing it is justified excellently by actual
experience.
3. On the other hand, in Maxwell's time, the term which was
\index{Continuity, equation of!electricity@{of electricity}}%
\index{Maxwell's!theory!(general case)}%
added to the fundamental equation of magnetism
\[
\curl \vH = \vs
\Tag{(9)}
\]
was purely hypothetical. In a moving field, such as in the discharge
of a \Typo{condensor}{condenser}, we cannot have $\div \vs = 0$, but in place of it
the ``equation of continuity''
\[
\frac{1}{c}\, \frac{\dd \rho}{\dd t} + \div \vs = 0
\Tag{(10)}
\]
\PageSep{162}
must hold. This gives expression to the fact that the current consists
of moving electricity. Since $\rho = \div \vD$, we find that not~$\vs$,
but $\vs + \dfrac{1}{c}\, \dfrac{\dd \vD}{\dd t}$ must be irrotational, and this immediately suggests
that instead of equation~\Eq{(9)} we must write for moving fields
\[
\curl \vH - \frac{1}{c}\, \frac{\dd \vD}{\dd t} = \vs\Add{.}
\Tag{(11)}
\]
Besides this, we have just as before
\[
\div \vD = \rho\Add{.}
\Tag{(11')}
\]
From \Eq{(11)} and~\Eq{(11')} we arrive conversely at the equation of continuity~\Eq{(10)}.
It is owing to the additional member $\dfrac{1}{c}\, \dfrac{\dd \vD}{\dd t}$ (Maxwell's
\Emph{displacement current}), a differential co-efficient with respect to
\index{Displacement current}%
the time, that electromagnetic disturbances are propagated in the
æther with the finite velocity~$c$. It is the basis of the electromagnetic
theory of light, which interprets optical phenomena with
such wonderful success, and which is experimentally verified in the
well-known experiments of Hertz and in wireless telegraphy, one of
its technical applications. This also makes it clear that these laws
are referred to the same reference-space as that for which the concentric
propagation of light holds, namely, the ``fixed'' æther. The
laws involving the specific characteristics of the matter under consideration
have yet to be added to Maxwell's field-equations \Eq{(8)} and~\Eq{(8')},
\Eq{(11)}~and~\Eq{(11')}.
We shall, however, here consider only the conditions in the
æther; in it
\[
\vD = \vE\quad\text{and}\quad
\vH = \vB,
\]
and Maxwell's equations are
\begin{alignat*}{3}
%[** TN: Omitted right brace]
\curl \vE &+ \frac{1}{c}\, \frac{\dd \vB}{\dd t} &&= \Typo{0}{\0},\qquad
\div \vB &&= 0\Add{,}
\Chg{\Tag{(12_{1})}}{\Tag{(12)}} \\
\curl \vB &- \frac{1}{c}\, \frac{\dd \vE}{\dd t} &&= \vs,\qquad
\div \vE &&= \rho\Add{.}
\Chg{\Tag{(12_{11})}}{\Tag{(12')}}
\end{alignat*}
According to the atomic theory of electrons these are generally
valid exact physical laws. This theory furthermore sets $\vs = \dfrac{\rho \vv}{c}$, in
which $\vv$~denotes the velocity of the matter with which the electric
charge is associated.
The \Emph{force} which acts on the masses consists of components
\index{Joule (heat-equivalent)}%
arising from the electrical and the magnetic field: its density is
\index{Electrical!displacement}%
\[
\vp = \rho \vE + [\vs\Com \vB]\Add{.}
\Tag{(13)}
\]
\PageSep{163}
\index{Divergence@{Divergence (\emph{div})}!(more general)}%
Since $\vs$~is parallel to~$\vv$, the work performed on the electrons per
unit of time and of volume is
\[
\vp · \vv = \rho \vE · \vv = c(\vs\Com \vE) = \vs · \vE'.
\]
It is used in increasing the kinetic energy of the electrons, which
is partly transferred to the neutral molecules as a result of collisions.
This augmented molecular motion in the interior of the conductor
expresses itself physically as the heat arising during this phenomenon,
as was pointed out by Joule. We find, in fact, experimentally
that $\vs · \vE'$ is the quantity of heat produced per unit of time
and per unit of volume by the current. The energy used up in
this way must be furnished by the instrument providing the current.
If we multiply equation~\Chg{\Eq{(12_{1})}}{\Eq{(12)}} by~$-\vB$, equation~\Chg{\Eq{(12_{11})}}{\Eq{(12')}} by~$\vE$ and add,
we get
\[
-c · \div [\vE\Com \vB]
- \frac{\dd}{\dd t}(\tfrac{1}{2}\vE^{2} + \tfrac{1}{2}\vB^{2})
= c(\vs\Com \vE).
\]
If we set
\[
[\vE\Com \vB] = \vs\Add{,}\qquad
\tfrac{1}{2}\vE^{2} + \tfrac{1}{2}\vB^{2} = W
\]
and integrate over any volume~$V$, this equation becomes
\[
-\frac{d}{dt} \int_{V} W\, dV
+ c \int_{\Omega} S_{n}\, do
= \int_{V} c(\vs\Com \vE)\, dV.
\]
The second member on the left is the integral, taken over the outer
surface of~$V_{1}$, of the component~$s_{n}$ of~$\vs$ along the inward normal.
On the right-hand side we have the work performed on the volume~$V$
per unit of time. It is compensated by the decrease of energy
$\Dint W\, dV$ contained in~$V$ and by the energy that flows into the portion
of space~$V$ from without. Our equation is thus an expression of
the \Emph{energy theorem}. \Emph{It confirms the assumption which we
made initially about the density~$W$ of the field-energy}, and
\index{Density!based@{(based on the notion of substance)}}%
we furthermore see that $\Typo{c\vS}{c\vs}$, familiarly known as Poynting's vector,
\index{Poynting's vector}%
\index{Vector!potential}%
represents the \Emph{energy stream or energy-flux}.
\index{Energy-steam or energy-flux}%
The field-equations~\Eq{(12)}\Add{,~\Eq{(12')}} have been integrated by Lorentz in the
following way, on the assumption that the distribution of charges
and currents are known. The equation $\div \vB = 0$ is satisfied by
setting
\[
-\vB = \curl \vf
\Tag{(14)}
\]
in which $-\vf$~is the vector potential. By substituting this in the
\index{Potential!vector-}%
first equation above we get that $\vE - \dfrac{1}{c}\, \Typo{\dfrac{d \vf}{dt}}{\dfrac{\dd \vf}{\dd t}}$ is irrotational, so that we
can set
\[
\vE - \frac{1}{c}\, \frac{\dd \vf}{\dd t} = \grad\phi\Add{,}
\Tag{(15)}
\]
\PageSep{164}
\index{Light!electromagnetic theory of}%
\index{Propagation!of electromagnetic disturbances}%
\index{Propagation!of light}%
\index{Retarded potential}%
in which $-\phi$~is the scalar potential. We may make use of the
\index{Potential!electrostatic}%
\index{Potential!retarded}%
arbitrary character yet possessed by~$\vf$ by making it fulfil the subsidiary
condition
\[
\frac{1}{c}\, \frac{\dd \phi}{\dd t} + \div \vf = 0.
\]
This is found to be expedient for our purpose (whereas for a
stationary field we assumed $\div \vf = 0$). If we introduce the
potentials in the two latter equations, we find by an easy
calculation
\begin{alignat*}{2}
-\frac{1}{c^{2}}\, \frac{\dd^{2} \phi}{\dd t^{2}} &+ \Delta\phi &&= \rho\Add{,}
\Tag{(16)} \\
-\frac{1}{c^{2}}\, \frac{\dd^{2} \vf}{\dd t^{2}} &+ \Delta\vf &&= \vs\Add{.}
\Tag{(16')}
\end{alignat*}
An equation of the form~\Eq{(16)} denotes a wave disturbance travelling
with the velocity~$c$. In fact, just as Poisson's equation $\Delta\phi = \rho$ has
\index{Velocity!light@{of light}}%
the solution
\[
-4\pi \phi = \int \frac{\rho}{r}\, dV
\]
so \Eq{(16)}~has the solution
\[
-4\pi \phi = \int \frac{\rho\left(t - \dfrac{r}{c}\right)}{r}\, dV;
\]
on the left-hand side of which $\phi$~is the value at a point~$O$ at time~$t$;
$r$~is the distance of the source~$P$, with respect to which we integrate,
from the point of emergence~$O$; and within the integral the value
of~$\rho$ is that at the point~$P$ at time $t - \dfrac{r}{c}$. Similarly \Eq{(16')}~has the
solution
\[
-4\pi \vf = \int \frac{\vs\left(t - \dfrac{r}{c}\right)}{r}\, dV.
\]
The field at a point does not depend on the distribution of charges
and currents at the same moment, but the determining factor for
every point is the moment that lies back just as many $\left(\dfrac{r}{c}\right)$'s as
the disturbance propagating itself with the velocity~$c$ takes to travel
from the source to the point of emergence.
Just as the expression for the potential (in Cartesian co-ordinates),
namely,
\[
\Delta\phi
= \frac{\dd^{2} \phi}{\dd x_{1}^{2}}
+ \frac{\dd^{2} \phi}{\dd x_{2}^{2}}
+ \frac{\dd^{2} \phi}{\dd x_{3}^{2}}
\]
\PageSep{165}
is invariant with respect to linear transformations of the variables
$x_{1}$,~$x_{2}$,~$x_{3}$, which are such that they convert the quadratic form
\[
x_{1}^{2} + x_{2}^{2} + x_{3}^{2}
\]
into itself, so the expression which takes the place of this expression
for the potential when we pass from statical to moving
\index{Potential!electromagnetic}%
\index{Potential!retarded}%
\index{Retarded potential}%
fields, namely,\Pagelabel{165}
\[
-\frac{1}{c^{2}}\, \frac{\dd^{2} \phi}{\dd t^{2}}
+ \frac{\dd^{2} \phi}{\dd x_{1}^{2}}
+ \frac{\dd^{2} \phi}{\dd x_{2}^{2}}
+ \frac{\dd^{2} \phi}{\dd x_{3}^{2}}
\quad\text{(\Emph{retarded potentials})}
\]
is an invariant for those linear transformations of the four co-ordinates,
$t$, $x_{1}$,~$x_{2}$,~$x_{3}$, the so-called Lorentz transformations, that
\index{Lorentz!Einstein@{-Einstein Theorem of Relativity}}%
transform the indefinite form
\[
-c^{2}t^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2}
\Tag{(17)}
\]
into itself. Lorentz and Einstein recognised that not only equation~\Eq{(16)}
but also the \emph{whole system of electromagnetic laws for the æther
has this property of invariance, namely, that these laws are the expression
of invariant relations between tensors which exist in a four-dimensional
affine space whose co-ordinates are $t$, $x_{1}$,~$x_{2}$,~$\Typo{x}{x_{3}}$ and upon
which a non-definite metrical structure is impressed by the form~\Eq{(17)}}.
This is the \Emph{Lorentz-Einstein Theorem of Relativity}.
\index{Relativity!theorem of (Lorentz-Einstein)}%
To prove the theorem we shall choose a new unit of time by
putting $ct = x_{0}$. The co-efficients of the metrical groundform are
then
\[
g_{ik} = 0\quad (i \neq k);\qquad
g_{ii} = \epsilon_{i},
\]
in which $\epsilon_{0} = -1$, $\epsilon_{1} = \epsilon_{2} = \epsilon_{3} = +1$; so that in passing from
components of a tensor that are co-variant with respect to an index~$i$
to the contra-variant components of that tensor we have only to
% [** TN: Ordinal]
multiply the $i$th~component by the sign of~$\epsilon_{i}$. The question of continuity
\index{Electromagnetic field!potential}%
for electricity~\Eq{(10)} assumes the desired invariant form
\[
\sum_{i=0}^{3} \frac{\dd s^{i}}{\dd x_{i}} = 0
\]
if we introduce $s^{0} = \rho$, and $s^{1}$,~$s^{2}$,~$s^{3}$, which are equal to the components
of~$\vs$, as the four contra-variant components of a vector
in the above four-dimensional space, namely, of the ``$4$-vector
current''. Parallel with this---as we see from \Eq{(16)}~and~\Eq{(16')}---we
\index{Four-current ($4$-current)}%
must combine
\[
\text{$\phi_{0} = \phi$ and the components of~$\vf$, namely, $\phi^{1}$, $\phi^{2}$, $\phi^{3}$,}
\]
to make up the contra-variant components of a four-dimensional
vector, which we call the electromagnetic potential; of its co-variant
components, the $0$th, i.e.\ $\phi_{0} = -\phi$, whereas the three
\PageSep{166}
\index{Field action of electricity!energy}%
others $\phi_{1}$,~$\phi_{2}$,~$\phi_{3}$ are equal to the components of~$\vf$. The equations
\Eq{(14)} and~\Eq{(15)}, by which the field-quantities $\vB$~and~$\vE$ are derived
from the potentials, may then be written in the invariant form
\[
\frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}} = F_{ik}
\Tag{(18)}
\]
in which we set
\[
\vE = (F_{10}, F_{20}, F_{30}),\qquad
\vB = (F_{23}, F_{31}, F_{12}).
\]
This is then how we may combine electric and magnetic intensity
of field to make up a single linear tensor of the second order~$F$,
the ``field''. From~\Eq{(18)} we get the invariant equations
\[
\frac{\dd F_{kl}}{\dd x_{i}}
+ \frac{\dd F_{li}}{\dd x_{k}}
+ \frac{\dd F_{ik}}{\dd x_{l}} = 0\Add{,}
\Tag{(19)}
\]
and this is Maxwell's first system of equations~\Chg{\Eq{(12_{1})}}{\Eq{(12)}}. We took a
circuitous route in using Lorentz's solution and the potentials
\index{Lorentz!transformation}%
only so as to be led naturally to the proper combination of the
three-dimensional quantities, which converts them into four-dimensional
vectors and tensors. By passing over to contra-variant
components we get
\[
\vE = (F^{01}, F^{02}, F^{03}),\qquad
\vB = (F^{23}, F^{31}, F^{12}).
\]
Maxwell's second system, expressed invariantly in terms of four-dimensional
tensors, is now
\[
\sum_{k} \frac{\dd F^{ik}}{\dd x_{k}} = s^{i}\Add{.}
\Tag{(20)}
\]
If we now introduce the four-dimensional vector with the co-variant
components
\[
p_{i} = F_{ik} s^{k}
\Tag{(21)}
\]
% [** TN: Next equation displayed in the original]
(and the contra-variant components $p^{i} = F^{ik} s_{k}$)%
---following our previous practice of omitting the signs of sum\-ma\-tion---then
$p^{0}$~is the ``work-density,'' that is, the work per
unit of time and per unit of volume: $p^{0} = (\vs\Com \vE)$ [the unit of time is
to be adapted to the new measure of time $x_{0} = ct$], and $p^{1}$,~$p^{2}$,~$p^{3}$ are
the components of the density of force.
This fully proves the Lorentz Theorem of Relativity. \emph{We
notice here that the laws that have been obtained are exactly the
same as those which hold in the stationary magnetic field \Inum{(§\,9 \Eq{(62)})}
except that they have been transposed from three-dimensional to four-dimensional
space.} There is no doubt that the real mathematical
harmony underlying these laws finds as complete an expression as
is possible in this formulation in terms of four-dimensional tensors.
\PageSep{167}
Further, we learn from the above that, exactly as in the case of
three-dimensions, we may derive the ``$4$-force'' $= p_{i}$ from a symmetrical
\index{Four-force ($4$-force)}%
four-dimensional ``stress-tensor''~$S$, thus
\begin{gather*}
-p_{i} = \frac{\dd S_{i}^{k}}{\dd x_{k}}
\quad\text{or}\quad
-p^{i} = \frac{\dd S^{ik}}{\dd x_{k}}\Add{,}
\Tag{(22)} \\
S_{i}^{k} = F_{ir} F^{kr} - \tfrac{1}{2} \delta_{i}^{k} |F|^{2}\Add{.}
\Tag{(22')}
\end{gather*}
The square of the numerical value of the field (which is not necessarily
positive here) is
\[
|F|^{2} = \tfrac{1}{2} F_{ik} F^{ik}.
\]
We shall verify formula~\Eq{(22)} by direct calculation. We have\Pagelabel{167}
\[
\frac{\dd S_{i}^{k}}{\dd x_{k}}
= F_{ir}\, \frac{\dd F^{kr}}{\dd x_{k}}
+ F^{kr}\, \frac{\dd F_{ir}}{\dd x_{k}}
- \tfrac{1}{2} F^{kr}\, \frac{\dd F_{kr}}{\dd x_{i}}.
\]
The first term on the right gives us
\[
-F_{ir} s^{r} = -p_{i}.
\]
If we write the co-efficient of~$F^{kr}$ skew-symmetrically we get for
the second term
\[
\tfrac{1}{2} F^{kr}
\left(\frac{\dd F_{ir}}{\dd \Typo{x}{x_{k}}}
- \frac{\dd F_{ik}}{\dd x_{r}}\right)
\]
which, combined with the third, gives
\[
-\tfrac{1}{2} F^{kr}
\left(\frac{\dd F_{ik}}{\dd x_{r}}
+ \frac{\dd F_{kr}}{\dd x_{i}}
+ \frac{\dd F_{ri}}{\dd x_{k}}\right).
\]
The expression consisting of three terms in the brackets $= 0$, by~\Eq{(19)}.
Now $|F|^{2} = \vB^{2} - \vE^{2}$. Let us examine what the individual
components of~$S_{ik}$ signify, by separating the index~$\Typo{o}{0}$ from the
others $1$,~$2$,~$3$, in conformity with the partition into space and time.
$S^{00} = \text{the energy-density } W = \frac{1}{2}(\vE^{2} + \vB^{2})$\Add{,}
\index{Density!electricity@{(of electricity and matter)}}%
\index{Energy-density!(in the electric field)}%
$S^{\Typo{o}{0}i} = \text{the components of } \vS = [\vE\Com \vB]$\quad $i,k = (1, 2, 3)$\Add{,}
$S^{ik} = \text{the components of the Maxwell stress-tensor}$, which is
composed of the electrical and magnetic parts given in §\,9. Accordingly
% [** TN: Ordinal; others set in-line]
the $0$th~equation of~\Eq{(22)} expresses the law of energy. The
$1$st, $2$nd, and $3$rd have a fully analogous form. If, for a
moment, we denote the components of the vector $\dfrac{1}{c} \vS$ by $G^{1}$,~$G^{2}$,~$G^{3}$
and take $\vt^{(i)}$ to stand for the vector with the components $S^{i1}$,~$S^{i2}$,~$S^{i3}$
we get
\[
-p_{i} = \frac{\dd G^{i}}{\dd t} + \div \vt^{(i)}\Add{,}\qquad
(i = 1, 2, 3)\Add{.}
\Tag{(23)}
\]
The force which acts on the electrons enclosed in a portion of
\PageSep{168}
\index{Field action of electricity!momentum}%
space~$V$ produces an increase in time of momentum equal to itself
\index{Momentum!density}%
\index{Momentum!flux}%
numerically\Add{.} This increase is balanced, according to~\Eq{(23)}, by a
corresponding decrease of the \Emph{field-momentum} distributed in the
field with a density~$\dfrac{\vS}{c}$, and the addition of field-momentum from
% [** TN: Ordinal]
without. The current of the $i$th~component of momentum is given
by~$\vt^{(i)}$, and thus the \Emph{momentum-flux} is nothing more than the
\index{Energy-momentum, tensor@{Energy-momentum, tensor (cf.\ Energy-momentum)}}%
\index{Energy-momentum, tensor!(in the electromagnetic field)}%
\index{Energy-momentum, tensor!theorem of (in the special theory of relativity)}%
Maxwell stress-tensor. \emph{The Theorem of the Conservation of
Energy is only one component, the time-component, of a law which
is invariant for Lorentz transformations, the other components being
the space-components which express the conservation of momentum.}
The total energy as well as the total momentum remains unchanged:
they merely stream from one part of the field to
another, and become transformed from field-energy and field-momentum
into kinetic-energy and kinetic-momentum of matter,
and \textit{vice versa}. That is the simple physical meaning of the
formulæ~\Eq{(22)}. In accordance with it we shall in future refer
to the tensor~$S$ of the four-dimensional world as the \Emph{energy-momentum-tensor}
or, more briefly, as the \Emph{energy-tensor}.
Its symmetry tells us that the \Emph{density of momentum $= \dfrac{1}{c^{2}}$ \emph{times}
the energy-flux}. The field-momentum is thus very weak,
but, nevertheless, it has been possible to prove its existence by
demonstrating the pressure of light on a reflecting surface.
A Lorentz transformation is linear. Hence (again suppressing
one space co-ordinate in our graphical picture) we see that it is
tantamount to introducing a new affine co-ordinate system. Let
us consider how the fundamental vectors $\ve_{0}'$,~$\ve_{1}'$,~$\ve_{2}'$ of the new
co-ordinate system lie relatively to the original fundamental vectors
$\ve_{0}$,~$\ve_{1}$,~$\ve_{2}$, that is to the unit vectors in the direction of the~$x_{0}$ (or~$t$),
$x_{1}$,~$x_{2}$ axes. Since, for
\[
\vx = x_{0} \ve_{0} + x_{1} \ve_{1} + x_{2} \ve_{2}
= x_{0}' \ve_{0}' + x_{1}' \ve_{1}' + x_{2}' \ve_{2}',
\]
we must have
\[
-x_{0}^{2} + x_{1}^{2} + x_{2}^{2}
= -x_{0}'^{2} + x_{1}'^{2} + x_{2}'^{2}
\bigl[ = Q(\vx)\bigr]
\]
we get $Q(\ve_{0}') = -1$. Accordingly, the vector~$\ve_{0}'$ starting from~$O$
(i.e.\ the $t'$-axis) lies within the cone of light-propagation; the
parallel planes $t' = \text{const.}$ lie so that they cut ellipses from the
cone, the middle points of which lie on the $t'$-axis (see \Fig{7}); the
$x_{1}'$-, $x_{2}'$-axis are in the direction of conjugate diameters of these
elliptical sections, so that the equation of each is
\[
x_{1}'^{2} + x_{2}'^{2} = \text{const.}
\]
\PageSep{169}
As long as we retain the picture of a material æther, capable of
executing vibrations, we can see in Lorentz's Theorem of Relativity
\index{Relativity!principle of!(Einstein's special)}%
only a remarkable property of mathematical transformations; the
relativity theorem of Galilei and Newton remains the truly valid
one. We are, however, confronted with the task of interpreting
not only optical phenomena but all electrodynamics and its laws
as the result of a mechanics of the æther which satisfies Galilei's
Theorem of Relativity. To achieve this we must bring the field-quantities
into definite relationship with the density and velocity of
the æther. Before the time of Maxwell's electromagnetic theory of
light, attempts were made to do this for optical phenomena; these
efforts were partly, but never wholly, crowned with success. This
attempt was not carried on (\textit{vide} \FNote{3}) in the case of the more
comprehensive domain into which Maxwell relegated optical phenomena.
On the contrary, \Emph{the idea of a field existing in empty
space and not requiring a medium to sustain it} gradually
began to win ground. Indeed, even Faraday had expressed in
unmistakable language that not the field should derive its meaning
through its association with matter, but, conversely, rather that
particles of matter are nothing more than singularities of the field.
\Section{21.}{Einstein's Principle of Relativity}
\index{Aether@{Æther}!(in a generalised sense)}%
\index{Special principle of relativity}%
Let us for the present retain our conception of the æther. It
should be possible to determine the motion of a body, for example,
the earth, relative to the fixed or motionless æther. We are not
helped by aberration, for this only shows that this relative motion
\Emph{changes} in the course of a year. Let $A_{1}$,~$O$,~$A_{2}$ be three fixed points
on the earth that share in its motion. Suppose them to lie in a
straight line along the direction of the earth's motion and to be
equidistant, so that $A_{1}O = OA_{2} = l$, and let $v$~be the velocity of
translation of the earth through the æther; let $\dfrac{v}{c} = q$, which we
shall assume to be a very small quantity. A light-signal emitted
at~$O$ will reach~$A_{2}$ after a time~$\dfrac{l}{c - v}$ has elapsed, and $A_{1}$~after a time~$\dfrac{l}{c + v}$.
Unfortunately, this difference cannot be demonstrated, as
we have no signal that is more rapid than light and that we could
use to communicate the time to another place.\footnote
{It might occur to us to transmit time from one world-point to another by
carrying a clock that is marking time from one place to the other. In practice,
this process is not sufficiently accurate for our purpose. Theoretically, it is by
no means certain that this transmission is independent of the traversed path.
In fact, the theory of relativity proves that, on the contrary, they are dependent
on one another; cf.~§\,22.}
We have recourse
\PageSep{170}
to Fizeau's idea, and set up little mirrors at $A_{1}$~and $A_{2}$ which reflect
the light-ray back to~$O$. If the light-signal is emitted at the
moment~$O$, then the ray reflected from~$A_{2}$ will reach~$A$ after a time
\[
\frac{l}{c - v} + \frac{l}{c + v} = \frac{2lc}{c^{2} - v^{2}}
\]
whereas that reflected from~$A_{1}$ reaches~$O$ after a time
\[
\frac{l}{c + v} + \frac{l}{c - v} = \frac{2lc}{c^{2} - v^{2}}.
\]
There is now no longer a difference in the times. Let us, however,
now assume a third point~$A$ which participates in the translational
motion through the æther, such that $OA = l$, but that $OA$~makes
an angle~$\theta$ with the direction of~$OA$. In \Fig{8}, $O$,~$O'$,~$O''$ are the
successive positions of the point~$O$ at the time~$0$ at which the signal
is emitted, at the time~$t'$ at which it is reflected from the mirror~$A$
\Figure{8}
placed at~$A'$, and finally at the time $t' + t''$ at which it again reaches~$O$,
respectively. From the figure we get the proportion
\[
OA' : O''A' = OO' : O''O'.
\]
Consequently the two angles at~$A'$ are equal to one another. The
reflecting mirror must be placed, just as when the system is at
rest, perpendicularly to the rigid connecting line~$OA$, in order that
the light-ray may return to~$O$. An elementary trigonometrical
calculation gives for the \Emph{apparent rate of transmission in the
direction~$\theta$}
\[
\frac{2l}{t' + t''}
= \frac{c^{2} - v^{2}}{\sqrt{c^{2} - v^{2} \sin^{2}\theta}}\Add{.}
\Tag{(24)}
\]
It is thus dependent on the angle~$\theta$, which gives the direction of
transmission. Observations of the value of~$\theta$ should enable us to
determine the direction and magnitude of~$v$.
{\Loosen These observations were attempted in the celebrated \Emph{Michelson-Morley
experiment} (\textit{vide} \FNote{4}). In this, two mirrors $A$,~$A'$ are
\index{Michelson-Morley experiment}%
rigidly fixed to~$O$ at distances $l$,~$l'$, the one along the line of motion
\PageSep{171}
\index{Contraction-hypothesis of Lorentz and Fitzgerald}%
the other perpendicular to it. The whole apparatus may be rotated
about~$O$. By means of a transparent glass plate, one-half of which
is silvered and which bisects the right angle at~$O$, a light-ray is split
up into two halves, one of which travels to~$A$, the other to~$A'$. They
are reflected at these two points; and at~$O$, owing to the partly
silvered mirror, they are again combined to a single composite ray.
We take $l$~and~$l'$ approximately equal; then, owing to the difference
in path given by~\Eq{(24)}, namely,}
\[
\frac{2l}{1 - q^{2}} - \frac{2l'}{\sqrt{1 - q^{2}}},
\]
interference occurs. If the whole apparatus is now turned slowly
through~$90°$ about~$O$ until $A'$~comes into the direction of motion,
this difference of path becomes
\[
\frac{2l}{\sqrt{1 - q^{2}}} - \frac{2l'}{1 - q^{2}}.
\]
Consequently, there is a shortening of the path by an amount
\[
2(l + l') \left(\frac{1}{1 - q^{2}} - \frac{1}{\sqrt{1 - q^{2}}}\right)
\sim (l + l')q^{2}.
\]
\Figure{9}
This should express itself in a shift of the initial interference fringes.
\emph{Although conditions were such that, numerically, even only $1$~per
cent.\ of the displacement of the fringes expected by Michelson could
not have escaped detection, no trace of it was to be found when the
experiment was performed.}
Lorentz (and Fitzgerald, independently) sought to explain this
\index{Lorentz!Fitzgerald@{-Fitzgerald contraction}}%
strange result by the bold hypothesis that a rigid body in moving
relatively to the æther undergoes a contraction in the direction of
the line of motion in the ratio $1 : \sqrt{1 - q^{2}}$. This would actually
account for the null result of the Michelson-Morley experiment.
For there, $OA$~has in the first position the true length $l\sqrt{1 - q^{2}}$,
\PageSep{172}
and $OA'$~the length~$l'$, whereas in the second position $OA$~has the
true length~$l$ but $OA'$~the length $l' · \sqrt{1 - q^{2}}$. The difference of path
would, in \Emph{each} case, be $\dfrac{2(l - l')}{\sqrt{1 - q^{2}}}$.
It was also found that, no matter into what direction a mirror
rigidly fixed to~$O$ was turned, the same apparent velocity of
transmission $\sqrt{c^{2} - v^{2}}$ was obtained for all directions; that is, that
this velocity did not depend on the direction~$\theta$, in the manner given
by~\Eq{(24)}. Nevertheless, theoretically, it still seemed possible to
demonstrate the decrease of the velocity of transmission from $c$ to~$\sqrt{c^{2} - v^{2}}$.
But if the æther shortens the measuring rods in the
direction of motion in the ratio $1 : \sqrt{1 - q^{2}}$, it need only retard
clocks in the same ratio to hide this effect, too. \emph{In fact, not only
the Michelson-Morley experiment but a whole series of further experiments
designed to demonstrate that the earth's motion has an influence
on combined mechanical and electromagnetic phenomena, have led to
a null result} (\textit{vide} \FNote{5}). Æther mechanics has thus to account
not only for Maxwell's laws but also for this remarkable interaction
between matter and æther. It seems that the æther has betaken
itself to the land of the shades in a final effort to elude the inquisitive
search of the physicist!
The only reasonable answer that was given to the question as
to why a translation in the æther cannot be distinguished from
rest was that of Einstein, namely, that \emph{there is no æther}! (The
æther has since the very beginning remained a vague hypothesis
and one, moreover, that has acted very poorly in the face of facts.)
The position is then this: for mechanics we get Galilei's Theorem
of Relativity, for electrodynamics, Lorentz's Theorem. If this
is really the case, they neutralise one another and thereby define
an absolute space of reference in which mechanical laws have the
Newtonian form, electrodynamical laws that given by Maxwell.
The difficulty of explaining the null result of the experiments whose
purpose was to distinguish translation from rest, is overcome only
by regarding \Emph{one or other} of these two principles of relativity as
being valid for \Emph{all} physical phenomena. That of Galilei does not
come into question for electrodynamics as this would mean that, in
Maxwell's theory, those terms by which we distinguish moving fields
from stationary ones would not occur: there would be no induction,
no light, and no wireless telegraphy. On the other hand, even
the contraction theory of Lorentz-Fitzgerald suggests that Newton's
mechanics may be modified so that it satisfies the Lorentz-Einstein
Theorem of Relativity, the deviations that occur being only of
\PageSep{173}
\index{Normal calibration of Riemann's space!system of co-ordinates}%
the order $\left(\dfrac{v}{c}\right)^{2}$; they are then easily within reach of observation for
all velocities~$v$ of planets or on the earth. The solution of Einstein
(\textit{vide} \FNote{6}), which at one stroke overcomes all difficulties, is then
this: \emph{the world is a four-dimensional affine space whose metrical
structure is determined by a non-definite quadratic form
\[
Q(\vx) = (\vx\Com \vx)
\]
which has one negative and three positive dimensions.} All physical
quantities are scalars and tensors of this four-dimensional world,
and all physical laws express invariant relations between them.
The simple concrete meaning of the form~$Q(\vx)$ is that a light-signal
which has been emitted at the world-point~$O$ arrives at all those and
only those world-points~$A$ for which $\vx = \Vector{OA}$ belongs to the one
of the two conical sheets defined by the equation $Q(\vx) = 0$ (cf.~§\,4).
Hence that sheet (of the two cones) which ``opens into the future''
namely, $Q(\vx) \leq 0$ is distinguished objectively from that which opens
into the past. By introducing an appropriate ``normal'' co-ordinate
system consisting of the zero point~$O$ and the fundamental vectors~$\ve_{i}$,
we may bring~$Q(\vx)$ into the normal form
\[
(\Vector{OA}, \Vector{OA}) = -x_{0}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2},
\]
in which the~$x_{i}$'s are the co-ordinates of~$A$; in addition, the
fundamental vector~$\ve_{0}$ is to belong to the cone opening into the
future. \Emph{It is impossible to narrow down the selection from
these normal co-ordinate systems any farther}: that is, none
\index{Co-ordinate systems!normal}%
are specially favoured; they are all equivalent. If we make use
of a particular one, then $x_{0}$~must be regarded as the time; $x_{1}$,~$x_{2}$,~$x_{3}$
as the Cartesian space co-ordinates; and all the ordinary expressions
referring to space and time are to be used in this system of reference
as usual. The adequate mathematical formulation of Einstein's
discovery was first given by Minkowski (\textit{vide} \FNote{7}): to him we
are indebted for the idea of four-dimensional world-geometry, on
which we based our argument from the outset.
How the null result of the Michelson-Morley experiment comes
about is now clear. For if the interactions of the cohesive forces
of matter as well as the transmission of light takes place according
to Einstein's Principle of Relativity, measuring rods must behave so
that no difference between rest and translation can be discovered by
means of objective determinations. Seeing that Maxwell's equations
satisfy Einstein's Principle of Relativity, as was recognised even by
Lorentz, we must indeed regard \emph{the Michelson-Morley experiment as
a proof that the mechanics of rigid bodies must, strictly speaking, be
\PageSep{174}
in accordance not with that of Galilei's Principle of Relativity, but
with that of Einstein}.
It is clear that this is mathematically much simpler and more
intelligible than the former: world-geometry has been brought into
closer touch with Euclidean space-geometry through Einstein and
Minkowski. Moreover, as may easily be shown, Galilei's principle
is found to be a limiting case of Einstein's world-geometry by
making $c$ converge to~$\infty$. The physical purport of this is that
\emph{we are to discard our belief in the objective meaning of
simultaneity; it was the great achievement of Einstein in the
\index{Simultaneity}%
field of the theory of knowledge that he banished this dogma from
our minds}, and this is what leads us to rank his name with that of
Copernicus. The graphical picture given at the end of the preceding
paragraph discloses immediately that the planes $x_{0}' = \text{const.}$
no longer coincide with the planes $x_{0} = \text{const}$. In consequence
of the metrical structure of the world, which is based on~$Q(\vx)$,
each plane $x_{0}' = \text{const.}$ has a measure-determination such that
the ellipse in which it intersects the ``light-cone,'' is a circle, and
that Euclidean geometry holds for it. The point at which it is
punctured by the $\Typo{\vx}{x}_{0}'$-axis is the mid-point of the elliptical section.
So the propagation of light takes place in the ``accented'' system
of reference, too, in concentric circles.
We shall next endeavour to eradicate the difficulties that seem
to our intuition, our inner knowledge of space and time, to be
involved in the revolution caused by Einstein in the conception of
time. According to the ordinary view the following is true. If I
shoot bullets out with all possible velocities in all directions from a
point~$O$, they will all reach world-points that are later than~$O$;
I cannot shoot back into the past. Similarly, an event which
happens at~$O$ has an influence only on what happens at later
world-points, whereas ``one can no longer undo'' the past: the
extreme limit is reached by gravitation, acting according to
Newton's law of attraction, as a result of which, for example, by
extending my arm, I at the identical moment produce an effect on
the planets, modifying their orbits ever so slightly. If we again
suppress a space-co-ordinate and use our graphical mode of representation,
then the absolute meaning of the plane $t = 0$ which
passes through~$O$ consists in the fact that it separates the ``future''
world-points, which can be influenced by actions at~$O$, from the
``past'' world-points from which an effect may be conveyed to or
conferred on~$O$. According to Einstein's Principle of Relativity, we
get in place of the plane of separation $t = 0$ the light cone
\[
x_{1}^{2} + x_{2}^{2} - c^{2}t^{2} = 0
\]
\PageSep{175}
\index{Active past and future}%
\index{Earlier@{\emph{Earlier} and \emph{later}}}%
\index{Passive past and future}%
\index{Past, active and passive}%
(which degenerates to the above double plane when $c = \infty$). This
makes the position clear in this way. The direction of all bodies
projected from~$O$ must point into the forward-cone, opening into
the future (so also the direction of the world-line of my own body,
my ``life-curve'' if I happen to be at~$O$). Events at~$O$ can influence
only happenings that occur at world-points that lie within this
forward-cone: the limits are marked out by the resulting propagation
of light into empty space.\footnote
{The propagation of gravitational force must, of course, likewise take place
with the speed of light, according to Einstein's Theory of Relativity. The law for
the gravitational potential must be modified in a manner analogous to that by
which electrostatic potential was modified in passing from statical to moving
fields.}
If I happen to be at~$O$, then $O$~divides
my life-curve into past and future; no change is thereby caused.
As far as my relationship to the world is concerned, however, the
forward-cone comprises all the world-points which are affected
by my active or passive doings at~$O$, whereas all events that are
complete in the past, that can no longer be altered, lie externally
to this cone. \Emph{The sheet of the forward-cone separates my
active future from my active past.} On the other hand, the
\Figure{10}
interior of the backward-cone includes all events in which I have
participated (either actively or as an observer) or of which I have
received knowledge of some kind or other, for only such events
may have had an influence on me; outside this cone are all
occurrences that I may yet experience or would yet experience if my
life were everlasting and nothing were shrouded from my gaze.
\Emph{The sheet of the backward-cone separates my passive past
from my passive future.} The sheet itself contains everything
on its surface that I see at this moment, or can see; it is thus
properly the picture of my external surroundings. In the fact that
we must in this way distinguish between \Emph{active} and \Emph{passive}, present,
\PageSep{176}
and future, there lies the fundamental importance of Römer's
discovery of the finite velocity of light to which Einstein's
Principle of Relativity first gave full expression. The plane $t = 0$
passing through~$O$ in an allowable co-ordinate system may be
placed so that it cuts the light-cone $Q(\Typo{x}{\vx}) = 0$ only at~$O$ and thereby
separates the cone of the active future from the cone of the passive
past.
For a body moving with uniform translation it is always
possible to choose an allowable co-ordinate system ($=$~normal co-ordinate
system) such that the body is at rest in it. The individual
parts of the body are then separated by definite distances from one
another, the straight lines connecting them make definite angles
with one another, and so forth, all of which may be calculated by
means of the formulæ of ordinary analytical geometry from the space-co-ordinates
$x_{1}$,~$x_{2}$,~$x_{3}$ of the points under consideration in the allowable
co-ordinate system chosen. I shall term them the \Emph{static
\index{Static!length}%
measures} of the body (this defines, in particular, the \Emph{static
length} of a measuring rod). If this body is a clock, in which a
periodical event occurs, there will be associated with this period in
the system of reference, in which the clock is at rest, a definite time,
determined by the increase of the co-ordinate~$x_{0}$ during a period;
we shall call this the ``proper time'' of the clock. If we push the
body at one and the same moment at different points, these points
will begin to move, but as the effect can at most be propagated
with the velocity of light, the motion will only gradually be communicated
to the whole body. As long as the expanding spheres
encircling each point of attack and travelling with the velocity of
light do not overlap, the parts surrounding these points that are
dragged along move independently of one another. It is evident
from this that, according to the theory of relativity, there cannot
be rigid bodies in the old sense; that is, no body exists which
remains objectively always the same no matter to what influences
it has been subjected. How is it that in spite of this we can use
our measuring rods for carrying out measurements in space? We
shall use an analogy. If a gas that is in equilibrium in a closed
vessel is heated at various points by small flames and is then removed
adiabatically, it will at first pass through a series of complicated
stages, which will not satisfy the equilibrium laws of
\Chg{thermo-dynamics}{thermodynamics}. Finally, however, it will attain a new state of
equilibrium corresponding to the new quantity of energy it contains,
which is now greater owing to the heating. We require of a rigid
body that is to be used for purposes of measurement (in particular,
\index{Measurement}%
a linear \Emph{measuring rod}) that, \Emph{after coming to rest in an
\PageSep{177}
\index{Future, active and passive}%
\index{Systems of reference}%
allowable system of reference}, it shall always remain exactly
the same as before, that is, that it shall have \Emph{the same static
measures} (or \Emph{static length}); and we require of a \Emph{clock} that
goes correctly \Emph{that it shall always have the same proper-time
when it has come to rest} (as a whole) \Emph{in an allowable
system of reference}. We may assume that the measuring rods
and clocks which we shall use satisfy this condition to a sufficient
degree of approximation. It is only when, in our analogy, the gas
is warmed sufficiently slowly (strictly speaking, infinitely slowly)
that it will pass through a series of \Chg{thermo-dynamic}{thermodynamic} states of
equilibrium; only when we move the measuring rods and clocks
steadily, without jerks, will they preserve their static lengths and
proper-times. The limits of acceleration within which this assumption
may be made without appreciable errors arising are
certainly very wide. Definite and exact statements about this
point can be made only when we have built up a \Emph{dynamics} based
on physical and mechanical laws.
To get a clear picture of the Lorentz-Fitzgerald contraction from
\index{Allowable systems}%
the point of view of Einstein's Theory of Relativity, we shall
imagine the following to take place in a plane. In an allowable
system of reference (co-ordinates $t$,~$x_{1}$,~$x_{2}$, one space-co-ordinate
being suppressed), to which the following space-time expressions
will be referred, there is at rest a plane sheet of paper (carrying
rectangular co-ordinates $x_{1}$,~$x_{2}$ marked on it), on which a closed
curve~$\vc$ is drawn. We have, besides, a circular plate carrying a
rigid clock-hand that rotates around its centre, so that its point
traces out the edge of the plate if it is rotated slowly, thus proving
that the edge is actually a circle. Let the plate now move along the
sheet of paper with uniform translation. If, at the same time, the
index rotates slowly, its point runs unceasingly along the edge of
the plate: in this sense the disc is circular during translation too.
Suppose the edge of the disc to coincide exactly with the curve~$\vc$
at a definite moment. If we measure~$\vc$ by means of measuring
rods that are at rest, we find that $\vc$~is not a circle but an ellipse.
This phenomenon is shown graphically in \Fig{11}. We have
added the system of reference $t'$,~$x_{1}'$,~$x_{2}'$ with respect to which the
disc is at rest. Any plane $t' = \text{const.}$ intersects the light cone
in this system of reference in a circle ``that exists for a single
moment''. The cylinder above it erected in the direction of the
$t'$-axis represents a circle that is at rest in the \Emph{accented} system,
and hence marks off that part of the world which is passed over
by our disc. The section of this cylinder and the plane $t = 0$ is
not a circle but an ellipse. The right-angled cylinder constructed
\PageSep{178}
on it in the direction of the $t$-axis represents the constantly present
curve traced on the paper.
If we now inquire what physical laws are necessary to distinguish
normal co-ordinate systems from all other co-ordinate
systems (in Riemann's sense), we learn that we require only
Galilei's Principle of Relativity and the law of the propagation of
light; by means of light-signals and point-masses moving under no
forces---even if we have only small limits of velocity within which
the latter may move---we are in a position to fix a co-ordinate
system of this kind. To see this we shall next add a corollary
to Galilei's Principle of Inertia. If a clock shares in the motion of
the point-mass moving under no forces, then its time-data are a
measure of the ``proper-time''~$s$ of the motion. Galilei's principle
\index{Proper-time}%
states that the world-line of the point is a straight line; we
elaborate this by stating further that the moments of the motion
\Figure{11}
characterised by $s = 0, 1, 2, 3, \dots$ (or by any arithmetical series
of values of~$s$) represent equidistant points along the straight line.
By introducing the parameter of proper-time to distinguish the
various stages of the motion we get not only a line in the four-dimensional
world but also a ``motion'' in it (cf.\ the definition on
\Pageref[p.]{105}) and according to Galilei this motion is a translation.
The world-points constitute a four-dimensional manifold; this is
perhaps the most certain fact of our empirical knowledge. We
shall call a system of four co-ordinates~$x_{i}$ ($i = 0, 1, 2, 3$), which are
used to fix these points in a certain portion of the world, a \Emph{linear
co-ordinate system}, if the motion of point-mass under no forces
and expressed in terms of the parameter~$s$ of the proper-time be
represented by formulæ in which the~$x_{i}$'s are linear functions of~$s$.
The fact that there are such co-ordinate systems is what the law of
inertia really asserts. After this condition of linearity, all that is
necessary to define the co-ordinate system fully is a linear transformation.
\PageSep{179}
That is, if $x_{i}$,~$x_{i}'$ are the co-ordinates respectively of
one and the same world-point in two different linear co-ordinate
systems, then the~$x_{i}'$'s a must be linear functions of the~$x$'s. By
simultaneously interpreting the~$x_{i}$'s as Cartesian co-ordinates in a
four-dimensional Euclidean space, the co-ordinate system furnishes
\index{Space!like@{-like} vector}%
us with a representation of the world (or of the portion of world
in which the $x_{i}$'s exist) on a Euclidean space of representation.
We may, therefore, formulate our proposition thus. A representation
of two Euclidean spaces by one another (or in other
words a transformation from one Euclidean space to another), such
that straight lines become straight lines and a series of equidistant
points become a series of equidistant points is necessarily an
affine transformation. \Fig{12} which represents Möbius' mesh-construction
(\textit{vide} \FNote{8}) may suffice to indicate the proof to
the reader. It is obvious that this mesh-system may be arranged
so that the three directions of the straight lines composing it may
be derived from a given, arbitrarily thin, cone carrying these
\Figure{12}
directions on it; the above geometrical theorem remains valid even
if we only know that the straight lines whose directions belong to
this cone become straight lines again as a result of the transformation.
Galilei's Principle of Inertia is sufficient in itself to prove
conclusively that the world is affine in character: it will not,
however, allow us deduce any further result. The metrical groundform~$(\vx\Com \vx)$
of the world is now accounted for by the process of light-propagation.
A light-signal emitted from~$O$ arrives at the world-point~$A$
if, and only if, $\vx = \Vector{OA}$ belongs to one of the two conical
sheets defined by $(\vx\Com \vx) = 0$. This determines the quadratic form
except for a constant factor; to fix the latter we must choose an
arbitrary unit-measure (cf.\ Appendix~I).\Pagelabel{179}
\Section{22.}{Relativistic Geometry, Kinematics, and Optics}
We shall call a world-vector~$\vx$ \Emph{space-like} or \Emph{time-like}, according
\index{Time!-like vectors}%
as $(\vx\Com \vx)$~is positive or negative. Time-like vectors are divided
\PageSep{180}
into those that point into the \Emph{future} and those that point into the
\Emph{past}. We shall call the invariant
\[
\Delta s = \sqrt{-(\vx\Com \vx)}
\Tag{(25)}
\]
of a time-like vector~$\vx$ which points into the future its \Emph{proper-time}.
\index{Proper-time}%
If we set
\[
\vx = \Delta s · \ve
\]
then~$\ve$, the direction of the time-like displacement, is a vector that
points into the future, and that satisfies the condition of normality
$(\ve\Com \ve) = -1$.
As in Galilean geometry, so in Einstein's world-geometry we
\index{Resolution of tensors into space and time of vectors}%
must \Emph{resolve the world into space and time} by projection
\index{Space!projection@{(as projection of the world)}}%
in the direction of a time-like vector~$\ve$ pointing into the future and
normalised by the condition $(\ve\Com \ve) = -1$. The process of projection
was discussed in detail in §\,19. The fundamental formulæ \Eq{(3)}, \Eq{(5)},
\Eq{(5')} that are set up must here be applied with $e = -1$.\footnote
{\Loosen Here the units of space and time are chosen so that the velocity of light
\textit{in~vacuo} becomes equal to~$1$. To arrive at the ordinary units of the c.g.s.\
systems, the equation of normality $(\ve\Com \ve) = -1$ must be replaced by $(\ve\Com \ve) = -c^{2}$,
and $e$~must be taken equal to~$-c^{2}$.}
World-points for which the vector connecting them is proportional to~$\ve$
coincide at a space-point which we may mark by means of a point-mass
at rest, and which we may represent graphically by a world-line
(straight) parallel to~$\ve$. The three-dimensional space~$\sfR_{\ve}$ that
is generated by the projection has a metrical character that is
Euclidean since, for every vector~$\vx^{*}$ which is orthogonal to~$\ve$, that
is, every vector~$\vx^{*}$ that satisfies the condition $(\vx^{*}\Com \ve) = 0$, $(\vx^{*}\Com \vx^{*})$~is
a positive quantity (except in the case in which $\vx^{*} = \Typo{0}{\0}$; cf.~§\,4).
Every displacement~$\vx$ of the world may be split up according to
the formula
\[
\vx = \Delta t \mid \sfx:
\]
$\Delta t$~is its duration (called ``height'' in §\,19): $\vx$~is the displacement
it produces in the space~$\sfR_{\ve}$.
If $e_{1}$,~$e_{2}$,~$e_{3}$ form a co-ordinate system in~$\sfR_{\ve}$, then the world-displacements
$\ve_{1}$,~$\ve_{2}$,~$\ve_{3}$ that are orthogonal to $\ve = \ve_{0}$, and that produce
the three given space-displacements, form in conjunction with~$\ve_{0}$
a \Emph{co-ordinate system, which belongs to~$\sfR_{\ve}$}, for the world-points.
It is normal if the three vectors~$\ve_{i}$ in~$\sfR_{\ve}$ form a Cartesian co-ordinate
system. In every case the system of co-efficients of the metrical
groundform has, in it, the form
\[
\left\lvert\begin{array}{@{}rccc@{}}
-1 & 0 & 0 & 0 \\
0 & g_{11} & g_{12} & g_{13} \\
0 & g_{21} & g_{22} & g_{23} \\
0 & g_{31} & g_{32} & g_{33} \\
\end{array}\right\rvert\Add{.}
\]
\PageSep{181}
The proper time~$\Delta s$ of a time-like vector~$\vx$ pointing into the
future (and for which $\vx = \Delta s · \ve$) is equal to the duration of~$\vx$ in the
space of reference~$\sfR_{\ve}$, in which $\vx$~calls forth no spatial displacement.
In the sequel we shall have to contrast several ways of splitting up
quantities into terms of the vectors $\ve$, $\ve'$,~\dots; $\ve$~(with or without
an index) is always to denote a time-like world-vector pointing into
the future and satisfying the condition of normality $(\ve\Com \ve) = -1$.
Let $K$ be a body at rest in~$\sfR_{\ve}$, $K'$~a body at rest in~$\sfR_{\ve}'$. $K'$~moves
with uniform translation in~$\sfR_{\ve}$. If, by splitting up~$\ve'$ into
terms of~$\ve$, we get in~$\sfR_{\ve}$
\[
e' = h \mid h\sfv
\Tag{(26)}
\]
then $K'$~undergoes the space-displacement~$h\sfv$ during the time (i.e.\
with the duration)~$h$ in~$\sfR_{\ve}$. Accordingly, $\sfv$~is the velocity of~$K'$ in~$\sfR_{\ve}$
or \Emph{the relative velocity of~$K'$ with respect to~$K$}. Its magnitude
is determined by $v^{2} = (\sfv\Com \sfv)$. By~\Eq{(3)} we have
\[
h = -(\ve'\Com \ve)\Add{;}
\Tag{(27)}
\]
on the other hand, by~\Eq{(5)}
\[
1 = -(\ve'\Com \ve') = h^{2} - h^{2}(\sfv\Com \sfv) = h^{2}(1 - v^{2}),
\]
thus we get
\[
h = \frac{1}{\sqrt{1 - v^{2}}}\Add{.}
\Tag{(28)}
\]
If, between two moments of $K'$'s~motion, it undergoes the world-displacement
$\Delta s · \ve'$, \Eq{(26)}~shows that $h · \Delta s = \Delta t$ is the duration of
this displacement in~$\sfR_{\ve}$. The proper time~$\Delta s$ and the duration~$\Delta t$ of
the displacement in~$\sfR_{\ve}$ are related by
\[
\Delta s = \Delta t \sqrt{1 - v^{2}}\Add{.}
\Tag{(29)}
\]
Since \Eq{(27)}~is symmetrical in $\ve$~and~$\ve'$, \Eq{(28)}~teaches us that the
\Emph{magnitude of the relative velocity of $K'$ with respect to~$K$ is
equal to that of $K$ with respect to~$K'$}. The vectorial relative
velocities \Emph{cannot} be compared with one another since the one
exists in the space~$\sfR_{\ve}$, the other in the space~$\sfR_{\ve}'$.
Let us consider a partition into three quantities $\ve$,~$\ve_{1}$,~$\ve_{2}$. Let
$K_{1}$,~$K_{2}$ be two bodies at rest in $\sfR_{\ve_{1}}$,~$\sfR_{\ve_{2}}$ respectively. Suppose we
have in~$\sfR_{\ve}$
\begin{align*}
\ve_{1} &= h_{1} \mid h_{1} \sfv_{1} & h_{1} &= \frac{1}{\sqrt{1 - v_{1}^{2}}}\Add{,} \\
\ve_{2} &= h_{2} \mid h_{2} \sfv_{2} & h_{2} &= \frac{1}{\sqrt{1 - v_{2}^{2}}}\Add{.} \\
\end{align*}
Then
\[
-(\ve_{1}\Com \ve_{2}) = h_{1}h_{2} \bigl\{1 - (v_{1}v_{2})\bigr\}.
\]
\PageSep{182}
Hence, if $K_{1}$~and $K_{2}$ have velocities $\sfv_{1}$,~$\sfv_{2}$ respectively in~$\sfR_{\ve}$, with
numerical values $v_{1}$,~$v_{2}$, then if these velocities $\sfv_{1}$,~$\sfv_{2}$ make an angle~$\theta$
with each other, and if $v_{12} = v_{21}$ is the magnitude of the velocity
of~$K_{2}$ relatively to~$K_{1}$ (or \textit{vice versa}), we find that the formula
\[
\frac{1 - v_{1}v_{2}\cos\theta}{\sqrt{1 - v_{1}^{2}} \sqrt{1 - v_{2}^{2}}}
= \frac{1}{\sqrt{1 - v_{12}^{2}}}
\Tag{(30)}
\]
holds: \Emph{it shows how the relative velocity of two bodies is
determined from their given velocities}. If, using hyperbolic
functions, we set $v = \tanh v$ for each of the values~$v$ of the velocity
($v$~being $< 1$), we get
\[
\cosh u_{1} \cosh u_{2} - \sinh u_{1} \sinh u_{2} \cos \theta = \cosh u_{12}.
\]
This formula becomes the cosine theorem of spherical geometry
if we replace the hyperbolic functions by their corresponding trigonometrical
functions; thus $u_{12}$~is the side opposite the angle~$\theta$ in a
\Figure{13}
triangle on the Bolyai-Lobatschefsky plane, the two remaining sides
being $u_{1}$~and~$u_{2}$.
Analogous to the relationship~\Eq{(29)} between time and proper-time,
there is one between length and statical-length. We shall
use~$\sfR_{\ve}$ as our space of reference. Let the individual point-masses
of the body at a \Emph{definite} moment be at the world-points
$O$,~$A$,~\dots\Add{.} The space-points $\sfO$,~$\sfA$,~\dots\ at~$\sfR_{\ve}$ at which they
are situated form a figure in~$\sfR_{\ve}$, on which we can confer duration, by
making the body leave behind it a copy of itself at the moment under
consideration in the space~$\sfR_{\ve}$; an example of this was presented in
the illustration given at the close of the preceding paragraph. If,
on the other hand, the world-points $O$,~$A$,~\dots\ are at the space-points
$\sfO'$,~$\sfA'$,~\dots\ in the space~$\sfR_{\ve}$ in which $K'$~is at rest, then
$O'$,~$A'$,~\dots\ constitute the statical shape of the body~$K'$ (cf.\ \Fig{13},
in which orthogonal world-distances are drawn perpendicularly).
\PageSep{183}
\index{Simultaneity}%
There is a transformation that connects the part of~$\sfR_{\ve}$, which receives
the imprint or copy, and the statical shape of the body in~$\sfR_{\ve}'$.
This transformation transforms the points $\sfA$,~$\Typo{A'}{\sfA'}$ into one
another. It is obviously affine (in fact, it is nothing more than
an orthogonal projection). Since the world-points $O$,~$A$ are \Emph{simultaneous}
for the partition into~$\ve$, we have
\[
\Vector{OA} = \vx = \Typo{0}{\0} \mid \sfx \text{ in } \sfR_{\ve},
\text{ and } \sfx = \Vector{OA}.
\]
By formula~\Eq{(5)}
\begin{align*}
%[** TN: Vectors rendered as bar accents in the original]
{\Vector{OA}}^{2} &= (\sfx\Com \sfx) = (\vx\Com \vx)\Add{,} \\
\Typo{O'A'^{2}}{{\Vector{O'A'}}^{2}} &= (\vx\Com \vx) + (\vx\Com \ve')^{2}.
\end{align*}
If, however, we determine $(\vx\Com \ve')$ in~$\sfR_{\ve}$ by~\Eq{(5')} we get
\[
(\vx\Com \ve') = h(\sfx\Com \sfv)\Add{,}
\]
and hence
\[
{\Vector{O'A'}}^{2} = (\sfx\Com \sfx) + \frac{(\sfx\Com \sfv)^{2}}{1 - v^{2}}.
\]
If we use a Cartesian co-ordinate system $x_{1}$,~$x_{2}$,~$x_{3}$ in~$\sfR_{\ve}$ with $\sfO$~as
origin, and having its $x_{1}$-axis in the direction of the velocity~$v$, then
if $x_{1}$,~$x_{2}$,~$x_{3}$ are the co-ordinates of~$\sfA$, we have
\begin{align*}
{\Vector{\sfO\sfA}}^{2} &= x_{1}^{2} + x_{2}^{2} + x_{3}^{2}\Add{,} \\
{\Vector{\sfO'\sfA'}}^{2} &= \frac{x_{1}^{2}}{1 - v^{2}} + x_{2}^{2} + x_{3}^{2}
= x_{1}'^{2} + x_{2}'^{2} + x_{3}'^{2}\Add{,} \\
\end{align*}
in the last term of which we have set
\[
x_{1}' = \frac{x_{1}}{\sqrt{1 - v^{2}}}\Add{,}\qquad
x_{2}' = x_{2}\Add{,}\qquad
x_{3}' = x_{3}\Add{.}
\Tag{(31)}
\]
By assigning to every point in~$\sfR_{\ve}$ with co-ordinates $(x_{1}, x_{2}, x_{3})$ the
point with co-ordinates $(x_{1}', x_{2}', x_{3}')$ as given by~\Eq{(31)}, we effect a
dilatation of the imprinted copy in the ratio $1 : \sqrt{1 - v^{2}}$ along the
direction of the body's motion. Our formulæ assert that the copy
thereby assumes a shape congruent to that of the body when at
rest; this is the \Emph{Lorentz-Fitzgerald contraction}. In particular,
the volume~$V$ that the body~$K'$ occupies at a definite moment in the
space~$\sfR_{\ve}$ is connected to its statical volume~$V_{0}$ by the relation
\index{Static!volume}%
\[
V = V_{0} \sqrt{1 - v^{2}}.
\]
Whenever we measure angles by optical means we determine
the angles formed by the light-rays for the system of reference in
\index{Light!ray}%
which the (rigid) measuring instrument is at rest. \emph{Again, when
our eyes take the place of these instruments it is these angles that
determine the visual form of objects that lie within the field of vision.}
To establish the relationship between geometry and the observation
\PageSep{184}
of geometrical magnitudes, we must therefore take optical considerations
into account. The solution of Maxwell's equations for
light-rays in the æther as well as in a homogeneous medium, which
is at rest in an allowable reference system, is of a form such that
the component of the ``phase'' quantities (in complex notation)
are all
\[
= \text{const. } e^{2\pi i \Theta(P)}
\]
in which $\Theta = \Theta(P)$ is, with the omission of an additive constant,
the phase determined by the conditions set down; it is a function
of the world-point which here occurs as the argument. If the
world co-ordinates are transformed linearly in any way, the components
in the new co-ordinate system will again have the same
form with the same phase-function~$\Theta$. The phase is accordingly
an invariant. For a plane wave it is a \Emph{linear} and (if we exclude
absorbing media) real function of the world-co-ordinates
of~$P$; hence the phase-difference at two arbitrary points $\Theta(B) - \Theta(A)$
is a linear form of the arbitrary displacement $\vx = \Vector{AB}$, that is,
a co-variant world-vector. If we represent this by the corresponding
displacement~$\vl$ (we shall allude to it briefly as the light-ray~$\vl$)
then
\[
\Theta(B) - \Theta(A) = (\vl\Com \vx).
\]
If we split it up by means of the time-like vector~$\ve$ into space and
time and set
\[
\vl = \nu \mid \frac{\nu}{q} \sfa
\Tag{(32)}
\]
so that the space-vector~$\sfa$ in~$\sfR_{\ve}$ is of unit length
\[
\vx = \Delta t \mid \sfx,
\]
then the phase-difference is
\[
\nu \left\{\frac{(\sfa\Com \sfx)}{q} - \Delta t\right\}.
\]
From this we see that $\nu$~signifies the frequency, $q$~the velocity of
transmission, and $\sfa$~the direction of the light-ray in the space~$\sfR_{\ve}$.
Maxwell's equations tell us that\Erratum{}{ in the æther} the velocity of transmission $q = 1$,
or that
\[
(\vl\Com \vl) = 0.
\]
If we split the world up into space and time in two ways,
firstly by means of~$\ve$, secondly by means of~$\ve'$, and distinguish the
magnitudes derived from the second process by accents we immediately
find as a result of the invariance of~$(\vl\Com \vl)$ the law
\[
\nu^{2}\left(\frac{1}{q^{2}} - 1\right)
= \nu'^{2}\left(\frac{1}{q'^{2}} - 1\right)\Add{.}
\Tag{(33)}
\]
\PageSep{185}
If we fix our attention on two light-rays $\vl_{1}$,~$\vl_{2}$ with frequencies
$\nu_{1}$,~$\nu_{2}$ and velocities of transmission $q_{1}$,~$q_{2}$ then
\[
(\vl_{1}\Com \vl_{2})
= \nu_{1}\nu_{2} \left\{\frac{\sfa_{1}\sfa_{2}}{q_{1}q_{2}} - 1\right\}.
\]
If they make an angle~$\omega$ to with one another, then
\[
\nu_{1}\nu_{2} \left\{\frac{\cos\omega}{q_{1}q_{2}} - 1\right\}
= \nu_{1}'\nu_{2}' \left\{\frac{\cos\omega'}{q_{1}'q_{2}'} - 1\right\}\Add{.}
\Tag{(34)}
\]
For the æther, these equations become
\[
q = q'\ (= 1),\qquad
\nu_{1}\nu_{2} \sin^{2} \frac{\omega}{2}
= \nu_{1}'\nu_{2}' \sin^{2} \frac{\omega'}{2}\Add{.}
\Tag{(35)}
\]
Finally, to get the relationship between the frequencies $\nu$~and~$\nu'$
we assume a body that is at rest in~$\sfR_{\ve}'$; let it have the velocity~$\sfv$
in the space~$\sfR_{\ve}$, then, as before, we must set
\[
\ve' = h \mid h\sfv \text{ in } \sfR_{\ve}\Add{.}
% [** TN: Repeated number]
\Tag{(26)}
\]
From \Eq{(26)}~and~\Eq{(32)} it follows that
\[
\nu' = -(\vl\Com \ve')
= \nu h \left\{1 - \frac{(\sfa\Com \sfv)}{q}\right\}.
\]
Accordingly, if the direction of the light-ray in~$\Typo{R}{\sfR}_{\ve}$ makes an angle~$\theta$ with the velocity of the body, then
\[
\frac{\nu'}{\nu}
= \frac{1 - \dfrac{v\cos\theta}{q}}{\sqrt{1 - v^{2}}}\Add{.}
\Tag{(36)}
\]
\Eq{(36)}~is Doppler's Principle. For example, since a sodium-molecule
\index{Doppler's Principle}%
which is at rest in an allowable system remains objectively the
same, this relationship~\Eq{(36)} will exist between the frequency~$\nu'$ of a
sodium-molecule which is at rest and $\nu$~the frequency of a sodium-molecule
moving with a velocity~$\nu$, both frequencies being observed
in a spectroscope which is at rest; $\theta$~is the angle between the
direction of motion of the molecule and the light-ray which enters
the spectroscope. If we substitute~\Eq{(36)} in~\Eq{(33)} we get an equation
between $q$~and~$q'$ which enables us to calculate the velocity of propagation~$q$
in a moving medium from the velocity of propagation~$q'$
in the same medium at rest; for example, in water, $v$~now represents
the rate of flow of the water; $\theta$~represents the angle that
the direction of flow of the water makes with the light-rays. If
we suppose these two directions to coincide, and then neglect powers
of~$v$ higher than the first (since $v$~is in practice very small compared
with the velocity of light), we get
\[
q = q' + v(1 - q'^{2})\Add{;}
\]
\PageSep{186}
that is, \Emph{not} the whole of the velocity~$v$ of the medium is added to
%[** TN: Large parentheses in the original]
the velocity of propagation, but only the fraction $1 - \dfrac{1}{n^{2}}$ (in which
$n = \dfrac{1}{q'}$ is the index of refraction of the medium). Fresnel's ``convection-co-efficient''
\index{Fresnel's convection co-efficient}%
$1 - \dfrac{1}{n^{2}}$ was determined experimentally by Fizeau
long before the advent of the theory of relativity by making two
light-rays from the same source interfere, after one had travelled
through water which was at rest whilst the other had travelled
through water which was in motion (\textit{vide} \FNote{9}). The fact that the
theory of relativity accounts for this remarkable result shows that
it is valid for the optics and electrodynamics of moving media
(and also that in such cases the relativity principle, which is derived
from that of Lorentz and Einstein by putting $q$ for~$c$, does not hold;
one might be tempted to believe this erroneously from the equation
of wave-motion that holds in such cases). We shall find the
special form of~\Eq{(34)} for the \emph{æther}, in which $q = q' = 1$ \Chg{(cf.~35)}{(cf.~\Eq{(35)})}, to be
\[
\sin^{2} \frac{\omega}{2}
= \frac{(1 - v\cos\theta_{1}) (1 - v\cos\theta_{2})}{1 - v^{2}} \sin^{2} \frac{\omega'}{2}.
\]
If the reference-space~$\sfR_{\ve}$ happens to be the one on which the
theory of planets is commonly founded (and in which the centre of
mass of the solar system is at rest), and if the body in question
is the earth (on which an observing instrument is situated), $v$~its
velocity in~$\sfR_{\ve}$, $\omega$~the angle in~$\sfR_{\ve}$ that two rays which reach the
solar system from two infinitely distant stars make with one another,
$\theta_{1}$,~$\theta_{2}$ the angles which these rays make with the direction of motion
of the earth in~$\sfR_{\ve}$, then the angle~$\omega'$, at which the stars are observed
from the earth, is determined by the preceding equation. We
cannot, of course, measure~$\omega$, but we note the changes in~$\omega'$ (the
\Emph{aberration}) by taking account of the changes in $\theta_{1}$~and~$\theta_{2}$ in the
\index{Aberration}%
course of a year.
The formulæ which give the relationship between time, proper-time,
volume and statical volume are also valid in the case of \Emph{non-uniform
motion}. If $d\vx$~is the infinitesimal displacement that a
moving point-mass experiences during an infinitesimal length of time
in the world, then
\[
d\vx = ds · \vu\Add{,}\qquad
(\vu\Com \vu) = -1,\qquad
ds > 0
\]
give the proper-time~$ds$ and the world-direction~$\vu$ of this displacement.
The integral
\[
\int ds = \int \sqrt{-(d\vx, d\vx)}
\]
\PageSep{187}
taken over a portion of the world-line is the proper-time that
elapses during this part of the motion: it is independent of the
manner in which the world has been split up into space and time
and, provided the motion is not too rapid, will be indicated by a
clock that is rigidly attached to the point-mass. If we use any
linear co-ordinates~$x_{i}$ whatsoever in the world, and the proper-time~$s$
as our parameters to represent our world-line analytically (just
as we use length of arc in three-dimensional geometry), then
\[
\frac{dx_{i}}{ds} = u^{i}
\]
{\Loosen are the (contra-variant) components of~$\vu$, and we get $\sum_{i} u_{i} u^{i} = -1$.
If we split up the world into space and time by means of~$\ve$, we find}
\[
\vu = \frac{1}{\sqrt{1 - v^{2}}} \mathrel{\bigg|}
\frac{\sfv}{\sqrt{1 - v^{2}}} \text{ in $\sfR_{\ve}$}
\]
in which $\sfv$~is the velocity of the mass-point; and we find that the
time~$dt$ that elapses during the displacement~$d\vx$ in~$\sfR_{\ve}$ and the
proper-time~$ds$ are connected by
\[
ds = dt \sqrt{1 - v^{2}}\Add{.}
\Tag{(37)}
\]
If two world-points $A$,~$B$ are so placed with respect to one another
that $\Vector{AB}$~is a time-like vector pointing into the future, then $A$~and~$B$
may be connected by world-lines, whose directions all likewise
satisfy this condition: in other words, point-masses that leave~$A$
may reach~$B$. The proper-time necessary for them to do this is
dependent on the world-line; it is longest for a point-mass that
passes from $A$ to~$B$ by uniform translation. For if we split up
the world into space and time in such a way that $A$~and~$B$ occupy
the same point in space, this motion degenerates simply to rest, and
we derive the proposition~\Eq{(37)} which states that the proper-time
lags behind the time~$t$. The life-processes of mankind may well
be compared to a clock. Suppose we have two twin-brothers who
take leave from one another at a world-point~$A$, and suppose one
remains at home (that is, permanently at rest in an allowable
reference-space), whilst the other sets out on voyages, during
which he moves with velocities (relative to ``home'') that approximate
to that of light. When the wanderer returns home in later
years he will appear appreciably younger than the one who stayed
at home.
An element of mass~$dm$ (of a continuously extended body) that
moves with a velocity whose numerical value is~$v$ occupies at a
\PageSep{188}
\index{Divergence@{Divergence (\emph{div})}!(more general)}%
particular moment a volume~$dV$ which is connected with its
statical volume~$dV_{0}$ by the formula
\[
dV = dV_{0} \sqrt{1 - v^{2}}\Add{.}
\]
Accordingly, we have the relation between the density $\dfrac{dm}{dV}= \mu$ and
the statical density $\dfrac{dm}{dV_{0}}= \mu_{0}$\Add{:}
\[
\mu_{0} = \mu \sqrt{1 - v^{2}}\Add{.}
\]
$\mu_{0}$~is an invariant, and $\mu_{0} \vu$~with components $\mu_{0}\Typo{u}{u^{i}}$~is thus a contra-variant
vector, the ``flux of matter,'' which is determined by the
\index{Continuity, equation of!mass@{of mass}}%
\index{Matter!flux of}%
motion of the mass independently of the co-ordinate system. It
satisfies the equation of continuity
\[
\sum_{i} \frac{\dd (\mu_{0} u^{i})}{\dd x_{i}} = 0.
\]
The same remarks apply to electricity. If it is associated with
matter so that $de$~is the electric charge of the element of mass~$dm$,
then the statical density $\rho_{0} = \dfrac{de}{dV_{0}}$ is connected to the density $\rho = \dfrac{de}{dV}$
by
\[
\rho_{0} = \rho \sqrt{1 - v^{2}},
\]
then
\[
s^{i} = \rho_{0} u^{i}
\]
are the contra-variant components of the electric current ($4$-vector);
this corresponds exactly to the results of §\,20. In Maxwell's
phenomenological theory of electricity, the concealed motions of
the electrons are not taken into account as motions of matter, consequently
electricity is not supposed attached to matter in his
theory. The only way to explain how it is that a piece of matter
carries a certain charge is to say this charge is that which is simultaneously
in the portion of space that is occupied by the matter
at the moment under consideration. From this we see that the
charge is not, as in the theory of electrons, an invariant determined
by the portion of matter, but is dependent on the way the world
has been split up into space and time.
\Section{23.}{The Electrodynamics of Moving Bodies}
By splitting up the world into space and time we split up all
tensors. We shall first of all investigate purely mathematically
how this comes about, and shall then apply the results to derive
\PageSep{189}
\index{World ($=$ space-time)}%
the fundamental equations of electrodynamics for moving bodies.
Let us take an $n$-dimensional metrical space, which we shall call
``world,'' based on the metrical groundform $(\vx\Com \vx)$. Let $\ve$~be a
vector in it, for which $(\ve\Com \ve) = e \neq 0$. We split up the world in the
usual way into space~$\sfR_{\ve}$ and time in terms of~$\ve$. Let $e_{1}$, $e_{2}$,~\dots\Add{,}
$e_{n-1}$ be any co-ordinate system in the space~$\sfR_{\ve}$, and let $\ve_{1}$, $\ve_{2}$,~\dots\Add{,}
$\ve_{n-1}$ be the displacements of the world that are orthogonal to
$\ve = \ve_{0}$ and that are produced in~$\sfR_{\ve}$ by $e_{1}$, $e_{2}$,~\dots\Add{,} $e_{n-1}$. In the
co-ordinate system~$\ve_{i}$ ($i = 0, 1, 2, \dots\Add{,} n - 1$) ``belonging to~$\sfR_{\ve}$''
and representing the world, the scheme of the co-variant components
of the metrical ground-tensor has the form
\[
\left\lvert\begin{array}{@{}ccc@{}}
e & 0 & 0 \\
0 & g_{11} & \Typo{g_{22}}{g_{12}} \\
0 & g_{21} & g_{22} \\
\end{array}\right\rvert
\qquad
(n = 3).
\]
As an example, we shall consider a tensor of the second order and
suppose it to have components~$T_{ik}$ in this co-ordinate system.
Now, we assert that it splits up, in a manner dependent only on~$\ve$,
according to the following scheme:
\[
\framebox{$\begin{array}{c|lc}
\Strut
T_{00} & T_{01}\quad\null & T_{02} \\
\hline
\Strut
T_{10} & T_{11} & T_{12} \\
T_{20} & T_{21} & T_{22} \\
\end{array}$}
\]
that is, into a scalar, two vectors and a tensor of the second order
existing in~$\sfR_{\ve}$, which are here characterised by their components in
the co-ordinate system~$e_{i}$ ($i = 1, 2, \dots\Add{,} n - 1$).
For if the arbitrary world-displacement~$\vx$ splits up in terms of~$\ve$
thus
\[
\vx = \xi \mid \sfx
\]
and if, when we divide~$\vx$ into two factors, one of which is proportional
to~$\ve$ and the other orthogonal to~$\ve$, we have
\[
\vx = \xi \ve + \vx^{*}
\]
then, if $\vx$~has components~$\xi^{i}$, we get
\[
\vx = \sum_{i=0}^{n-1} \xi^{i} \ve_{i},\qquad
\xi = \xi^{0},\qquad
\vx^{*} = \sum_{i=1}^{n-1} \xi^{i} \ve_{i},\qquad
\sfx = \sum_{i=1}^{n-1} \xi^{i} e_{i}.
\]
Thus, without using a co-ordinate system we may represent the
splitting up of a tensor in the following manner. If $\vx$,~$\vy$ are any
two arbitrary displacements of the world, and if we set
\[
\vx = \xi \ve + \vx^{*},\qquad
\vy = \eta\ve + \vy^{*}\Add{,}
\Tag{(38)}
\]
\PageSep{190}
so that $\vx^{*}$~and $\vy^{*}$ are orthogonal to~$\ve$, then the bilinear form
belonging to the tensor of the second order is
\[
T(\vx\Com \vy)
= \xi\eta T(\ve\Com \ve)
+ \eta T(\vx^{*}\Com \ve)
+ \xi T(\ve\Com \vy^{*})
+ T(\vx^{*}\Com \vy^{*}).
\]
Hence, if we interpret $\vx^{*}$,~$\vy^{*}$ as the displacements of the world
orthogonal to~$\ve$, which produce the two arbitrary displacements
$\sfx$,~$\sfy$ of the space, we get
1. a scalar $T(\ve\Com \ve) = J = \sfJ$,
2. two linear forms (vectors) in the space~$\sfR_{\ve}$, defined by
\[
\sfL(\vx) = T(\vx^{*}\Com \ve),\qquad
\sfL'(\sfx) = T(\ve\Com \vx^{*}),
\]
3. a bilinear form (tensor) in the space~$\sfR_{\ve}$, defined by
\[
T(\sfx\Com \sfy) = T(\vx^{*}\Com \vy^{*}).
\]
If $\vx$,~$\vy$ are arbitrary world-displacements that produce $\sfx$,~$\sfy$,
respectively in~$\sfR_{\ve}$ we must replace $\vx^{*}$,~$\vy^{*}$ in this definition by
$\vx - \xi\ve$, $\vy - \eta\ve$ in accordance with~\Eq{(38)}; in these,
\[
\xi = \frac{1}{e}(\vx\Com \ve),\qquad
\eta = \frac{1}{e}(\vy\Com \ve).
\]
If we now set
\[
T(\vx\Com \ve) = L(\vx),\qquad
T(\ve\Com \vx) = L'(\vx),
\]
we get
\[
\left.
\begin{gathered}
\sfL(\sfx) = L(\vx) - \frac{J}{e}(\vx\Com \ve),\qquad
\sfL'(\sfx) = L'(\Typo{\sfx}{\vx}) - \frac{J}{e}(\vx\Com \ve)\Add{,} \\
%
\Squeeze[0.875]{T(\sfx\Com \sfy)
= T(\vx\Com \vy)
- \frac{1}{e}(\vy\Com \ve) L(\vx)
- \frac{1}{e}(\vx\Com \ve) L'(\vy)
+ \frac{J}{e^{2}} (\vx\Com \ve) (\vy\Com \ve)\Add{.}}
\end{gathered}
\right\}
\Tag{(39)}
\]
The linear and bilinear forms (vectors and tensors) of~$\sfR_{\ve}$ on the left
may be represented by the world-vectors and world-tensors on the
right which are derived uniquely from them. In the above representation
by means of components, this amounts to the following:
that, for example,
\[
\sfT = \left\lvert\begin{array}{@{}cc@{}}
T_{11} & T_{12} \\
T_{21} & T_{22} \\
\end{array}\right\rvert
\quad\text{is represented by}\quad
\left\lvert\begin{array}{@{}ccc@{}}
0 & 0 & 0 \\
0 & T_{11} & T_{12} \\
0 & T_{21} & T_{22} \\
\end{array}\right\rvert.
\]
It is immediately clear that in all calculations the tensors of space
may be replaced by the representative world-tensors. We shall,
however, use this device only in the case when, if one space-tensor
is $\lambda$~times another, the same is true of the representative world-tensors.
If we base our calculations of components on an \Emph{arbitrary}
co-ordinate system, in which
\[
\ve = (e^{0}, e^{1}, \dots\Add{,} e^{n-1})
\]
then the invariant is
\[
J = T_{ik} e^{i} e^{k}
\quad\text{and}\quad
e = e^{i} e_{i}.
\]
\PageSep{191}
But the two vectors and the tensor in~$\sfR_{\ve}$ have as their representatives
in the world, according to~\Eq{(39)}, the two vectors and the tensor with
components:
\begin{align*}
\sfL &: L_{i} - \frac{J}{e} e_{i}\qquad
\Typo{L}{L_{i}} = T_{ik} e^{k}, \\
\sfL' &: L_{i}' - \frac{J}{e} e_{i}\qquad
L_{i}' = T_{ki} e^{k}; \\
\sfT &: T_{ik} - \frac{e_{k} L_{i} + e_{i} L_{k}'}{e} + \frac{J}{e^{2}} e_{i} e_{k}.
\end{align*}
In the case of a skew-symmetrical tensor, $J$~becomes $= 0$ and
$\sfL' = -\sfL$; our formulæ degenerate into
\begin{align*}
\sfL &: L_{i} = T_{ik} e^{k}\Add{,} \\
\sfT &: T_{ik} + \frac{e_{i} L_{k} - e_{k} L_{i}}{e}.
\end{align*}
A linear world-tensor of the second order splits up in space into a
vector and a linear space-tensor of the second order.
Maxwell's field-equations for bodies at rest have been set out in
\index{Induction, magnetic!law of}%
§\,20. H.~Hertz was the first to attempt to extend them so that
they might apply generally for moving bodies. Faraday's Law of
Induction states that the time-decrement of the flux of induction
enclosed in a conductor is equal to the induced electromotive force,
that is
\[
-\frac{1}{c}\, \frac{d}{dt} \int B_{n}\, do = \int \vE\, d\vr\Add{.}
\Tag{(40)}
\]
The surface-integral on the left, if the conductor be in motion, must
be taken over a surface stretched out inside the conductor and
moving with it. Since Faraday's Law of Induction has been proved
\index{Faraday's Law of Induction}%
for just those cases in which the time-change of the flux of induction
within the conductor is brought about by the motion of the conductor,
Hertz did not doubt that this law was equally valid for
the case, too, when the conductor was in motion. The equation
$\div \vB = 0$ remains unaffected. From vector analysis we know that,
taking this equation into consideration, the law of induction~\Eq{(40)}
may be expressed in the differential form:
\[
\curl \vE = -\frac{1}{c}\, \frac{\dd \vB}{\dd t} + \frac{1}{c} \curl [\vv\Com \vB]
\Tag{(41)}
\]
in which $\dfrac{\dd \vB}{\dd t}$ denotes the differential co-efficient of~$\vB$ with respect
to the time for a fixed point in space, and $\vv$~denotes the velocity of
the matter.
Remarkable inferences may be drawn from~\Eq{(41)}. As in Wilson's
\PageSep{192}
experiment (\textit{vide} \FNote{10}), we suppose a homogeneous dielectric between
the two plates of a condenser, and assume that this dielectric
moves with a constant velocity of magnitude~$\vv$ between these plates,
which we shall take to be connected by means of a conducting
wire. Suppose, further, that there is a homogeneous magnetic field~$H$
parallel to the plates and perpendicular to~$\vv$. We shall imagine
the dielectric separated from the plates of the condenser by a
narrow empty space, whose thickness we shall assume $\to 0$ in the
limit. It then follows from~\Eq{(41)} that, in the space between the
plates, $\vE - \dfrac{1}{c} [\vv\Com \vB]$ is derivable from a potential; since the latter
must be zero at the plates which are connected by a conducting
wire it is easily seen that we must have $\vE = \dfrac{1}{c} [\vv\Com \vB]$. Hence a
homogeneous electric field of intensity $E = \dfrac{\mu}{c} vH$ (in which $\mu$~denotes
permeability) arises which acts perpendicularly to the plates.
Consequently, a statical charge of surface-density $\dfrac{\epsilon\mu}{c} vH$ ($\epsilon = $ dielectric
constant) must be called up on the
plates.
%[** TN: Width-dependent line break and fake \par]
\WrapFigure{1.25in}{14}
\noindent If the dielectric is a gas, this effect
should manifest itself, no matter to what degree
the gas has been rarefied, since $\epsilon\mu$~converges,
\Emph{not} towards~$0$, but towards~$1$, at infinite rarefaction.
This can have only one meaning if
we are to retain our belief in the æther,
namely, that the effect must occur if the
æther between the plates is moving relatively
to the plates and to the æther outside them.
To explain induction we should, however,
be compelled to assume that the æther is
dragged along by the connecting wire.\footnote
{In~\Eq{(41)} $\vv$~signified the velocity of the æther, \Emph{not} relative to the matter
but relative to what?}
General observations, Fizeau's experiment
dealing with the propagation of light in flowing water, and
Wilson's experiment itself, prove that this assumption is incorrect.
\index{Wilson's experiment}%
Just as in Fizeau's experiment the convection-co-efficient
$1 - \dfrac{1}{n^{2}}$ appears, so in the present experiment we observe only a
change of magnitude
\[
\frac{\epsilon\mu - 1}{c} vH
\]
\PageSep{193}
which vanishes when $\epsilon\mu = 1$. This seems to be an inexplicable
contradiction to the phenomenon of induction in the moving
conductor.
The theory of relativity offers a full explanation of this. If, as
in §\,20, we again set $ct = x_{0}$, and if we again build up a field~$F$
out of $\vE$~and~$\vB$, and a skew-symmetrical tensor~$H$ of the second
order out of $\vD$~and~$\vH$, we have the field-equations
\[
\left.
\begin{aligned}
\frac{\dd F_{kl}}{\dd x_{i}}
+ \frac{\dd F_{li}}{\dd x_{k}}
+ \frac{\dd F_{ik}}{\dd x_{l}} &= 0\Add{,} \\
\sum_{k} \frac{\dd H^{ik}}{\dd x_{k}} &= s^{i}\Add{.}
\end{aligned}
\right\}
\Tag{(42)}
\]
These hold if we regard the~$F_{ik}$'s as co-variant, the~$H^{ik}$'s as contra-variant
components, in each case, of a tensor of the second order,
but the~$s^{i}$'s as the contra-variant components of a vector in the
four-dimensional world, since the latter are invariant in any
arbitrary linear co-ordinate system. The laws of matter
\[
\vD = \epsilon \vE\Add{,}\qquad
\vB = \mu \vH\Add{,}\qquad
\vs = \sigma \vE
\]
signify, however, that if we split up the world into space and time
in such a way that matter is at rest, and if $F$~splits up into $\vE \mid \vB$,
$H$~into $\vD \mid \vH$, and $s$~into $\rho \mid \vs$, then the above relations hold. If
we now use any arbitrary co-ordinate system, and if the world-direction
of the matter has the components~$u^{i}$ in it then, after our
explanations above, these facts assume the form
\begin{flalign*}
(a) && H_{i}^{*} &= \epsilon F_{i}^{*} &&
\Tag{(43)}
\end{flalign*}
in which
\[
F_{i}^{*} = F_{ik} u^{k}\quad\text{and}\quad H_{i}^{*} = H_{ik} u^{k}\Add{;}
\]
\begin{flalign*}
(b) && F_{ik} - (u_{i}F_{k}^{*} - u_{k} F_{i}^{*})
&= \mu \bigl\{H_{ik} - (u_{i} H_{k}^{*} - u_{k} H_{i}^{*})\bigr\}\Add{;} &&
\Tag{(44)}
\end{flalign*}
\begin{flalign*}
\text{and }(c) && s_{i} + u_{i}(s_{k} u^{k}) = \sigma F_{i}^{*}\Add{.} &&
\Tag{(45)}
\end{flalign*}
This is the invariant form of these laws. For purposes of calculation
it is convenient to replace~\Eq{(44)} by the equations
\[
F_{kl} u_{i} + F_{li} u_{k} + F_{ik} u_{l}
= \mu \{H_{kl} u_{i} + H_{li} u_{k} + H_{ik} u_{l}\}
\Tag{(46)}
\]
which are derived directly from them. Our manner of deriving
them makes it clear that they hold only for matter which is in
uniform translation. We may, however, consider them as being
valid also for a single body in uniform translation, if it is separated
by empty space from bodies moving with velocities differing from
its own.\footnote
{This is the essential point in most applications. By applying Maxwell's
statical laws to a region composed, in each case, of a body~$K$ and the empty
space surrounding it and referred to the system of reference in which $K$~is at
rest, we find no discrepancies occurring in empty space when we derive results
from different bodies moving relatively to one another, \Emph{because the principle
of relativity holds for empty space}.}
Finally, they may also be considered to hold for matter
\PageSep{194}
\index{Field action of electricity!electromagnetic@{(electromagnetic)}}%
\index{Ponderomotive force!of the electric, magnetic and electromagnetic field}%
moving in any manner whatsoever, provided that its velocity does
not fluctuate too rapidly. After having obtained the invariant form
in this way, we may now split up the world in terms of any
arbitrary~$\ve$. Suppose the measuring instruments that are used to
determine the ponderomotive effects of field to be at rest in~$\Typo{R}{\sfR}_{\ve}$.
We shall use a co-ordinate system belonging to~$\Typo{R}{\sfR}_{\ve}$ and thus set
\begin{gather*}
\begin{array}{@{}rrr@{\,}c@{\,}lcr@{\,}c@{\,}c}
(F_{10}, & F_{20}, & F_{30}) & = & (\sfE_{1}, &\sfE_{2}, &\sfE_{3}) & = & \vE\Add{,} \\
(F_{23}, & F_{31}, & F_{12}) & = & (\sfB_{23}, &\sfB_{31}, &\sfB_{12}) & = & \vB\Add{,} \\
\hline
(H_{10}, & H_{20}, & H_{30}) & = & (\sfD_{1}, &\sfD_{2}, &\sfD_{3}) & = & \vD\Add{,} \\
(H_{23}, & H_{31}, & H_{12}) & = & (\sfH_{23}, &\sfH_{31}, &\sfH_{12}) & = & \vH\Add{,} \\
\end{array}\displaybreak[0] \\
\begin{aligned}
s^{0} &= \rho; & (s^{1}, s^{2}, s^{3}) = (\sfs^{1}, \sfs^{2}, \sfs^{3}) &= \vs\Add{,} \\
u^{0} &= \frac{1}{\sqrt{1 - v^{2}}}\quad &
(u^{1}, u^{2}, u^{3}) = \frac{(\sfv^{1}, \sfv^{2}, \sfv^{3})}{\sqrt{1 - v^{2}}} &= \frac{\vv}{\sqrt{1 - v^{2}}}\Add{,}
\end{aligned}
\end{gather*}
we hereby again arrive at \Emph{Maxwell's field-equations, which are
thus valid in a totally unchanged form, not only for static,
but also for moving matter}. Does this not, however, conflict
violently with the observations of induction, which appear to
require the addition of a term as in~\Eq{(41)}? No; for these
observations do not really determine the intensity of field~$\vE$, but
only the current which flows in the conductor; for moving bodies,
however, the connection between the two is given by a different
equation, namely, by~\Eq{(45)}.
If we write down those equations of \Eq{(43)}, \Eq{(45)}, which correspond
to the components with indices $i = 1, 2, 3$, and those of~\Eq{(46)}, which
correspond to
\[
(i\Com k\Com l) = (2\Com 3\Com 0),\quad (3\Com 1\Com 0),\quad (1\Com 2\Com 0)
\]
(the others are superfluous), the following results, as is easily seen,
come about. If we set
\begin{alignat*}{5}
\vE &+ [\vv\Com \vB] &&= \vE^{*}\Add{,} \qquad & \vD &&+ [\vv\Com \vH] &&= \vD^{*}\Add{,} \\
\vB &- [\vv\Com \vE] &&= \vB^{*}\Add{,} & \vH &&- [\vv\Com \vD] &&= \vH^{*}\Add{,}
\end{alignat*}
then
\[
\vD^{*} = \epsilon \vE^{*}\Add{,}\qquad
\vB^{*} = \mu \vH^{*}.
\]
If, in addition, we resolve~$\vs$ into the ``convection-current''~$\vc$ and
the ``conduction-current''~$\vs^{*}$, that is,
\begin{gather*}
\vs = \vc + \vs^{*}\Add{,} \\
\vc = \rho^{*} \vv\Add{,}\qquad
\rho^{*} = \frac{\rho - (\vv\Com \vs)}{1 - v^{2}} = \rho - (\vv\Com \vs^{*})\Add{,}
\end{gather*}
\PageSep{195}
then
\[
\vs^{*} = \frac{\sigma \vE^{*}}{\sqrt{1 - v^{2}}}.
\]
Everything now becomes clear: the current is composed partly of
\index{Convection currents}%
\index{Current!convection}%
a convection-current which is due to the motion of charged matter,
and partly of a conduction-current, which is determined by the
\index{Conduction}%
conductivity~$\sigma$ of the substance. The conduction-current is calculated
from Ohm's Law, if the electromotive force is defined
by the line-integral, not of~$\vE$, but of~$\vE^{*}$. An equation exactly
analogous to~\Eq{(41)} holds for~$\vE^{*}$, namely:
\[
\curl \vE^{*} = -\frac{\dd \vB}{\dd t} + \curl [\vv\Com \vB]
\quad\text{(we now always take $c = 1$)}
\]
or expressed in integrals, as in~\Eq{(40)},
\[
-\frac{d}{dt} \int B_{n}\, do = \int \vE^{*}\, d\vr.
\]
This explains fully Faraday's phenomenon of induction in moving
conductors. For Wilson's experiment, according to the present
theory, $\curl \vE = \Typo{0}{\0}$, that is, $\vE$~will be zero between the plates. This
gives us the constant values of the individual vectors (of which the
electrical ones are perpendicular to the plates, whilst the magnetic
ones are directed parallel to the plates and perpendicular to the
velocity): these values are:
%[** TN: Left-aligned in the original]
\begin{gather*}
E^{*} = vB^{*} = v\mu H^{*} = \mu v(H + vD)\Add{,} \\
D = D^{*} - vH = \epsilon E^{*} - vH.
\end{gather*}
If we substitute the expression for~$E^{*}$ in the first equation, we get
\begin{gather*}
D = v\bigl\{(\epsilon\mu - 1)H + \epsilon\mu vD\bigr\}\Add{,} \\
D = \frac{\epsilon\mu - 1}{1 - \epsilon\mu v^{2}} vH.
\end{gather*}
This is the value of the superficial density of charge that is called
up on the condenser plates: it agrees with our observations since,
on account of $v$~being very small, the denominator in our formula
differs very little from unity.
The boundary conditions at the boundary between the matter
and the æther are obtained from the consideration that the field-magnitudes
$F$~and~$H$ must not suffer any sudden (discontinuous)
changes in moving along with the matter; but, in general, they will
undergo a sudden change, at some fixed space-point imagined
in the æther for the sake of clearness, at the instant at which the
matter passes over this point. If $s$~is the proper-time of an elementary
particle of matter then
\[
\frac{dF_{ik}}{ds} = \frac{\dd F_{ik}}{\dd x_{l}} u^{l}
\]
\PageSep{196}
must remain finite everywhere. If we set
\[
\frac{\dd F_{ik}}{\dd x_{l}}
= -\left(\frac{\dd F_{kl}}{\dd x_{i}} + \frac{\dd F_{li}}{\dd x_{k}}\right)
\]
we see that this expression
\[
= \frac{\dd F_{i}^{*}}{\dd x_{k}} - \frac{\dd F_{k}^{*}}{\dd x_{i}}.
\]
Consequently, $\vE^{*}$~cannot have a surface-curl (and $\vB$~cannot have a
surface-divergence).
The fundamental equations for moving bodies were deduced by
Lorentz from the theory of electrons in a form equivalent to the
above before the discovery of the principle of relativity. This is
not surprising, seeing that Maxwell's fundamental laws for the
æther satisfy the principle of relativity, and that the theory of
electrons derives those governing the behaviour of matter by building
up mean values from these laws. Fizeau's and Wilson's experiments
and another analogous one, that of Röntgen and Eichwald
(\textit{vide} \FNote{11}), prove that the electromagnetic behaviour of matter is
in accordance with the principle of relativity; the problems of the
electrodynamics of moving bodies first led Einstein to enunciate it.
We are indebted to Minkowski for recognising clearly that the
fundamental equations for moving bodies are determined uniquely
by the principle of relativity if Maxwell's theory for matter at rest
is taken for granted. He it was, also, who formulated it in its
final form (\textit{vide} \FNote{12}).
Our next aim will be to subjugate \Emph{mechanics}, which does not
obey the principle in its classical form, to the principle of relativity
of Einstein, and to inquire whether the modifications that the latter
demands can be made to harmonise with the facts of experiment.
\Section{24.}{Mechanics according to the Principle of Relativity}
On the theory of electrons we found the mechanical effect of the
electromagnetic field to depend on a vector~$\vp$ whose contra-variant
components are
\[
p^{i} = F^{ik} s_{k} = \rho_{0} F^{ik} u_{k}.
\]
It therefore satisfies the equation
\[
p^{i} u_{i} = (\vp\Com \vu) = 0
\Tag{(47)}
\]
in which $\vu$~is the world-direction of the matter. If we split up $\vp$
and~$\vu$ in any way into space and time thus
\[
\left.
\begin{aligned}
\vu &= h \mid h\sfv\Add{,} \\
\vp &= \lambda \mid \sfp\Add{,}
\end{aligned}
\right\}
\Tag{(48)}
\]
\PageSep{197}
we get $\sfp$ as the force-density and, as we see from~\Eq{(47)} or from
\index{Density!general@{(general conception)}}%
\[
h\bigl\{\lambda - (\sfp\Com \sfv)\bigr\} = 0
\]
that $\lambda$~is the work-density.
We arrive at the fundamental law of the mechanics which
\index{Mechanics!fundamental law of!special@{(in special theory of relativity)}}%
agrees with Einstein's Principle of Relativity by the same method
as that by which we obtain the fundamental equations of electromagnetics.
We assume that Newton's Law remains valid in the
system of reference in which the matter is at rest. We fix our
attention on the point-mass~$m$, which is situated at a definite world-point~$O$
and split up our quantities in terms of its world-direction~$\vu$
into space and time. $m$~is momentarily at rest in~$\sfR_{\vu}$. Let $\mu_{0}$~be
the density in~$\sfR_{\vu}$ of the matter at the point~$O$. Suppose that, after
an infinitesimal element of time~$ds$ has elapsed, $m$~has the world-direction
$\vu + d\vu$. It follows from $(\vu\Com \vu) = -1$ that $(\vu · d\vu) = 0$.
Hence, splitting up with respect to~$\vu$, we get
\[
\vu = 1 \mid \sfO,\qquad
d\vu = 0 \mid d\sfv,\qquad
\vp = 0 \mid \sfp.
\]
It follows from
\[
\vu + d\vu = 1 \mid d\sfv
\]
that $d\sfv$~is the relative velocity acquired by~$m$ (in~$\sfR_{\vu}$) during the
time~$ds$. Thus there can be no doubt that the fundamental law of
mechanics is
\[
\mu_{0}\, \frac{d\sfv}{ds} = \sfp.
\]
From this we derive at once the invariant form
\[
\mu_{0}\, \frac{d\vu}{ds} = \vp\Add{,}
\Tag{(49)}
\]
which is quite independent of the manner of splitting up. In it, $\mu_{0}$~is
the \Emph{statical density}, that is, the density of the mass when at
\index{Static!density}%
rest; $ds$~is the \Emph{proper-time} that elapses during the infinitesimal
\index{Proper-time}%
displacement of the particle of matter, during which its world-direction
increases by~$d\vu$.
Resolution into terms of~$\vu$ is a partition which would alter
during the motion of the particle of matter. If we now split up
our quantities, however, into space and time by means of some
fixed time-like vector~$\ve$ that points into the future and satisfies the
condition of normality $(\ve\Com \ve) = -1$, then, by \Eq{(48)},~\Eq{(49)} resolves into
\[
\left.
\begin{aligned}
\mu_{0}\, \frac{d}{ds} \left(\frac{1}{\sqrt{1 - v^{2}}}\right) &= \lambda\Add{,} \\
\mu_{0}\, \frac{d}{ds} \left(\frac{\sfv}{\sqrt{1 - v^{2}}}\right) &= \sfp\Add{.} \\
\end{aligned}
\right\}
\Tag{(50)}
\]
\PageSep{198}
If, in this partition or resolution, $t$~denotes the time, $dV$~the volume,
and $dV_{0}$~the static volume of the particle of matter at a definite
moment, its mass, however, being $m = \mu_{0}\, dV_{0}$, and if
\[
\sfp\, dv = \sfP,\qquad
\lambda\, dV = \sfL
\]
denotes the force acting on the particle and its work, respectively,
then if we multiply our equations by~$dV$ and take into account that
\[
\mu_{0}\, dV · \frac{d}{ds}
= m \sqrt{1 - v^{2}} · \frac{d}{ds}
= m · \frac{d}{dt}
\]
and that the mass~$m$ remains constant during the motion, we get
finally
\begin{align*}
\frac{d}{dt} \left(\frac{m}{\sqrt{1 - v^{2}}}\right) &= \sfL\Add{,}
\Tag{(51)} \\
\frac{d}{dt} \left(\frac{m\sfv}{\sqrt{1 - v^{2}}}\right) &= \sfP\Add{.}
\Tag{(52)}
\end{align*}
These are the equations for the mechanics of the point-mass. The
equation of momentum~\Eq{(52)} differs from that of Newton only in
that the (kinetic) momentum of the point-mass is not~$m\sfv$ but
$= \dfrac{m\sfv}{\sqrt{1 - v^{2}}}$. The equation of energy~\Eq{(51)} seems strange at first:
if we expand it into powers of~$v$, we get
\[
\frac{m}{\sqrt{1 - v^{2}}} = m + \frac{mv^{2}}{2} + \dots,
\]
so that if we neglect higher powers of~$v$ and also the constant~$m$
we find that the expression for the kinetic energy degenerates into
the one given by classical mechanics.
This shows that the deviations from the mechanics of Newton
are, as we suspected, of only the second order of magnitude in the
velocity of the point-masses as compared with the velocity of light.
Consequently, in the case of the small velocities with which we
usually deal in mechanics, no difference can be demonstrated experimentally.
It will become perceptible only for velocities that
approximate to that of light; in such cases the inertial resistance of
matter against the accelerating force will increase to such an extent
that the possibility of actually reaching the velocity of light is excluded.
\Emph{Cathode rays} and the $\beta$-radiations emitted by radioactive
\index{Cathode rays}%
substances have made us familiar with free negative electrons
whose velocity is comparable to that of light. Experiments by
Kaufmann, Bucherer, Ratnowsky, Hupka, and others, have shown in
actual fact that the longitudinal acceleration caused in the electrons
by an electric field or the transverse acceleration caused by a magnetic
field is just that which is demanded by the theory of relativity. A
\PageSep{199}
further confirmation based on the motion of the electrons circulating
in the atom has been found recently in the \emph{fine structure} of the
spectral lines emitted by the atom (\textit{vide} \FNote{13}). Only when we
have added to the fundamental equations of the electron theory,
which, in §\,20, was brought into an invariant form agreeing with
the principle of relativity, the equation $s^{i} = \rho_{0} u^{i}$, namely, the assertion
that electricity is associated with matter, and also the fundamental
equations of mechanics, do we get a complete cycle of
connected laws, in which a statement of the actual unfolding of
natural phenomena is contained, independent of all conventions of
notation. Now that this final stage has been carried out, we may
at last claim to have proved the validity of the principle of relativity
for a certain region, that of electromagnetic phenomena.
In the electromagnetic field the ponderomotive vector~$p_{i}$ is
derived from a tensor~$S_{ik}$, dependent only on the local values of
the phase-quantities, by the formulæ:
\[
p^{i} = -\frac{\dd S_{i}^{k}}{\dd x_{k}}.
\]
In accordance with the universal meaning ascribed to the conception
\index{Energy-momentum, tensor!(general)}%
\index{Energy-momentum, tensor!(kinetic and potential)}%
\index{Potential!energy-momentum tensor of}%
\emph{energy} in physics, we must assume that this holds not only for the
electromagnetic field but for every region of physical phenomena,
and that it is expedient to regard this tensor instead of the ponderomotive
force as the primary quantity. Our purpose is to discover
for every region of phenomena in what manner the energy-momentum-tensor
(whose components~$S_{ik}$ must always satisfy the condition
of symmetry) depends on the characteristic field- or phase-quantities.
The left-hand side of the mechanical equations\Pagelabel{199}
\[
\mu_{0}\, \frac{du^{i}}{ds} = p_{i}
\]
may be reduced directly to terms of a ``kinetic'' energy-momentum-tensor
thus:
\[
U_{ik} = \mu_{0} u_{i} u_{k}.
\]
For
\[
\frac{\dd U_{i}^{k}}{\dd x_{k}}
= u_{i}\, \frac{\dd (\mu_{0} u^{k})}{\dd x_{k}}
+ \mu_{0} u^{k}\, \frac{\dd u_{i}}{\dd x_{k}}.
\]
The first term on the right $= 0$, on account of the equation of continuity
for matter; the second $= \mu_{0}\, \dfrac{du^{i}}{ds}$ because
\[
u^{k}\, \frac{\dd u_{i}}{\dd x_{k}}
= \frac{\dd u_{i}}{\dd x_{k}}\, \frac{\dd x_{k}}{\dd s}
= \frac{du_{i}}{ds}.
\]
Accordingly, the equations of mechanics assert that the complete
energy-momentum-tensor $T_{ik} = U_{ik} + S_{ik}$ composed of the kinetic
\PageSep{200}
\index{Moment!mechanical}%
tensor~$U$ and the potential tensor~$S$ satisfies the theorems of conservation
\index{Potential!energy-momentum tensor of}%
\[
\frac{\dd T_{i}^{k}}{\dd x_{k}} = 0.
\]
The Principle of the Conservation of Energy is here expressed in
its clearest form. But, according to the theory of relativity, it is
indissolubly connected with the principle of the conservation of
momentum and \Emph{the conception \emph{momentum} (or \emph{impulse}) must
\index{Momentum}%
claim just as universal a significance as that of energy}.
If we express the kinetic tensor at a world-point in terms of a
normal co-ordinate system such that, relatively to it, the matter itself
is momentarily at rest, its components assume a particularly simple
form, namely, $U_{00} = \mu_{0}$ (or $= c^{2} \mu_{0}$, if we use the c.g.s.\ system, in
which $c$~is not $= 1$), and all the remaining components vanish.
This suggests the idea that mass is to be regarded as concentrated
potential energy that moves on through space.
\Section{25.}{Mass and Energy}
To interpret the idea expressed in the preceding sentence we
shall take up the thread by returning to the consideration of the
motion of the electron. So far, we have imagined that we have to
write for the force~$\vP$ in its equation of motion~\Eq{(52)} the following:
\[
\vP = e\bigl(\vE + [\vv\Com \vH]\bigr)\quad
(e = \text{charge of the electron})
\]
that is, that $\vP$ is composed of the impressed electric and magnetic
fields $\vE$ and~$\vH$. Actually, however, the electron is subject not
only to the influence of these external fields during its motion but
also to the accompanying field which it itself generates. A
difficulty arises, however, in the circumstance that we do not
know the constitution of the electron, and that we do not know the
nature and laws of the cohesive pressure that keeps the electron
together against the enormous centrifugal forces of the negative
charge compressed in it. In any case the electron at rest and its
electric field (which we consider as part of it) is a physical system,
which is in a state of statical equilibrium---and that is the essential
point. Let us choose a normal co-ordinate system in which the
electron is at rest. Suppose its energy-tensor to have components~$t_{ik}$.
The fact that the electron is at rest is expressed by the vanishing
of the energy-flux of whose components are~$t_{\Typo{o}{0}i}$ ($i = 1, 2, 3$).
% [** TN: Ordinal]
The $0$th~condition of equilibrium
\[
\frac{\dd t_{i}^{k}}{\dd x_{k}} = 0
\Tag{(53)}
\]
\PageSep{201}
then tells us that the energy-density~$t_{00}$ is independent of the time~$x_{0}$.
On account of symmetry the components~$t_{i\Typo{o}{0}}$ ($i = 1, 2, 3$) of
the momentum-density each also vanish. If $\vt^{(1)}$ is the vector whose
components are $t_{11}$,~$t_{12}$,~$t_{13}$, the condition for equilibrium~\Eq{(53)},
($i = 1$), gives
\[
\div \vt^{(1)} = 0.
\]
Hence we have, for example,
\[
\div (x_{2} \vt^{(1)}) = x_{2} \div \vt^{(1)} + t_{12} = t_{12}
\]
and since the integral of a divergence is zero (we may assume that
the~$t$'s vanish at infinity at least as far as to the fourth order) we get
\[
\int t_{12}\, dx_{1}\, dx_{2}\, dx_{3} = 0.
\]
In the same way we find that, although the~$t_{ik}$'s (for $i, k = 1, 2, 3$)
do not vanish, their volume integrals $\Dint t_{ik}\, dV_{0}$ do so. We may
regard these circumstances as existing for every system in statical
equilibrium. The result obtained may be expressed by invariant
formulæ for the case of any arbitrary co-ordinate system thus:
\[
\int t_{ik}\, dV_{0} = E_{0} u_{i} u_{k}\quad (i, k = 0, 1, 2, 3)\Add{.}
\Tag{(54)}
\]
$E_{0}$~is the energy-content (measured in the space of reference for
which the electron is at rest), $u_{i}$~are the co-variant components of
the world-direction of the electron, and $dV_{0}$~the statical volume of
an element of space (calculated on the supposition that the whole
of space participates in the motion of the electron). \Eq{(54)}~is
rigorously true for uniform translation. We may also apply the
formula in the case of non-uniform motion if $\vu$~does not change
too suddenly in space or in time. The components
\[
\bar{p}^{i} = -\frac{\dd t^{ik}}{\dd x_{k}}
\]
of the ponderomotive effect, exerted on the electron by itself, are
however, then no longer $= 0$.
If we assume the electron to be entirely without mass, and if
$p^{i}$~is the ``$4$-force'' acting from without, then equilibrium demands
that
\[
\bar{p}^{i} + p^{i} = 0\Add{.}
\Tag{(55)}
\]
We split up $\vu$ and~$\vp$ into space and time in terms of a fixed~$\ve$, getting
\[
\vu = h \mid h\sfv,\qquad
\vp = (p^{i}) = \lambda \mid \sfp
\]
and we integrate~\Eq{(55)} with respect to the volume $dV\!\! =\! dV_{0} \sqrt{1 - v^{2}}$.
Since, if we use a normal co-ordinate system
corresponding to~$\sfR_{\ve}$, we have
\PageSep{202}
\begin{align*}
\int \bar{p}^{i}\, dV
&= \int \bar{p}^{i}\, dx_{1}\, dx_{2}\, dx_{3}
= -\frac{d}{dx_{0}} \int t^{i\Typo{o}{0}}\, dx_{1}\, dx_{2}\, dx_{3} \\
&= -\frac{d}{dx_{0}} (E_{0} u^{0} u^{i} \sqrt{1 - v^{2}})
= -\frac{d}{dt} (E_{0} u^{i})
\end{align*}
(in which $x_{0} = t$, the time), we get
\begin{align*}
\frac{\Typo{t}{d}}{dt} \left(\frac{E_{0}}{\sqrt{1 - v^{2}}}\right)
&= \sfL\ \left(= \int \lambda\, dV\right), \\
\frac{\Typo{t}{d}}{dt} \left(\frac{E_{0}\sfv}{\sqrt{1 - v^{2}}}\right)
&= \sfP\ \left(= \int \sfp\, dV\right).
\end{align*}
These equations hold if the force~$\sfP$ acting from without is not too
great compared with~$\dfrac{E_{0}}{a}$, $a$~being the radius of the electron, and if
its density in the neighbourhood of the electron is practically
constant. They agree exactly with the fundamental equations of
mechanics if the mass~$m$ is replaced by~$E$. In other words,
\Emph{inertia is a property of energy}. In mechanics we ascribe to
\index{Inertia!(as property of energy)}%
every material body an invariable mass~$m$ which, in consequence of
the manner in which it occurs in the fundamental law of mechanics,
represents the inertia of matter, that is, its resistance to the
accelerating forces. Mechanics accepts this inertial mass as given
and as requiring no further explanation. We now recognise that the
potential energy contained in material bodies is the cause of this
inertia, and that the value of the mass corresponding to the energy~$E_{0}$
expressed in the c.g.s.\ system, in which the velocity of light is
\Emph{not} unity, is
\[
m = \frac{E_{0}}{c^{2}}\Add{.}
\Tag{(56)}
\]
We have thus attained a new, purely dynamical view of matter.\footnote
{Even Kant in his \Title{Metaphysischen Anfangsgründen der Naturwissenschaft},
teaches the doctrine that matter fills space not by its mere existence but in
virtue of the repulsive forces of all its parts.}
Just as the theory of relativity has taught us to reject the belief that
we can recognise one and the same point in space at different times,
\Emph{so now we see that there is no longer a meaning in speaking
of the same position of matter at different times}. The
electron, which was formerly regarded as a body of foreign
substance in the non-material electromagnetic field, now no longer
seems to us a very small region marked off distinctly from the
field, but to be such that, for it, the field-quantities and the
electrical densities assume enormously high values. An ``energy-knot''
of this type propagates itself in empty space in a manner no
different from that in which a water-wave advances over the surface
\PageSep{203}
of the sea; there is no ``one and the same substance'' of which the
electron is composed at all times. There is only a potential; and
no kinetic energy-momentum-tensor becomes added to it. The
resolution into these two, which occurs in mechanics, is only
the separation of the thinly distributed energy in the field
from that concentrated in the energy-knots, electrons and
atoms; the boundary between the two is quite indeterminate.
The theory of fields has to explain why the field is granular in
structure and why these energy-knots preserve themselves permanently
from energy and momentum in their passage to and fro
(although they do not remain fully unchanged, they retain their
identity to an extraordinary degree of accuracy); therein lies the
\Emph{problem of matter}. The theory of Maxwell and Lorentz is
\index{Matter}%
incapable of solving it for the primary reason that the force of
cohesion holding the electron together is wanting in it. \Emph{What is
commonly called matter is by its very nature atomic}; for
we do not usually call diffusely distributed energy matter. \Emph{Atoms
and electrons are not}, of course, \Emph{ultimate invariable elements},
which natural forces attack from without, pushing them hither and
thither, but they are themselves distributed continuously and subject
to minute changes of a fluid character in their smallest parts. It is
not the field that requires matter as its carrier in order to be able to
exist itself, but \Emph{matter} is, on the contrary, \Emph{an offspring of the
field}. The formulæ that express the components of the energy-tensor~$T_{ik}$
in terms of phase-quantities of the field tell us \emph{the laws
according to which} the field is associated with energy and momentum,
that is, with matter. Since there is no sharp line of demarcation
between diffuse field-energy and that of electrons and atoms,
we must broaden our conception of matter, if it is still to retain an
\emph{exact} meaning. In future we shall assign the term matter to that
real thing, which is represented by the energy-momentum-tensor.
In this sense, the optical field, for example, is also associated with
matter. Just as in this way matter is merged into the field, so
mechanics is expanded into physics. For the law of conservation of
matter, the fundamental law of mechanics
\[
\frac{\dd \Typo{T_{k}^{i}}{T_{i}^{k}}}{\dd x_{k}} = 0\Add{,}
\Tag{(57)}
\]
in which the~$T_{ik}$'s are expressed in terms of the field-quantities,
represents a differential relationship between these quantities, and
must therefore follow from the field-equations. In the wide sense,
in which we now use the word, matter is that of which we take
cognisance directly through our senses. If I seize hold of a piece
of ice, I experience the energy-flux flowing between the ice and
my body as warmth, and the momentum-flux as pressure. The
\PageSep{204}
energy-flux of light on the surface of the epithelium of my eye
\index{Energy!(possesses inertia)}%
determines the optical sensations that I experience. Hidden behind
the matter thus revealed directly to our organs of sense there is,
however, the \Emph{field}. To discover the laws governing the latter
itself and also the laws by which it determines matter we have a
first brilliant beginning in Maxwell's Theory, but this is not our
final destination in the quest of knowledge.\footnote
{Later we shall once again modify our views of matter; the idea of the
existence of substance has, however, been finally quashed.}
To account for the inertia of matter we must, according to
formula~\Eq{(56)}, ascribe a very considerable amount of energy-content
to it: one kilogram of water is to contain $9 · 10^{23}$~ergs. A small portion
of this energy is energy of cohesion, that keeps the molecules
or atoms associated together in the body. Another portion is the
chemical energy that binds the atoms together in the molecule and
the sudden liberation of which we observe in an explosion (in solid
bodies this chemical energy cannot be distinguished from the energy
of cohesion). Changes in the chemical constitution of bodies or in
the grouping of atoms or electrons involve the energies due to the
electric forces that bind together the negatively charged electrons
and the positive nucleus; all ionisation phenomena are included
in this category. The energy of the composite atomic nucleus, of
which a part is set free during radioactive disintegration, far exceeds
the amounts mentioned above. The greater part of this, again,
consists of the intrinsic energy of the elements of the atomic nucleus
and of the electrons. We know of it only through inertial effects
as we have hitherto---owing to a merciful Providence---not discovered
a means of bringing it to ``explosion''. \Emph{Inertial mass
\index{Mass!energy@{(as energy)}}%
varies with the contained energy.} If a body is heated, its
inertial mass increases; if it is cooled, it decreases; this effect is, of
course, too small to be observed directly.
The foregoing treatment of systems in statical equilibrium, in
which we have in general followed Laue,\footnote
{\textit{Vide} \FNote{14}.}
was applied to the electron
with special assumptions concerning its constitution, even before
Einstein's discovery of the principle of relativity. The electron was
assumed to be a sphere with a uniform charge either on its surface
or distributed evenly throughout its volume, and held together by
a cohesive pressure composed of forces equal in all directions and
directed towards the centre. The resultant ``electromagnetic mass''
$\dfrac{E_{0}}{c^{2}}$ agrees numerically with the results of observation, if one
ascribes a radius of the order of magnitude $10^{-13}$~cms.\ to the
electron. There is no cause for surprise at the fact that even before
\PageSep{205}
the advent of the theory of relativity this interpretation of electronic
inertia was possible; for, in treating electrodynamics after the
manner of Maxwell, one was already unconsciously treading in the
steps of the principle of relativity as far as this branch of phenomena
is concerned. We are indebted to Einstein and Planck,
above all, for the enunciation of the inertia of energy (\textit{vide} \FNote{15}).
Planck, in his development of dynamics, started from a ``test body''
which, contrary to the electron, was fully known although it was
not in the ordinary sense material, namely, cavity-radiation in
\Chg{thermo-dynamical}{thermodynamical} equilibrium, as produced according to Kirchoff's
Law, in every cavity enclosed by walls at the same uniform
temperature.
In the phenomenological theories in which the atomic structure
\index{Energy-momentum, tensor!(of an incompressible fluid)}%
\index{Hydrostatic pressure}% [** TN: Hyphenated (but text usage inconsistent)]
\index{Pressure, on all sides!hydrostatic}%
of matter is disregarded we imagine the energy that is stored up
in the electrons, atoms, etc., to be distributed uniformly over the
bodies. We need take it into consideration only by introducing the
statical density of mass~$\mu$ as the density of energy in the energy-momentum-tensor---referred
to a co-ordinate system in which the
matter is at rest. Thus, if in \Chg{hydro-dynamics}{hydrodynamics} we limit ourselves to
\index{Hydrodynamics}% [** TN: Hyphenated (but text usage inconsistent)]
adiabatic phenomena, we must set
\[
|T_{i}^{k}|
= \left\lvert\begin{array}{@{}c|ccc@{}}
-\mu_{0} & 0 & 0 & 0 \\
\hline
\Strut
0 & p & 0 & 0 \\
0 & 0 & p & 0 \\
0 & 0 & 0 & p \\
\end{array}\right\rvert
\]
in which $p$~is the homogeneous pressure; the energy-flux is zero
in adiabatic phenomena. To enable us to write down the components
of this tensor in any arbitrary co-ordinate system, we must
set $\mu_{0} = \mu^{*} - p$, in addition. We then get the invariant equations
\begin{align*}
T_{i}^{k} &= \mu^{*} u_{i} u^{k} + p\delta_{i}^{k}\Add{,} \\
\text{or}\quad
T_{ik} &= \mu^{*} u_{i} u_{k} + p · g_{ik}\Add{.}
\Tag{(58)}
\end{align*}
The statical density of mass is
\[
T_{ik} u^{i} u^{k} = \mu^{*} - p = \mu_{0}
\]
and hence we must put~$\mu_{0}$, and \Emph{not}~$\mu^{*}$, equal to a constant in the
case of incompressible fluids. If no forces act on the fluid, the
hydrodynamical equations become
\[
\frac{\dd T_{i}^{k}}{\dd x_{k}} = 0.
\]
Just as is here done for hydrodynamics so we may find a form for
the theory of elasticity based on the principle of relativity (\textit{vide}
\FNote{16}). There still remains the task of making the law of
\PageSep{206}
gravitation, which, in Newton's form, is entirely bound to the
principle of relativity of Newton and Galilei, conform to that of
Einstein. This, however, involves special problems of its own to
which we shall return in the last chapter.
\Section{26.}{Mie's Theory}
\index{Mie's Theory}%
The theory of Maxwell and Lorentz cannot hold for the interior
of the electron; therefore, from the point of view of the ordinary
theory of electrons we must treat the electron as something given
\textit{a~priori}, as a foreign body in the field. A more general theory
of electrodynamics has been proposed by \Emph{Mie}, by which it seems
possible to derive the matter from the field (\textit{vide} \FNote{17}). We
shall sketch its outlines briefly here---as an example of a physical
theory fully conforming with the new ideas of matter, and one that
will be of good service later. It will give us an opportunity of
formulating the problem of matter a little more clearly.
We shall retain the view that the following phase-quantities
are of account: \Eq{(1)}~the four-dimensional current-vector~$s$, the
``electricity''; \Eq{(2)}~the linear tensor of the second order~$F$, the
``field''. Their properties are expressed in the equations
\begin{alignat*}{2}
(1)&& \frac{\dd s^{i}}{\dd x_{i}} &= 0, \\
(2)&&\quad
\frac{\dd F_{kl}}{\dd x_{i}}
+ \frac{\dd F_{li}}{\dd x_{k}}
+ \frac{\dd F_{ik}}{\dd x_{l}} &= 0.
\end{alignat*}
Equations~\Eq{(2)} hold if $F$~is derivable from a vector~$\phi_{i}$ according to
the formulæ
\[
\llap{(3)\quad}
F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}.
\]
Conversely, it follows from~\Eq{(2)} that a vector~$\phi$ must exist such that
equations~\Eq{(3)} hold. In the same way \Eq{(1)}~is fulfilled if $s$~is derivable
from a skew-symmetrical tensor~$H$ of the second order according to
\[
\llap{(4)\qquad}
s^{i} = \frac{\dd H^{ik}}{\dd x_{k}}.
\]
Conversely, it follows from~\Eq{(1)} that a tensor~$H$ satisfying these
conditions must exist. Lorentz assumed generally, not only for
the æther, but also for the domain of electrons, that $H = F$.
Following Mie, we shall make the more general assumption that
$H$~is not a mere number of calculation but has a real significance,
and that its components are, therefore, universal functions of the
primary phase-quantities $s$~and~$F$. To be logical we must then
\PageSep{207}
\index{Causality, principle of}%
\index{Metrics or metrical structure!(general)}%
make the same assumption about~$\phi$. The resultant scheme of
quantities
\[
\begin{array}{@{}c|c@{}}
\phi & F \\
\hline
\Strut s & H
\end{array}
\]
contains the quantities of intensity in the first row; they are connected
with one another by the differential equations~\Eq{(3)}. In the
second row we have the quantities of magnitude, for which the
differential quantities~\Eq{(4)} hold. If we perform the resolution into
space and time and use the same terms as in §\,20 we arrive at the
well-known equations
\begin{alignat*}{4}
(1)&\quad& \frac{d\rho}{dt} &+ \div s &&= 0, &&\displaybreak[0] \\
(2)&& \frac{d\sfB}{dt} &+ \curl \sfE &&= \Typo{0}{\0} & (\div \sfB &= 0),\displaybreak[0] \\
(3)&& \frac{df}{dt} &+ \grad \phi &&= \sfE & (-\curl f &= \sfB),\displaybreak[0] \\
(4)&& \frac{d\sfD}{dt} &- \curl \sfH &&= -s & (\div \sfD &= \rho).
\end{alignat*}
If we know the universal functions, which express $\phi$~and~$H$ in
terms of $s$~and~$F$, then, excluding the equations in brackets,
and counting each component separately, we have ten ``principal
equations'' before us, in which the derivatives of the ten phase-quantities
with respect to the time are expressed in relation to
themselves and their spatial derivatives; that is, we have physical
laws in the form that is demanded by the \Emph{principle of causality}.
The principle of relativity that here appears as an antithesis, in
a certain sense, to the principle of causality, demands that the
principal equations be accompanied by the bracketed ``subsidiary
equations,'' in which no time derivatives occur. The conflict is
avoided by noticing that the subsidiary equations are superfluous.
For it follows from the principal equations \Eq{(2)}~and~\Eq{(3)} that
\[
\frac{\dd}{\dd t} (\sfB + \curl f) = \Typo{0}{\0},
\]
and from \Eq{(1)}~and~\Eq{(4)} that
\[
\frac{\dd \rho}{\dd t} = \frac{\dd}{\dd t}(\div \sfD).
\]
It is instructive to compare Mie's Theory with Lorentz's fundamental
equations of the theory of electrons. In the latter, \Eq{(1)},~\Eq{(2)},
and~\Eq{(4)} occur, whilst the law by which $H$~is determined from the
primary phase-quantities is simply expressed by $\sfD = \sfE$, $\sfH = \sfB$.
On the other hand, in Mie's theory, $\phi$~and~$f$ are defined in~\Eq{(3)} as
\PageSep{208}
the result of a \emph{process of calculation}, and there is no law that
determines how these potentials depend on the phase-quantities of
the field and on the electricity. In place of this we find the formula
giving the density of the mechanical force and the law of mechanics,
\index{Force!(ponderomotive, of electromagnetic field)}%
which governs the motion of electrons under the influence of this
force. Since, however, according to the new view which we have put
forward, the mechanical law must follow from the field-equations,
an addendum becomes necessary; for this purpose, Mie makes the
assumption that $\phi$~and~$f$ acquire a physical meaning in the sense
indicated. We may, however, enunciate Mie's equation~\Eq{(3)} in a
form fully analogous to that of the fundamental law of mechanics.
We contrast the ponderomotive force occurring in it with the ``electrical
force''~$\sfE$ in this case. In the statical case \Eq{(3)}~states that
\[
\sfE - \grad \phi = \Typo{0}{\0}\Add{,}
\Tag{(59)}
\]
that is, the electric force~$\sfE$ is counterbalanced in the æther by an
\index{Electrical!momentum}%
\index{Electrical!pressure}%
\index{Moment!electrical}%
\index{Pressure, on all sides!electrical}%
``\Emph{electrical pressure}''~$\phi$. In general, however, a resulting electrical
force arises which, by~\Eq{(3)}, now belongs to the magnitude~$f$
as the ``\Emph{electrical momentum}''. It inspires us with wonder to
see how, in Mie's Theory, the fundamental equation of electrostatics~\Eq{(59)}
which stands at the commencement of electrical theory,
suddenly acquires a much more vivid meaning by the appearance
of potential as an electrical pressure; this is the required cohesive
pressure that keeps the electron together.
The foregoing presents only an empty scheme that has to be
filled in by the yet unknown universal functions that connect the
quantities of magnitude with those of intensity. Up to a certain
degree they may be determined purely speculatively by means of
the postulate that the theorem of conservation~\Eq{(57)} must hold for
the energy-momentum-tensor~$T_{ik}$ (that is, that the principle of
energy must be valid). For this is certainly a necessary condition,
if we are to arrive at some relationship with experiment at all.
The energy-law must be of the form
\[
\frac{\dd W}{\dd t} + \div \Typo{S}{\sfs} = 0
\]
in which $W$~is the density of energy, and $\sfs$~the energy-flux. We
get at Maxwell's Theory by multiplying~\Eq{(2)} by~$\sfH$ and \Eq{(4)}~by~$\sfE$, and
then adding, which gives
\[
\sfH\, \frac{\dd \sfB}{\dd t}
+ \sfE\, \frac{\dd \sfD}{\dd t}
+ \div [\sfE\Com \sfH]
= -(\sfE\Com \sfs)\Add{.}
\Tag{(60)}
\]
In this relation~\Eq{(60)} we have also on the right, the work, which is
used in increasing the kinetic energy of the electrons or, according
to our present view, in increasing the potential energy of the field
\PageSep{209}
of electrons. Hence this term must also be composed of a term
differentiated with respect to the time, and of a divergence. If we
now treat equations \Eq{(1)} and~\Eq{(3)} in the same way as we just above
treated \Eq{(2)}~and~\Eq{(4)}, that is, multiply~\Eq{(1)} by~$\phi$ and \Eq{(3)}~scalarly by~$\Typo{s}{\sfs}$,
we get
\[
\phi\, \frac{\dd \rho}{\dd t} + \sfs\, \frac{\dd f}{\dd t} + \div(\phi \sfs)
= (\sfE\Com \sfs)\Add{.}
\Tag{(61)}
\]
\Eq{(60)}~and~\Eq{(61)} together give the energy theorem; accordingly the
energy-flux must be
\[
\sfS = [\sfE\Com \sfH] + \phi \sfs\Add{,}
\]
and
\[
\phi\, \delta\rho + \sfs\, \delta f + \sfH\, \delta\sfB + \sfE\, \delta\sfD
= \delta W
\]
is the total differential of the energy-density. It is easy to see why
a term proportional to~$\sfs$, namely~$\phi \sfs$, has to be added to the term~$(\sfE\Com \sfH)$
which holds in the æther. For when the electron that
generates the convection-current~$\sfs$ moves, its energy-content flows
also. In the æther the term~$(\sfE\Com \sfH)$ is overpowered by~$\sfS$, but in the
electron the other~$\phi \sfs$ easily gains the upper hand. The quantities
$\rho$,~$f$, $\sfB$,~$\sfD$ occur in the formula for the total differential of the
energy-density as independent differentiated phase-quantities. For
the sake of clearness we shall introduce $\phi$~and~$\sfE$ as independent
variables in place of $\rho$~and~$\sfD$. By this means all the quantities of
intensity are made to act as independent variables. We must
build up
\[
L = W - \sfE\sfD - \rho\phi\Add{,}
\Tag{(62)}
\]
and then we get
\[
\delta L = (\sfH\, \delta\sfB - \sfD\, \delta\sfE) + (\sfs\, \delta f - \rho\, \delta\phi).
\]
If $L$~is known as a function of the quantities of intensity, then
these equations express the quantities of magnitude as functions of
the quantities of intensity. \Emph{In place of the ten unknown universal
functions we have now only one},~$L$; this is accomplished
by the \Emph{principle of energy}.
Let us again return to four-dimensional notation, we then have
\[
\delta L = \tfrac{1}{2} H^{ik}\, \delta F_{ik} + s^{i}\, \delta\phi_{i}\Add{.}
\Tag{(63)}
\]
From this it follows that~$\delta L$, and hence~$L$, the ``\Emph{Hamiltonian
Function}'' is an invariant. The simplest invariants that may be
\index{Hamilton's!function}%
\index{Hamilton's!principle!Mie@{(according to Mie)}}%
formed from a vector having components~$\phi_{i}$ and a linear tensor of
the second order having components~$F_{ik}$ are the squares of the
following expressions:
\begin{flalign*}
&\text{the vector~$\phi^{i}$,} &&
\phi_{i} \phi^{i}\Add{,} && && \\
&\text{the tensor~$F_{ik}$,} &
2L^{0} &= \tfrac{1}{2} F_{ik} F^{ik}\Add{,} && &&
\end{flalign*}
\PageSep{210}
the linear tensor of the fourth order with components $\sum ± F_{ik} F_{lm}$
(the summation extends over the $24$~permutations of the indices
$i$,~$k$, $l$,~$m$; the upper sign applies to the even permutations, the lower
ones to the odd); and finally of the vector~$F_{ik} \phi^{k}$.
Just as in three-dimensional geometry the most important
theorem of congruence is that a vector-pair $\va$,~$\vb$ is fully characterised
in respect to congruence by means of the invariants $\va^{2}$, $\va\vb$,
$\vb^{2}$, so it may be shown in four-dimensional geometry that the invariants
quoted determine fully in respect to congruence the figure
composed of a vector~$\phi$ and a linear tensor of the second order~$F$.
Every invariant, in particular the Hamiltonian Function~$L$, must
therefore be expressible algebraically in terms of the above four
quantities. Mie's Theory thus resolves the problem of matter into
a determination of this expression. Maxwell's Theory of the æther
which, of course, precludes the possibility of electrons, is contained
in it as the special case $L = L^{0}$. If we also express $W$ and the
components of~$\sfS$ in terms of four-dimensional quantities, we see
% [** TN: Ordinal]
that they are the negative ($0$th)~row in the scheme
\[
T_{i}^{k} = F_{ir} H^{kr} + \phi_{i} s^{k} - L · \delta_{i}^{k}\Add{.}
\Tag{(64)}
\]
The $T_{i}^{k}$'s are thus the mixed components of the energy-momentum-tensor,
which, according to our calculations, fulfil the theorem of
conservation~\Eq{(57)} for $i = 0$ and hence also for $i = 1, 2, 3$. In the
next chapter we shall add the proof that its \Typo{convariant}{co-variant} components
satisfy the condition of symmetry $T_{ki} = T_{ik}$.
The laws for the field may be summarised in a very simple
principle of variation, Hamilton's Principle. For this we regard
only the potential with components~$\phi_{i}$ as an independent phase-quantity,
and \emph{define} the field by the equation
\[
F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}.
\]
Hamilton's invariant function~$L$ which depends on the potential
and the field enters into these laws. We \emph{define} the current-vector~$\Typo{3}{s}$
and the skew-symmetrical tensor~$H$ by means of~\Eq{(63)}. If in an
arbitrary linear co-ordinate system
\[
d\omega = \sqrt{g}\, dx_{0}\, dx_{1}\, dx_{2}\, dx_{3}
\]
is the four-dimensional ``volume-element'' of the world ($-g$~is the
\index{Volume-element}%
determinant of the metrical groundform) then the integral $\Dint L\, d\omega$
taken over any region of the world is an invariant. It is called the
\Emph{Action} contained in the region in question. Hamilton's Principle
\index{Action@\emph{Action}!(cf.\ Hamilton's Function)}%
states that the change in the total \emph{Action} for each infinitesimal
\PageSep{211}
variation of the state of the field, which vanishes outside a finite
region, is zero, that is,
\[
\delta \int L\, d\omega = \int \delta L\, d\omega = 0\Add{.}
\Tag{(65)}
\]
This integral is to be taken over the whole world or, what comes to
the same thing, over a finite region beyond which the variation of
the phase vanishes. This variation is represented by the infinitesimal
increments~$\delta \phi_{i}$ of the potential-components and the accompanying
infinitesimal change of the field
\[
\delta F_{ik}
= \frac{\dd (\delta \phi_{i})}{\dd x_{k}}
- \frac{\dd (\delta \phi_{k})}{\dd x_{i}}
\]
in which $\delta \phi_{i}$~are space-time functions that only differ from zero
within a finite region. If we insert for~$\delta L$ the expression~\Eq{(63)}, we
get
\[
\delta L = s^{i}\, \delta \phi_{i}
+ H^{ik}\, \frac{\dd(\delta \phi_{i})}{\dd x_{k}}.
\]
By the principle of partial integration (\textit{vide} \Pageref{111}) we get
\[
\int H^{ik}\, \frac{\dd(\delta \phi_{i})}{\dd x_{k}}\, d\omega
= -\int \frac{\dd H^{ik}}{\dd x_{k}}\, \delta \phi_{i}\, d\omega,
\]
and, accordingly,
\[
\delta \int L\, d\omega
= \int \left\{s^{i} - \frac{\dd H^{ik}}{\dd x_{k}}\right\} \delta \phi_{i}\, d\omega\Add{.}
\Tag{(66)}
\]
Whereas \Eq{(3)}~is given by definition, we see that Hamilton's Principle
furnishes the field-equations~\Eq{(4)}. In point of fact, if, for instance,
\[
s - \frac{\dd H^{ik}}{\dd x_{k}} \neq 0
\]
but is $> 0$ at a certain point, then we could mark off a small region
encircling this point, such that, for it, this difference is positive
throughout. If we then choose a non-negative function for~$\delta \phi_{1}$ that
vanishes outside the region marked off, and if $\delta \phi_{2} = \delta \phi_{3} = \delta \phi_{4} = 0$,
we arrive at a contradiction to equation~\Eq{(65)}---\Eq{(1)} and~\Eq{(2)} follow
from \Eq{(3)}~and~\Eq{(4)}.
We find, then, \Emph{that Mie's Electrodynamics exists in a compressed
\index{Action@\emph{Action}!principle of}%
form in Hamilton's Principle~\Eq{(65)}}---analogously to the
manner in which the development of mechanics attains its zenith
in the principle of action. Whereas in mechanics, however, a
definite function~$L$ of action corresponds to every given mechanical
system and has to be \Erratum{deducted}{deduced} from the constitution of the system,
we are here concerned with a single system, the world. This is
where the real problem of matter takes its beginning: we have to
determine the ``function of action,'' the world-function~$L$, belonging to
\PageSep{212}
the world. For the present it leaves us in perplexity. If we choose
an arbitrary~$L$, we get a ``possible'' world governed by this function
of action, which will be perfectly intelligible to us---more so than
the actual world---provided that our mathematical analysis does not
fail us. We are, of course, then concerned in discovering the only
existing world, the \Emph{real} world for us. Judging from what we know
of physical laws, we may expect the~$L$ which belongs to it to be
distinguished by having simple mathematical properties. Physics,
this time as a physics of fields, is again pursuing the object of reducing
the totality of natural phenomena to \Emph{a single physical law}: it
was believed that this goal was almost within reach once before
when Newton's \Typo{Principia}{\Title{Principia}}, founded on the physics of mechanical
point-masses was celebrating its triumphs. But the treasures of
knowledge are not like ripe fruits that may be plucked from a tree.
For the present we do not yet know whether the phase-quantities
on which Mie's Theory is founded will suffice to describe matter or
whether matter is purely ``electrical'' in nature. Above all, the
ominous clouds of those phenomena that we are with varying
success seeking to explain by means of the quantum of action, are
throwing their shadows over the sphere of physical knowledge,
threatening no one knows what new revolution.
Let us try the following hypothesis for~$L$:
\[
L = \tfrac{1}{2} |F|^{2} + w(\sqrt{-\phi_{i} \phi^{i}})
\Tag{(67)}
\]
($w$~is the symbol for a function of one variable); it suggests itself
as being the simplest of those that go beyond Maxwell's Theory.
We have no grounds for assuming that the world-function has
\index{World ($=$ space-time)!-law}%
actually this form. We shall confine ourselves to a consideration
of statical solutions, for which we have
\begin{align*}
\sfB &= \sfH = \Typo{0}{\0}, && \sfs = \sff = \Typo{0}{\0}\Add{,} \\
\sfE &= \grad \phi, && \div \sfD = \rho\Add{,} \\
\sfD &= \sfE, && \rho = -w'(\phi)
\end{align*}
(the accent denoting the derivative). In comparison with the
ordinary electrostatics of the æther we have here the new circumstance
that the density~$\rho$ is a universal function of the potential, the
electrical pressure~$\phi$. We get for Poisson's equation
\[
\Delta \phi + w'(\phi) = 0\Add{.}
\Tag{(68)}
\]
If $w(\phi)$~is not an even function of~$\phi$, this equation no longer holds
after the transition from $\phi$ to~$-\phi$; this would account for \Emph{the
difference between the natures of positive and negative
\index{Electricity, positive and negative}%
electricity}. Yet it certainly leads to a remarkable difficulty in the
case of non-statical fields. If charges having opposite signs are to
occur in the latter, the root in~\Eq{(67)} must have different signs at
\PageSep{213}
\index{Reality}%
different points of the field. Hence there must be points in the
field, for which $\phi_{i} \phi^{i}$~vanishes. In the neighbourhood of such a
point $\phi_{i} \phi^{i}$~must be able to assume positive and negative values
(this does not follow in the statical case, as the minimum of the
function~$\phi_{0}^{2}$ for~$\phi_{0}$ is zero). The solutions of our field-equations
must, therefore, become imaginary at regular distances apart. It
would be difficult to interpret a degeneration of the field into
separate portions in this way, each portion containing only charges
of one sign, and separated from one another by regions in which
the field becomes imaginary.
A solution (vanishing at infinity) of equation~\Eq{(68)} represents
a possible state of electrical equilibrium, or a possible corpuscle
capable of existing individually in the world that we now proceed
to construct. The equilibrium can be stable, only if the solution
is radially symmetrical. In this case, if $r$~denotes the radius
vector, the equation becomes
\[
\frac{1}{r^{2}}\, \frac{d}{dr} \left(r^{2}\, \frac{d\phi}{dr}\right)
+ w'(\phi) = 0\Add{.}
\Tag{(69)}
\]
If \Eq{(69)}~is to have a regular solution
\[
-\phi = \frac{e_{0}}{r} + \frac{e_{1}}{r^{2}} + \dots
\Tag{(70)}
\]
at $r = \infty$, we find by substituting this power series for the first term
of the equation that the series for~$w'(\phi)$ begins with the power~$r^{-4}$
or one with a still higher negative index, and hence that $w(x)$~must
be a zero of at least the fifth order for $x = 0$. On this assumption
the equations must have a single infinity of regular solutions at
$r = 0$ and also a \Erratum{singular}{single} infinity of regular solutions at $r = \infty$.
We may (in the ``general'' case) expect these two \Emph{one-dimensional}
families of solutions (included in the two-dimensional complete
family of all the solutions) to have a finite or, at any rate, a discrete
number of solutions. These would represent the various possible
corpuscles. (Electrons and elements of the atomic nucleus?) \emph{One}
electron or \emph{one} atomic nucleus does not, of course, exist alone in
\index{Electron}%
the world; but the distances between them are so great in comparison
with their own size that they do not bring about an
appreciable modification of the structure of the field within the
i interior of an individual electron or atomic nucleus. If $\phi$~is a
solution of~\Eq{(69)} that represents such a corpuscle in~\Eq{(70)} then its
total charge
\[
= 4\pi \int_{0}^{\infty} w'(\phi) r^{2}\, dr
= -4\pi · r^{2}\, \frac{d\phi}{dr}\bigg|_{r = \infty}
= 4\pi c_{0},
\]
\PageSep{214}
but its mass is calculated as the integral of the energy-density~$W$
\index{Density!electricity@{(of electricity and matter)}}%
that is given by~\Eq{(62)}:
\begin{align*}
\text{Mass}
&= 4\pi \int_{0}^{\infty} \bigl\{\tfrac{1}{2}(\grad \phi)^{2}
+ w(\phi) - \phi w'(\phi)\bigr\}r^{2}\, dr \\
&= 4\pi \int_{0}^{\infty} \bigl\{w(\phi)
- \tfrac{1}{2} \phi w'(\phi)\bigr\}r^{2}\, dr.
\end{align*}
\emph{These physical laws, then, enable us to calculate the mass and
charge of the electrons, and the atomic weights and atomic charges
\index{Charge!(\emph{as a substance})}%
of the individual existing elements} whereas, hitherto, we have always
accepted these ultimate constituents of matter as things given with
their numerical properties. All this, of course, is merely a suggested
\emph{plan of action} as long as the world-function~$L$ is not known. The
special hypothesis~\Eq{(67)} from which we just now started was
assumed only to show what a deep and thorough knowledge of
matter and its constituents as based on laws would be exposed to
our gaze if we could but discover the action-function. For the
rest, the discussion of such arbitrarily chosen hypotheses cannot
lead to any proper progress; new physical knowledge and principles
will be required to show us the right way to determine the
Hamiltonian Function.
To make clear, \textit{ex contrario}, the nature of pure physics of fields,
which was made feasible by Mie for the realm of electrodynamics
as far as its general character furnishes hypotheses, the principle
of action~\Eq{(65)} holding in it will be contrasted with that by which
the theory of Maxwell and Lorentz is governed; the latter theory
recognises, besides the electromagnetic field, a substance moving in
\index{Substance}%
it. This substance is a three-dimensional continuum; hence its
parts may be referred in a continuous manner to the system of
values of three co-ordinates $\alpha$,~$\beta$,~$\gamma$. Let us imagine the substance
divided up into infinitesimal elements. Every element of substance
has then a definite invariable positive mass~$dm$ and an invariable
electrical charge~$de$. As an expression of its history there corresponds
\index{Electrical!charge!substance@{(as a substance)}}%
to it then a world-line with a definite direction of traverse
or, in better words, an infinitely thin ``world-filament''. If we again
divide this up into small portions, and if
\[
ds = \sqrt{-g_{ik}\, dx_{i}\, dx_{k}}
\]
is the proper-time length of such a portion, then we may introduce
the space-time function~$\mu_{0}$ of the statical mass-density by means of
the invariant equation
\[
dm\, ds = \mu_{0}\, d\omega\Add{.}
\Tag{(71)}
\]
\PageSep{215}
We shall call the integral
%[** TN: Original symbol is bold X with a horizontal line through the middle]
\[
\int_{\rX} \mu_{0}\, d\omega
= \int dm\, ds
= \int dm \int \sqrt{-g_{ik}\, dx_{i}\, dx_{k}}
\]
taken over a region~$\rX$ of the world the \Emph{substance-action of mass}.
\index{Substance-action of electricity and gravitation}%
In the last integral the inside integration refers to that part of the
world-line of any arbitrary element of substance of mass~$dm$, which
belongs to the region~$\rX$, the outer integral signifies summation
taken for all elements of the substance. In purely mathematical
language this transition from substance-proper-time integrals to
space-time integrals occurs as follows. We first introduce the
substance-density~$\Typo{v}{\nu}$ of the mass thus:
\[
dm = \nu\, d\alpha\, d\beta\, d\gamma
\]
($\nu$~behaves as a scalar-density for arbitrary transformations of the
substance co-ordinates $\alpha$,~$\beta$,~$\gamma$). On each world-line of a substance-point
$\alpha$,~$\beta$,~$\gamma$ we reckon the proper-time~$s$ from a definite initial
point (which must, of course, vary \Emph{continuously} from substance-point
to substance-point). The co-ordinates~$x_{i}$ of the world-point
at which the substance-point $\alpha$,~$\beta$,~$\gamma$\Typo{,}{} happens to be at the moment~$s$
of its motion (after the proper-time~$s$ has elapsed), are then
continuous functions of $\alpha$,~$\beta$,~$\gamma$,~$s$, whose functional determinant
\[
\frac{\dd (x_{0}\Com x_{1}\Com x_{2}\Com x_{3})}
{\dd (\alpha\Com \beta\Com \gamma\Com s)}
\]
we shall suppose to have the absolute value~$\Delta$. The equation~\Eq{(71)}
then states that
\[
\mu_{0} \sqrt{g} = \frac{\nu}{\Delta}.
\]
In an analogous manner we may account for the statical density~$\rho_{0}$
of the electrical charge. We shall set down
\[
%[** TN: Small parentheses in the original]
\int \left(de \int \phi_{i}\, dx_{i}\right)
\]
as \Emph{substance-action of electricity}; in it the outer integration
is again taken over all the substance-elements, but the inner one in
each case over that part of the world-line of a substance-element
carrying the charge~$de$ whose path lies in the interior of the world-region~$\rX$.
We may therefore also write
\[
\int de\, ds · \phi u
= \int \rho_{0} u^{i} \phi_{i}\, d\omega
= \int s^{i} \phi_{i}\, d\omega
\]
{\Loosen if $u^{i} = \dfrac{dx_{i}}{ds}$ are the components of the world-direction, and $s^{i} = \rho_{0} u^{i}$
are the components of the $4$-current (a pure convection current).
\PageSep{216}
\index{Field action of electricity}%
Finally, in addition to the substance-action there is also a \Emph{field-action
of electricity}, for which Maxwell's Theory makes the simple
convention}
\[
\tfrac{1}{4} \int F_{ik} F^{ik}\, d\omega\qquad
\left(F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}\right).
\]
Hamilton's Principle, which gives a condensed statement of the
\index{Hamilton's!principle!special@{(in the special theory of relativity)}}%
Maxwell-Lorentz Laws, may then be expressed thus:
\emph{The total action, that is, the sum of the field-action and substance-action
of electricity plus the substance-action of the mass for any
arbitrary variation (vanishing for points beyond a finite region) of
the field-phase (of the~$\phi_{i}$'s) and for a similarly conditioned space-time
displacement of the world-lines described by the individual substance-points
undergoes no change.}
This principle clearly gives us the equations
\[
\frac{\dd F^{ik}}{\dd x_{k}} = s^{i} = \rho_{0} u^{i},
\]
if we vary the~$\phi_{i}$'s. If, however, we keep the $\phi_{i}$'s constant, and
perform variations on the world-lines of the substance-points, we
get, by interchanging differentiation and variation (as in §\,17 in
determining the shortest lines), and then integrating partially:
\begin{align*}
\int \phi_{i}\, dx_{i}
&= \int (\delta \phi_{i}\, dx_{i} + \phi_{i}\, d\delta \phi_{i})
= \int (\delta \phi_{i}\, dx_{i} - \delta x_{i}\, d\phi_{i}) \\
&= \int \left(\frac{\dd \phi_{i}}{\dd x_{k}}
- \frac{\dd \phi_{k}}{\dd x_{i}}\right)
\delta x_{k} · dx_{i}\Add{.}
\end{align*}
In this the $\delta x_{i}$'s are the components of the infinitesimal displacement,
which the individual points of the world-line undergo.
Accordingly, we get
\[
%[** TN: Small parentheses in the original]
\delta \int \left(de \int \phi_{i}\, dx_{i}\right)
= \int de\, ds · F_{ik} u^{i}\, \delta x_{k}
= \int \rho_{0} F_{ik} u^{i}\, \delta x_{k} · d\omega.
\]
If we likewise perform variation on the substance-action of the
mass (this has already been done in §\,17 for a more general case,
in which the~$g_{ik}$'s were variable), we arrive at the mechanical
equations which are added to the field-equations in Maxwell's
Theory; namely
\[
\mu_{0}\, \frac{du_{i}}{ds} = p_{i}\qquad
p_{i} = \rho_{0} F_{ik} u^{k} = F_{ik} s^{k}.
\]
This completes the cycle of laws which were mentioned on \Pageref{199}.
This theory does not, of course, explain the existence of the
electron, since the cohesive forces are lacking in it.
A striking feature of the principle of action just formulated is
that a field-action does not associate itself with the substance-action
\PageSep{217}
of the mass, as happens in the case of electricity. This gap will
be filled in the next chapter, in which it will be shown that the
\Emph{gravitational field} is what corresponds to mass in the same way
as the electromagnetic field corresponds to the electrical charge.
\medskip
The great advance in our knowledge described in this chapter
consists in recognising that the scene of action of reality is not a
three-dimensional Euclidean space but rather a \Emph{four-dimensional
world, in which space and time are linked together indissolubly}.
However deep the chasm may be that separates the
intuitive nature of space from that of time in our experience,
nothing of this qualitative difference enters into the objective world
which physics endeavours to crystallise out of direct experience.
It is a four-dimensional continuum, which is neither ``time'' nor
``space''. Only the consciousness that passes on in one portion
of this world experiences the detached piece which comes to meet
it and passes behind it, as \Emph{history}, that is, as a process that is
going forward in time and takes place in space.
This four-dimensional space is \Emph{metrical} like Euclidean space,
but the quadratic form which determines its metrical structure is
not definitely positive, but has \Emph{one} negative dimension. This circumstance
is certainly of no mathematical importance, but has a
deep significance for reality and the relationship of its action. It
was necessary to grasp the idea of the metrical four-dimensional
world, which is so simple from the mathematical point of view, not
only in isolated abstraction but also to pursue the weightiest inferences
that can be drawn from it towards setting up the view of
physical phenomena, so that we might arrive at a proper understanding
of its content and the range of its influence: that was
what we aimed to do in a short account. It is remarkable that
the three-dimensional geometry of the statical world that was put
into a complete axiomatic system by Euclid has such a translucent
character, whereas we have been able to assume command
over the four-dimensional geometry only after a prolonged struggle
and by referring to an extensive set of physical phenomena and
empirical data. Only now the theory of relativity has succeeded
in enabling our knowledge of physical nature to get a full grasp of
the fact of motion, of change in the world.
\PageSep{218}
\Chapter{IV}
{The General Theory of Relativity}
\Section[The Relativity of Motion, Metrical Fields, Gravitation]
{27.}{The Relativity of Motion, Metrical Fields, Gravitation\protect\footnotemark}
\footnotetext{\textit{Vide} \FNote{1}.}
\First{However} successfully the Principle of Relativity of Einstein
worked out in the preceding chapter marshals the physical
laws which are derived from experience and which define
the relationship of action in the world, we cannot express ourselves
as satisfied from the point of view of the theory of knowledge.
Let us again revert to the beginning of the foregoing chapter.
There we were introduced to a ``kinematical'' principle of relativity;
$x_{1}$,~$x_{2}$,~$x_{3}$,~$t$ were the space-time co-ordinates of a world-point
referred to a definite permanent Cartesian co-ordinate system in
space; $x_{1}'$,~$x_{2}'$,~$x_{3}'$,~$t'$ were the co-ordinates of the same point relative
to a second such system, that may be moving arbitrarily with respect
to the first; they are connected by the transformation formulæ~\textEq{(II)},
\Pageref{152}. It was made quite clear that two series of physical
states or phases cannot be distinguished from one another in an
objective manner, if the phase-quantities of the one are represented
by the same mathematical functions of $x_{1}'$,~$x_{2}'$,~$x_{3}'$,~$t'$ as those that
describe the first series in terms of the arguments $x_{1}$,~$x_{2}$,~$x_{3}$,~$t$.
Hence the physical laws must have exactly the same form in the
one system of independent space-time arguments as in the other.
It must certainly be admitted that the facts of dynamics are
apparently in direct contradiction to Einstein's postulate, and it is
just these facts that, since the time of Newton, have forced us to
attribute an absolute meaning, not to translation, but to rotation.
Yet our minds have never succeeded in accepting unreservedly
this torso thrust on them by reality (in spite of all the attempts
that have been made by philosophers to justify it, as, for example,
Kant's \Title{Metaphysische Anfangsgründe der Naturwissenschaften}),
and the problem of centrifugal force has always been felt to be an
unsolved enigma (\textit{vide} \FNote{2}).
Where do the centrifugal and other inertial forces take their
origin? Newton's answer was: in absolute space. The answer
\PageSep{219}
given by the special theory of relativity does not differ essentially
from that of Newton. It recognises as the source of these forces
the metrical structure of the world and considers this structure as
a formal property of the world. But that which expresses itself as
force must itself be real. We can, however, recognise the metrical
structure as something real, if it is itself capable of undergoing
changes and reacts in response to matter. Hence our only way
out of the dilemma---and this way, too, was opened up by
Einstein---is to apply Riemann's ideas, as set forth in Chapter~II,
to the four-dimensional Einstein-Minkowski world which was
treated in Chapter~III instead of to three-dimensional Euclidean
space. In doing this we shall not for the present make use of the
most general conception of the metrical manifold, but shall retain
Riemann's view. According to this, we must assume the world-points
to form a four-dimensional manifold, on which a measure-determination
is impressed by a non-degenerate quadratic differential
form~$Q$ having one positive and three negative dimensions.\footnote
{We have made a change in the notation, as compared with that of the
preceding chapter, by placing reversed signs before the metrical groundform.
The former convention was more convenient for representing the splitting up
of the world into space and time, the present one is found more expedient in
the general theory.}
In
any co-ordinate system~$x_{i}$ ($i = 0, 1, 2, 3$), in Riemann's sense, let
\[
Q = \sum_{i\Com k} g_{ik}\, dx_{i}\, dx_{k}\Add{.}
\Tag{(1)}
\]
Physical laws will then be expressed by tensor relations that are
invariant for arbitrary continuous transformations of the arguments~$x_{i}$.
In them the co-efficients~$g_{ik}$ of the quadratic differential form~\Eq{(1)}
will occur in conjunction with the other physical phase-quantities.
\index{Phase}%
Hence we shall satisfy the postulate of relativity
enunciated above, without violating the facts of experience, \Emph{if we
regard the~$g_{ik}$'s}\Typo{,}{} in exactly the same way as we regarded the components~$\phi_{i}$
of the electromagnetic potential (which are formed by
the co-efficients of an invariant \Emph{linear} differential form $\sum \phi_{i}\, dx_{i}$), \Emph{as
physical phase-quantities, to which there corresponds something
real, namely, the ``metrical field''}. Under these circumstances
invariance exists not only with respect to the transformations
mentioned~\textEq{(II)}, which have a fully arbitrary (non-linear)
character only for the time-co-ordinate, but for any transformations
whatsoever. This special distinction conferred on the time-co-ordinate
by~\textEq{(II)}, is, indeed, incompatible with the knowledge gained
\PageSep{220}
from Einstein's Principle of Relativity. By allowing any arbitrary
transformations in place of~\textEq{(II)}, that is, also such as are non-linear
with respect to the space-co-ordinates, we affirm that Cartesian
co-ordinate systems are in no wise more favoured than any
``curvilinear'' co-ordinate system. \Emph{This seals the doom of the
idea that a geometry may exist independently of physics} in the
traditional sense, and it is just because we had not emancipated ourselves
from the dogma that such a geometry existed that we arrived
by logical considerations at the relativity principle~\textEq{(II)}, and not at
once at the principle of invariance for arbitrary transformations of
the four world-co-ordinates. Actually, however, spatial measurement
is based on a physical event: the reaction of light-rays and
rigid measuring rods on our whole physical world. We have
already encountered this view in §\,21, but we may, above all, take
up the thread from our discussion in §\,12, for we have, indeed, here
arrived at Riemann's ``dynamical'' view as a necessary consequence
of the relativity of all motion. The behaviour of light-rays and
measuring rods, besides being determined by their own natures, is
also conditioned by the ``metrical field,'' just as the behaviour of an
electric charge depends not only on it, itself, but also on the electric
field. Again, just as the electric field, for its part, depends on the
charges and is instrumental in producing a mechanical interaction
between the charges, so we must assume here that \Emph{the metrical
field} (or, in mathematical language, the tensor with components~$g_{ik}$)
\Emph{is related to the material content filling the world}.
We again call attention to the principle of action set forth at the
conclusion of the preceding paragraph; in both of the parts which
refer to substance, the metrical field takes up the same position
towards mass as the electrical field does towards the electric charge.
The assumption, which was made in the preceding chapter, concerning
the metrical structure of the world (corresponding to that
of Euclidean geometry in three-dimensional space), namely, that
there are specially favoured co-ordinate systems, ``linear'' ones, in
which the metrical groundform has constant co-efficients, can no
longer be maintained in the face of this view.
A simple illustration will suffice to show how geometrical
conditions are involved when motion takes place. Let us set a
plane disc spinning uniformly. I affirm that if we consider
Euclidean geometry valid for the reference-space relative to which
we speak of uniform rotation, then it is no longer valid for the
rotating disc itself, if the latter be measured by means of measuring
rods moving with it. For let us consider a circle on the disc
described with its centre at the centre of rotation. Its radius
\PageSep{221}
remains the same no matter whether the measuring rods with
which I measure it are at rest or not, since its direction of motion
is perpendicular to the measuring rod when in the position required
for measuring the radius, that is, along its length. On the other
hand, I get a value greater for the circumference of the circle than
that obtained when the disc is at rest when I apply the measuring
rods, owing to the Lorentz-Fitzgerald contraction which the latter
undergoes. The Euclidean theorem which states that the circumference
of the circle $= 2\pi$~times the radius thus no longer holds
on the disc when it rotates.
The falling over of glasses in a dining-car that is passing
round a sharp curve and the bursting of a fly-wheel in rapid rotation
are not, according to the view just expressed, effects of ``an absolute
rotation'' as Newton would state but whose existence we deny;
they are effects of the ``metrical field'' or rather of the affine
relationship associated with it. Galilei's principle of inertia shows
that there is a sort of ``forcible guidance'' which compels a body
that is projected with a definite velocity to move in a definite way
which can be altered only by external forces. This ``guiding field,''
which is physically real, was called ``affine relationship'' above.
When a body is diverted by external forces the guidance by forces
such as centrifugal reaction asserts itself. In so far as the state of
the guiding field does not persist, and the present one has emerged
from the past ones under the influence of the masses existing in
the world, namely, the fixed stars, the phenomena cited above are
partly an effect of the fixed stars, \Emph{relative to which} the rotation
takes place.\footnote
{We say ``partly'' because the distribution of matter in the world does
not define the ``guiding field'' uniquely, for both are \Emph{at one moment} independent
of one another and accidental (analogously to charge and electric
field). Physical laws tell us merely how, when such an initial state is given,
all other states (past and future) necessarily arise from them. At least, this is
how we must judge, if we are to maintain the standpoint of pure physics of
fields. The statement that the world in the form we perceive it taken as a
whole is stationary (i.e.\ at rest) can be interpreted, if it is to have a meaning at
all, as signifying that it is in statistical equilibrium. Cf.~§\,34.}
Following Einstein by starting from the special theory of
relativity described in the preceding chapter, we may arrive at the
general theory of relativity in two successive stages.
I\@. In conformity with the principle of continuity we take the
same step in the four-dimensional world that, in Chapter~II,
brought us from Euclidean geometry to Riemann's geometry. This
causes a quadratic differential form~\Eq{(1)} to appear. There is no
difficulty in adapting the physical laws to this generalisation. It is
\PageSep{222}
expedient to represent the magnitude quantities by tensor-densities
instead of by tensors as in Chapter~III; we can do this by multiplying
throughout by~$\sqrt{g}$ (in which $g$~is the negative determinant of
the~$g_{ik}$'s). Thus, in particular, the mass- and charge-densities $\mu$~and~$\rho$,
instead of being given by formula~\Eq{(71)} of §\,26, will be
given by
\[
dm\, ds = \mu\, dx,\qquad
de\, ds = \rho\, dx\qquad
(dx = dx_{0}\, dx_{1}\, dx_{2}\, dx_{3}).
\]
The proper time~$ds$ along the world-line is determined from
\[
ds^{2} = g_{ik}\, dx_{i}\, dx_{k}\Add{.}
\]
Maxwell's equations will be
\index{Maxwell's!theory!(in the light of the general theory of relativity)}%
\[
F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}},
\qquad \frac{\dd \Typo{\vF}{\vF^{ik}}}{\dd x_{k}} = \vs^{i},
\]
in which the~$\phi_{i}$'s are the co-efficients of an invariant linear
differential form~$\phi_{i}\, dx_{i}$, and $\vF^{ik}$~denotes $\sqrt{g} · F^{ik}$ according to our
convention above. In Lorentz's Theory we set
\[
\vs^{i} = \rho u^{i}\qquad
\left(u^{i} = \frac{dx_{i}}{ds}\right).
\]
The mechanical force per unit of volume (a co-variant vector-density
\index{Centrifugal forces}%
\index{Force!(ponderomotive, of gravitational field)}%
\index{Mechanics!fundamental law of!general@{(in general theory of relativity)}}%
\index{Ponderomotive force!of the gravitational field}%
in the four-dimensional world) is given by:\footnote
{The sign is reversed on account of the reversal of sign in the metrical
groundform.}
\[
\vp_{i} = -F_{ik} \vs^{k}\Add{,}
\Tag{(2)}
\]
and the mechanical equations are in general
\[
\mu \left(\frac{du_{i}}{ds} - \Chr{i\beta}{\alpha} u_{\alpha} u^{\beta}\right)
= \vp_{i}
\Tag{(3)}
\]
with the condition that $\vp_{i} u^{i}$ always $= 0$. We may put them into
the same form as we found for them earlier by introducing, in
addition to the~$\vp_{i}$'s, the quantities
\[
\Chr{i\beta}{\alpha} · \mu u_{\alpha} u^{\beta}
= \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} · \mu u^{\alpha} u^{\beta}
\Tag{(4)}
\]
(cf.\ §\,17, equation~\Eq{(64)}) as the density components~$\bar{\vp}_{i}$ of a
``pseudo-force'' (force of reaction of the guiding field). The
equations then become
\[
\mu\, \frac{du_{i}}{ds} = \vp_{i} + \bar{\vp}_{i}.
\]
The simplest examples of such ``pseudo-forces'' are centrifugal
forces and Coriolis forces. If we compare formula~\Eq{(4)} for the
\index{Coriolis forces}%
``pseudo-force'' arising from the metrical field with that for the
mechanical force of the electromagnetic field, we find them fully
\PageSep{223}
\index{Centrifugal forces}%
\index{Ponderomotive force!of the gravitational field}%
analogous. For just as the vector-density with the contra-variant
components~$\vs^{i}$ characterises electricity so, as we shall presently
see, moving matter is described by the tensor-density which has
the components $\vT_{i}^{k} = \mu u_{i} u^{k}$. The quantities
\[
\Gamma_{i\beta}^{\alpha} = \Chr{i\beta}{\alpha}
\]
correspond as components of the metrical field to the components~$F_{ik}$
of the electric field. Just as the field-components~$F$
are derived by differentiation from the electromagnetic potential~$\phi_{i}$,
so also the~$\Gamma$'s from the~$g_{ik}$'s; these thus constitute the potential of
the metrical field. The force-density is the product of the electric
field and electricity on the one hand, and of the metrical field and
matter on the other, thus
\[
\vp_{i} = -F_{ik} \vs^{k},\qquad
\bar{\vp}_{i} = \Gamma_{i\beta}^{\alpha} \vT_{\alpha}^{\beta}.
\]
If we abandon the idea of a substance existing independently of
physical states, we get instead the general energy-momentum-density~$\vT_{i}^{k}$
which is determined by the state of the field. According
to the special theory of relativity it satisfies the Law of Conservation
\[
\frac{\dd \vT_{i}^{k}}{\dd x_{k}} = 0\Add{.}
\]
This equation is now to be replaced, in accordance with formula~\Eq{(37)}
§\,14, by the general invariant
\[
\frac{\dd \vT_{i}^{k}}{\dd x_{k}} - \Gamma_{i\beta}^{\alpha} \vT_{\alpha}^{\beta} = 0\Add{.}
\Tag{(5)}
\]
If the left-hand side consisted only of the first member, $\vT$~would
now again satisfy the laws of conservation. But we have, in this
case, a second term. The ``real'' total force
\[
\vp_{i} = -\frac{\dd \vT_{i}^{k}}{\dd x_{k}}
\]
does not vanish but must be counterbalanced by the ``pseudo-force''
which has its origin in the metrical field, namely
\[
\bar{\vp}_{i}
= \Gamma_{i\beta}^{\alpha} \vT_{\alpha}^{\beta}
= \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta}\Add{.}
\Tag{(6)}
\]
These formulæ were found to be expedient in the special theory
of relativity when we used curvilinear co-ordinate systems, or such
as move curvilinearly or with acceleration. To make clear the
simple meaning of these considerations we shall use this method
to determine the \Emph{centrifugal force} that asserts itself in a rotating
system of reference. If we use a normal co-ordinate system
\PageSep{224}
for the world, namely, $t$,~$x_{1}$,~$x_{2}$,~$x_{3}$, but introduce $r$,~$z$,~$\theta$, in place
of the Cartesian space co-ordinates, we get
\[
ds^{2} = dt^{2} - (dz^{2} + dr^{2} + r^{2}\, d\theta^{2}).
\]
Using $\omega$ to denote a constant angular velocity, we make the
substitution
\[
\theta = \theta' + \omega t',\qquad
t = t'
\]
and, after the substitution, drop the accents. We then get
\[
ds^{2} = dt^{2}(1 - r^{2} \omega^{2}) - 2r^{2} \omega\, d\theta\, dt - (dz^{2} + dr^{2} + r^{2}\, d\theta^{2}).
\]
If we now put
\[
t = x_{0},\qquad
\theta = x_{1},\qquad
z = x_{2},\qquad
r = x_{3},
\]
we get for a point-mass which is at rest in the system of reference
now used
\[
u^{1} = u^{2} = u^{3} = 0;\quad
\text{and hence } (u^{0})^{2} (1 - r^{2} \omega^{2}) = 1.
\]
The components of the centrifugal force satisfy formula~\Eq{(4)}
\[
\bar{\vp}_{i}
= \tfrac{1}{2}\, \frac{\dd g_{00}}{\dd x_{i}} · \mu(u^{0})^{2}
\]
and since the derivatives with respect to $x_{0}$,~$x_{1}$,~$x_{2}$ of~$g_{00}$, which is
equal to $1 - r^{2} \omega^{2}$, vanish and since
\[
\frac{\dd g_{00}}{\dd x_{3}} = \frac{\dd g_{00}}{\dd r} = -2r \omega^{2}
\]
then, if we return to the usual units, in which the velocity of light
is \Emph{not} unity, and if we use contra-variant components instead of
co-variant ones, and instead of the indices $0, 1, 2, 3$ the more
indicative ones $t$,~$\theta$,~$z$,~$r$, we obtain
\[
\bar{\vp}^{t} = \bar{\vp}^{\theta} = \bar{\vp}^{z} = 0,\qquad
\bar{\vp}^{r} = \frac{\mu r \omega^{2}}{1 - \left(\dfrac{r\omega}{c}\right)^{2}}\Add{.}
\Tag{(7)}
\]
Two closely related circumstances characterise the ``pseudo-forces''
of the metrical field. \emph{Firstly}, the acceleration which they
impart to a point-mass situated at a definite space-time point (or,
more exactly, one passing through this point with a definite velocity)
is independent of its mass, i.e.\ the force itself is proportional to the
inertial mass of the point-mass at which it acts. \emph{Secondly}, if we
use an appropriate co-ordinate system, namely, a geodetic one, at
a definite space-time point, these forces vanish (cf.\ §\,14). If the
special theory of relativity is to be maintained, this vanishing can
be effected simultaneously for all space-time points by the introduction
of a linear co-ordinate system, but in the general case it is
possible to make the whole $40$~components $\Gamma_{i\beta}^{\alpha}$ of the affine relationship
\PageSep{225}
vanish at least for each individual point by choosing an
appropriate co-ordinate system at this point.\footnote
{Hence we see that it is in the nature of the metrical field that it cannot be
described by a field-tensor~$\Gamma$ which is invariant with respect to arbitrary transformations.}
Now the two related circumstances just mentioned are true, as
\index{Eotvos@{Eötvös' experiment}}%
\index{Inertial force!mass}%
we know, of the \Emph{force of gravitation}. The fact that a given
gravitational field imparts the same acceleration to every mass that
\index{Gravitational!mass}%
\index{Mass!inertial and gravitational}%
is brought into the field constitutes the real essence of the problem
of gravitation. In the electrostatic field a slightly charged particle
is acted on by the force~$e · \vE$, the electric charge~$e$ depending only
on the particle, and~$\vE$, the electric intensity of field, depending
only on the field. If no other forces are acting, this force imparts
to the particle whose inertial mass is~$m$ an acceleration which is
given by the fundamental equation of mechanics $m\vb = e\vE$. There
is something fully analogous to this in the gravitational field. The
force that acts on the particle is equal to~$g\vG$, in which~$g$, the
``gravitational charge,'' depends only on the particle, whereas $\vG$~depends
only on the field: the acceleration is determined here again
by the equation $m\vb = g\vG$. The curious fact now manifests itself
that the ``gravitational charge'' or \Emph{the ``gravitational mass''~$g$
is equal to the ``inertial mass''~$m$}. Eötvös has comparatively
recently tested the accuracy of this law by actual experiments of
the greatest refinement (\textit{vide} \FNote{3}). The centrifugal force imparted
to a body at the earth's surface by the earth's rotation is
proportional to its inertial mass but its weight is proportional to its
gravitational mass. The resultant of these two, the \emph{apparent} weight,
would have different directions for different bodies if gravitational and
inertial mass were not proportional throughout. The absence of this
difference of direction was demonstrated by Eötvös by means of the
exceedingly sensitive instrument known as the torsion-balance: it
enables the inertial mass of a body to be measured to the same
degree of accuracy as that to which its weight may be determined
by the most sensitive balance. The proportionality between gravitational
and inertial mass holds in cases, too, in which a diminution
of mass is occasioned not by an escape of substance in the old sense,
but by an emission of radioactive energy.
The inertial mass of a body has, according to the fundamental
law of mechanics, a \Emph{universal} significance. It is the inertial mass
that regulates the behaviour of the body under the influence of any
forces acting on it, of whatever physical nature they may be; the
inertial mass of the body is, however, according to the usual view
associated only with a special physical field of force, namely, that
\PageSep{226}
of gravitation. From this point of view, however, the identity
between inertial and gravitational mass remains fully incomprehensible.
Due account can be taken of it only by a mechanics which
\index{Mechanics!fundamental law of!general@{(in general theory of relativity)}}%
from the outset takes into consideration gravitational as well as inertial
mass. This occurs in the case of the mechanics given by the
general theory of relativity, in which we assume that \Emph{gravitation,
just like centrifugal and Coriolis forces, is included in the
``pseudo-force'' which has its origin in the metrical field}.
We shall find actually that the planets pursue the courses mapped
out for them by the guiding field, and that we need not have recourse
to a special ``force of gravitation,'' as did Newton, to account
for the influence which diverts the planets from their paths as
prescribed by Galilei's Principle (or Newton's first law of motion).
The gravitational forces satisfy the second postulate also; that is,
they may be made to vanish at a space-time point if we introduce
an appropriate co-ordinate system. A closed box, such as a lift, whose
suspension wire has snapped, and which descends without friction
in the gravitational field of the earth, is a striking example of such
a system of reference. All bodies that are falling freely will appear
to be at rest to an observer in the box, and physical events will
happen in the box in just the same way as if the box were at rest
and there were no gravitational field, in spite of the fact that the
gravitational force is acting.
II\@. The transition from the special to the general theory of
relativity, as described in~\Inum{I}, is a purely mathematical process. By
introducing the metrical groundform~\Eq{(1)}, we may formulate physical
laws so that they remain invariant for arbitrary transformations;
this is a possibility that is purely mathematical in essence and
denotes no particular peculiarity of these laws. A new physical
factor appears only when it is assumed that the metrical structure
of the world is not given \textit{a~priori}, but that the above quadratic form
is related to matter by generally invariant laws. Only this fact
justifies us in assigning the name ``general theory of relativity'' to
our reasoning; we are not simply giving it to a theory which has
merely borrowed the mathematical form of relativity. The same
fact is indispensable if we wish to solve the problem of the relativity
of motion; it also enables us to complete the analogy mentioned in~\Inum{I},
according to which the metrical field is related to matter in the
same way as the electric field to electricity. Only if we accept
this fact does the theory briefly quoted at the end of the previous
section become possible, according to which \Emph{gravitation is a
mode of expression of the metrical field}; for we know by experience
that the gravitational field is determined (in accordance
\PageSep{227}
with Newton's law of attraction) by the distribution of matter.
This assumption, rather than the postulate of general invariance,
seems to the author to be the real pivot of the general theory of
relativity. If we adopt this standpoint we are no longer justified
\index{General principle of relativity}%
\index{Relativity!principle of!(general)}%
in calling the forces that have their origin in the metrical field
pseudo-forces. They then have just as real a meaning as the
mechanical forces of the electromagnetic field. Coriolis or centrifugal
forces are real force effects, which the gravitational or
guiding field exerts on matter. Whereas, in~\Inum{I}, we were confronted
with the easy problem of extending known physical laws (such as
Maxwell's equations) from the special case of a constant metrical
fundamental tensor to the general case, we have, in following the
ideas set out just above, to discover the \Emph{invariant law of gravitation,
according to which matter determines the components~$\Gamma_{\beta i}^{\alpha}$
of the gravitational field}, and which replaces the Newtonian
law of attraction in Einstein's Theory. The well-known laws of the
field do not furnish a starting-point for this. Nevertheless Einstein
succeeded in solving this problem in a convincing fashion, and in
showing that the course of planetary motions may be explained just
as well by the new law as by the old one of Newton; indeed, that
the only discrepancy which the planetary system discloses towards
Newton's Theory, and which has hitherto remained inexplicable,
namely, the gradual advance of Mercury's perihelion by $43''$~per
century, is accounted for accurately by Einstein's theory of gravitation.
Thus this theory, which is one of the greatest examples of the
power of speculative thought, presents a solution not only of the
problem of the relativity of all motion (the only solution which
satisfies the demands of logic), but also of the problem of gravitation
(\textit{vide} \FNote{4}). We see how cogent arguments added to those in
Chapter~II bring the ideas of Riemann and Einstein to a successful
issue. It may also be asserted that their point of view is the first
to give due importance to the circumstance that space and time,
in contrast with the material content of the world, are \Emph{forms} of
phenomena. Only physical phase-quantities can be measured,
that is, read off from the behaviour of matter in motion; but we
cannot measure the four world-co-ordinates that we assign \textit{a~priori}
arbitrarily to the world-points so as to be able to represent the
phase-quantities extending throughout the world by means of
mathematical functions (of four independent variables).
Whereas the potential of the electromagnetic field is built up
from the co-efficients of an invariant \Emph{linear} differential form of
the world-co-ordinates~$\phi_{i}\, dx_{i}$, the potential of the gravitational field
\PageSep{228}
is made up of the co-efficients of an invariant \Emph{quadratic} differential
form. This fact, which is of fundamental importance, constitutes
the form of \Emph{Pythagoras' Theorem} to which it has gradually been
\index{Pythagoras' Theorem}%
transformed by the stages outlined above. It does not actually
spring from the observation of gravitational phenomena in the true
sense (Newton accounted for these observations by introducing a
single gravitational potential), but from geometry, from the observations
of measurement. Einstein's theory of gravitation is the result
of the fusion of two realms of knowledge which have hitherto been
developed fully independently of one another; this synthesis may
be indicated by the scheme
\[
\underbrace{\text{Pythagoras}\quad\text{Newton}}_{\mbox{Einstein}}
\]
\Emph{To derive the values of the quantities~$g_{ik}$ from directly
observed phenomena}, we use light-signals and point-masses which
are moving under no forces, as in the special theory of relativity.
Let the world-points be referred to any co-ordinates~$x_{i}$ in some way.
The geodetic lines passing through a world-point~$O$, namely,
\begin{gather*}
\frac{d^{2} x_{i}}{ds^{2}}
+ \Chr{\alpha\beta}{i} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0\Add{,}
\Tag{(8)} \\
g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds} = C = \text{const.}\Add{,}
\Tag{(9)}
\end{gather*}
split up into two classes; \Inum{(\ia)}~those with a \Emph{space-like} direction,
\Inum{(\ib)}~those with a \Emph{time-like} direction ($C < 0$ or $C > 0$ respectively).
The latter fill a ``double'' cone with the common vertex at~$O$ and
which, at~$O$, separates into two simple cones, of which one opens
into the future and the other into the past. The first comprises
all world-points that belong to the ``active future'' of~$O$, the second
all world-points that constitute the ``passive past'' of~$O$. The
limiting sheet of the cone is formed by the geodetic null-lines
($C = 0$); the ``future'' half of the sheet contains all the world-points
at which a light-signal emitted from~$O$ arrives, or, more
generally, the exact initial points of every effect emanating from~$O$.
The metrical groundform thus determines in general what world-points
are related to one another in effects. If $dx_{i}$ are the relative
co-ordinates of a point~$O'$ infinitely near~$O$, then $O'$~will be traversed
by a light-signal emitted from~$O$ if, and only if, $g_{ik}\, dx_{i}\, dx_{k} = 0$.
By observing the arrival of light at the points neighbouring
to~$O$ we can thus determine the ratios of the values of the~$g_{ik}$'s at
the point~$O$; and, as for~$O$, so for any other point. It is impossible,
however, to derive any further results from the phenomenon of the
propagation of light, for it follows from a remark on \Pageref{127} that
\PageSep{229}
the geodetic null-lines are dependent only on the ratios of the~$g_{ik}$'s.
The optical ``direction'' picture that an observer (``point-eye''
as on \Pageref[p.]{99}) receives, for instance, from the stars in the heavens,
is to be constructed as follows. From the world-point~$O$ at which
the observer is stationed those geodetic null-lines (light-lines) are to
be drawn on the backward cone which cuts the world-lines of the
stars. The direction of every light-line at~$O$ is to be resolved into
one component which lies along the direction~$\ve$ of the world-line of
the observer and another~$\vs$ which is perpendicular to it (the meaning
of perpendicular is defined by the metrical structure of the world
as given on \Pageref[p.]{121}); $\vs$~is the spatial direction of the light-ray.
Within the three-dimensional linear manifold of the line-elements
at~$O$ perpendicular to~$\ve$, $-ds^{2}$~is a definitely positive form. The
angles (that arise from it when it is taken as the metrical groundform,
and which are to be calculated from formula~\Eq{(15)}, §\,11)
between the spatial directions~$\vs$ of the light-rays are those that
determine the positions of the stars as perceived by the observer.
The factor of proportionality of the~$g_{ik}$'s which could not be
derived from the phenomenon of the transmission of light may be
determined from the motion of point-masses which carry a clock
\index{Motion!(under no forces)}%
with them. For if we assume that---at least for unaccelerated
motion under no forces---the time read off from such a clock is the
proper-time~$s$, equation~\Eq{(9)} clearly makes it possible to apply the
unit of measure along the world-line of the motion (cf.\ Appendix~I).\Pagelabel{229}
\Section{28.}{Einstein's Fundamental Law of Gravitation}
\index{Gravitation!Newton's Law of}%
\index{Newton's Law of Gravitation}%
According to the Newtonian Theory the condition (or phase) of
matter is characterised by a \Emph{scalar}, the mass-density~$\mu$; and the
gravitational potential is also a scalar~$\Phi$: Poisson's equation holds,
that is,
\[
\Delta \Phi = 4\pi k\mu
\Tag{(10)}
\]
($\Delta = \div \grad$; $k = $ the gravitational constant). This is the law
according to which matter determines the gravitational field. But
according to the theory of relativity matter can be described
'rigorously only by a symmetrical \Emph{tensor} of the second order~$T_{ik}$,
or better still by the corresponding mixed tensor-density~$\vT_{i}^{k}$;
in harmony with this the potential of the gravitational field
consists of the components of a symmetrical \Emph{tensor}~$g_{ik}$. Therefore,
in Einstein's Theory we expect equation~\Eq{(10)} to be replaced by a
system of equations of which the left side consists of differential
expressions of the second order in the~$g_{ik}$'s, and the right side of
components of the energy-density; this system has to be invariant
with respect to arbitrary transformations of the co-ordinates. To
\PageSep{230}
\index{Potential!of the gravitational field}%
find the law of gravitation we shall do best by taking up the thread
from Hamilton's Principle formulated at the close of §\,26. The
\emph{Action} there consisted of three parts: the substance-action of
electricity, the field-action of electricity, and the substance-action of
mass or gravitation. In it there is lacking a fourth term, the field-action
of gravitation, which we have now to find. Before doing
this, however, we shall calculate the change in the sum of the first
three terms already known, when we leave the potentials~$\phi_{i}$ of the
electromagnetic field and the world-lines of the substance-elements
unchanged but subject the~$g_{ik}$'s, \Emph{the potentials of the metrical
field, to an infinitesimal virtual variation~$\delta$}. This is possible
only from the point of view of the general theory of relativity.
This causes no change in the substance-action of electricity, but
the change in the integrands that occur in the field-action, namely
\[
\tfrac{1}{2} \vS = \tfrac{1}{4} F_{ik} \vF^{ik}
\]
is
\[
\tfrac{1}{4}\bigl\{\sqrt{g} \delta(F_{ik} F^{ik}) + (F_{ik} F^{ik}) \delta \sqrt{g}\bigr\}.
\]
The first summand in the curved bracket here $= \vF_{rs}\, \delta F^{rs}$ and hence,
since
\[
F^{rs} = g^{ri} g^{sk} F_{ik}\Add{,}
\]
we immediately get the value
\[
2\sqrt{g} F_{ir} F_{k}^{r}\, \delta g^{ik}.
\]
The second summand, by~\Eq{(58')} §\,17,
\[
= -\vS g_{ik}\, \delta g^{ik}.
\]
Thus, finally, we find the variation in the field-action to be
\[
= \int \tfrac{1}{2} \vS\, \delta g^{ik}\, dx
= \int \tfrac{1}{2} \vS^{ik}\, \delta g_{ik}\, dx
\quad\text{(cf.\ \Eq{(59)}, §\,17)}
\]
if\Pagelabel{230}
\[
\vS_{i}^{k} = \tfrac{1}{2} \vS \delta_{i}^{k} = F_{ir} \vF^{kr}
\Tag{(11)}
\]
are the components of the energy-density of the electromagnetic
field.\footnote
{The signs are the reverse of those used in Chapter~III on account of the
change in the sign of the metrical groundform.}
It suddenly becomes clear to us now (and only now that we
have succeeded in calculating the variation of the world's metrical
field) what is the origin of the complicated expressions~\Eq{(11)} for the
energy-momentum density of the electromagnetic field.
We get a corresponding result for the substance-action of the
mass; for we have
\[
\delta \sqrt{g_{ik}\, dx_{i}\, dx_{k}}
= \tfrac{1}{2}\, \frac{dx_{i}\, dx_{k}\, \delta g_{ik}}{ds}
= \tfrac{1}{2} ds\, u^{i} u^{k}\, \delta g_{ik},
\]
\PageSep{231}
and hence
\[
\delta \int \left(dm \int \sqrt{g_{ik}\, dx_{i}\, dx_{k}}\right)
= \int \tfrac{1}{2} \mu u^{i} u_{k}\, \delta g_{ik}\, dx.
\]
Hence the total change in the \emph{Action} so far known to us is, for
a variation of the metrical field,
\[
\int \tfrac{1}{2} \vT^{ik}\, \delta g_{ik}\, dx
\Tag{(12)}
\]
in which $\vT_{i}^{k}$~denotes the tensor-density of the total energy.
\Emph{The absent fourth term of the \emph{Action}, namely, the field-action
of gravitation}, must be an invariant integral, $\Dint \vG\, dx$, of
\index{Field action of electricity!gravitation@{of gravitation}}%
which the integrand~$\vG$ is composed of the potentials~$g_{ik}$ and of the
field-components~$\dChr{ik}{r}$ of the gravitational field, built up from the
$g_{ik}$'s and their first derivatives. It would seem to us that only under
such circumstances do we obtain differential equations of order
not higher than the second for our gravitational laws. If the total
differential of this function is
\[
\Squeeze{\delta \vG = \tfrac{1}{2} \vG^{ik}\, \delta g_{ik} + \tfrac{1}{2} \vG^{ik, r}\, \delta g_{ik, r}\qquad
(\vG^{ki} = \vG^{ik} \text{ and } \vG^{ki, r} = \vG^{ik, r})}
\Tag{(13)}
\]
we get, for an infinitesimal variation~$\delta g_{ik}$ which disappears for
regions beyond a finite limit, by partial integration, that
\[
\delta \int \vG\, dx
= \int \tfrac{1}{2}[\vG]^{ik}\, \delta g_{ik}\, dx
\Tag{(14)}
\]
in which the ``Lagrange derivatives'' $[\vG]^{ik}$, which are symmetrical
in $i$~and~$k$, are to be calculated according to the formula
\[
[\vG] = \vG^{ik} - \frac{\dd \vG^{ik, r}}{\dd x_{r}}.
\]
The gravitational equations will then actually assume the form
which was predicted, namely
\[
[\vG]_{i}^{k} = -\vT_{i}^{k}\Add{.}
\Tag{(15)}
\]
There is no longer any cause for surprise that it happens to be the
energy-momentum components that appear as co-efficients when
we vary the~$g_{ik}$'s in the first three factors of the \emph{Action} in accordance
with~\Eq{(12)}. Unfortunately a scalar-density~$\vG$, of the type we wish,
does not exist at all; for we can make all the~$\dChr{ik}{r}$'s vanish at any
given point by choosing the appropriate co-ordinate system. Yet
the scalar~$R$, the curvature defined by Riemann, has made us
familiar with an invariant which involves the second derivatives
of the~$g_{ik}$'s only \Emph{linearly}: it may even be shown that it is the
\PageSep{232}
only invariant of this kind (\textit{vide} Appendix~II,\Pagelabel{232} in which the proof is
given). In consequence of this linearity we may use the invariant
integral $\Dint \frac{1}{2} R \sqrt{g}\, dx$ to get the derivatives of the second order by
partial integration. We then get
\[
\int \tfrac{1}{2} R \sqrt{g}\, dx = \int \vG\, dx
\]
$+$~a divergence integral, that is, an integral whose integrand is of
the form~$\dfrac{\dd \vw^{i}}{\dd x_{i}}$: $\vG$~here depends only on the~$g_{ik}$'s and their first
derivatives. Hence, for variations~$\delta g_{ik}$, that vanish outside a finite
region, we get
\[
\delta \int \tfrac{1}{2} R \sqrt{g}\, dx = \delta \int \vG\, dx
\]
since, according to the principle of partial integration,
\[
\int \frac{\dd (\delta \vw^{i})}{\dd x_{i}}\, dx = 0.
\]
Not $\Dint \vG\, dx$ itself is an invariant, but the variation $\delta \Dint \vG\, dx$, and this is
the essential feature of Hamilton's Principle. \emph{We need not, therefore,
have fears about introducing $\Dint \vG\, dx$ as the \emph{Action} of the gravitational
field; and this hypothesis is found to be the only possible one.}
We are thus led under compulsion, as it were, to the unique
gravitational equations~\Eq{(15)}. It follows from them that \Emph{every kind
of energy exerts a gravitational effect}: this is true not only
\index{Energy!(acts gravitationally)}%
of the energy concentrated in the electrons and atoms, that is of
matter in the restricted sense, but also of diffuse field-energy (for
the~$\vT_{i}^{k}$'s are the components of the total energy).
Before we carry out the calculations that are necessary if we
wish to be able to write down the gravitational equations explicitly,
we must first test whether we get analogous results \Emph{in the case of
Mie's Theory}. The \emph{Action}, $\Dint \vL\, dx$, which occurs in it is an invariant
not only for linear, but also for arbitrary transformations. For $\vL$~is
composed algebraically (not as a result of tensor analysis) of the
components~$\phi_{i}$ of a co-variant vector (namely, of the electromagnetic
potential), of the components~$F_{ik}$ of a linear tensor of the second
order (namely, of the electromagnetic field), and of the components~$g_{ik}$
of the fundamental metrical tensor. We set the total differential~$\delta \vL$
of this function
\PageSep{233}
equal to
\begin{gather*}
\tfrac{1}{2} \vT^{ik}\, \delta g_{ik} + \delta_{0} \vL,
\quad\text{in which }
\delta_{0} \vL = \tfrac{1}{2} \vH^{ik}\, \delta F_{ik} + \vs^{i}\, \delta \phi_{i} \\
(\vT^{ki} = \vT^{ik},\quad \vH^{ki} = -\vH^{ik})\Add{.}
\Tag{(16)}
\end{gather*}
We then call the tensor-density~$\vT_{i}^{k}$ the energy or matter. By doing
this, we affirm once again that the metrical field (with the potentials~$g_{ik}$)
is related to matter~($\vT^{ik}$) in the same way as the electromagnetic
field (with the potentials~$\phi_{i}$) is related to the electric current~$\vs^{i}$.
We are now obliged to prove that the present explanation leads
accurately to the expressions given in~\Eq{(64)}, §\,26, for energy and
momentum. This will furnish the proof, which was omitted above,
of the symmetry of the energy-tensor. To do this we cannot use
the method of direct calculation as above in the particular case of
Maxwell's Theory, but we must apply the following elegant considerations,
the nucleus of which is to be found in Lagrange, but
which were discussed with due regard to formal perfection by F.~Klein
(\textit{vide} \FNote{5}).
We subject the world-continuum to an infinitesimal deformation,
as a result of which in general the point~$(x_{i})$ becomes transformed
into the point~$(\bar{x}_{j})$
\[
\bar{x}_{i} = x_{i} + \epsilon · \xi^{i}(x_{0}\Com x_{1}\Com x_{2}\Com x_{3})
\Tag{(17)}
\]
(in which $\epsilon$~is the constant infinitesimal parameter, all of whose
higher powers are to be struck out). We imagine the phase-quantities
to follow the deformation so that at its conclusion the
new~$\phi_{i}$'s (we call them~$\bar{\phi}_{i}$) are functions of the co-ordinates of
such a kind that, in consequence of~\Eq{(17)}, the equations
\[
\phi_{i}(x)\, dx_{i} = \bar{\phi}_{i}(\bar{x})\, d\bar{x}_{i}
\Tag{(18)}
\]
hold; and in the same sense the symmetrical and skew-symmetrical
bilinear differential form with the co-efficients $g_{ik}$,~$F_{ik}$, respectively,
remains unchanged. The changes $\bar{\phi}_{i}(x) - \phi_{i}(x)$ which the quantities
$\phi_{i}$~undergo at a fixed world-point~$(x_{i})$ as a result of the deformation
will be denoted by~$\delta \phi_{i}$; $\delta g_{ik}$~and $\delta F_{ik}$ have a corresponding meaning.
{\Loosen If we replace the old quantities~$\phi_{i}$ in the function~$\vL$ by the $\bar{\phi}_{i}$
arising from the deformation, we shall suppose the function $\bar{\vL} = \vL + \delta \vL$
to result; the~$\delta \vL$ in it is given by~\Eq{(16)}. Furthermore, let
$\rX$~be an arbitrary region of the world which, owing to the deformation,
becomes~$\Bar{\rX}$. The deformation causes the \emph{Action} $\Dint_{\rX} \vL\, dx$ to
undergo a change $\delta' \Dint_{\rX} \vL\, dx$ which is equal to the difference between
\PageSep{234}
the integral~$\bar{\vL}$ taken over~$\rX$ and the integral~$\vL$ taken over~$\Bar{\rX}$. The
invariance of the \emph{Action} is expressed by the equation}
\[
\delta' \int_{\rX} \vL\, dx = 0\Add{.}
\Tag{(19)}
\]
We make a natural division of this difference into two parts: (1)~the
difference between the integrals of $\bar{\vL}$~and $\vL$ over~$\Bar{\rX}$\Add{,} (2)~the
difference between the integral of~$\vL$ over $\Bar{\rX}$ and~$\rX$. Since $\Bar{\rX}$~differs
from~$\rX$ only by an infinitesimal amount, we may set
\[
\delta \int_{\rX} \vL\, dx = \int_{\rX} \delta \vL\, dx
\]
for the first part. On \Pageref{111} we found the second part to be
\[
\epsilon \int_{\rX} \frac{\dd (\vL \xi^{i})}{\dd x_{i}}\, dx.
\]
To be able to complete the argument we must next calculate the
variations $\delta \phi_{i}$, $\delta g_{ik}$,~$\delta F_{ik}$. If we set $\bar{\phi}_{i}(\bar{x}) - \phi_{i}(x) = \delta' \phi_{i}$ for a
moment, then, owing to~\Eq{(18)}, we get
\[
\delta' \phi_{i} · dx_{i} + \epsilon \phi_{r}\, d\xi^{r} = 0
\]
and hence
\[
\delta' \phi_{i} = -\epsilon · \phi_{r}\, \frac{\dd \xi^{i}}{\dd x^{i}}.
\]
Moreover, since
\[
\delta \phi_{i}
= \delta' \phi_{i} - \bigl\{\bar{\phi}_{i}(\bar{x}) - \bar{\phi}_{i}(x)\bigr\}
= \delta' \phi_{i} - \epsilon · \frac{\dd \phi}{\dd x_{r}}\, \xi^{r}
\]
we get, suppressing the self-evident factor~$\epsilon$,
\[
-\delta \phi_{i}
= \phi_{r}\, \frac{\dd \xi^{r}}{\dd x_{i}}
+ \frac{\dd \phi_{i}}{\dd x_{r}}\, \xi^{r}\Add{.}
\Tag{(20)}
\]
In the same way, we get
\begin{alignat*}{3}
-\delta g_{ik}
&= g_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
&&+ g_{rk}\, \frac{\dd \xi^{r}}{\dd x_{i}}
&&+ \frac{\dd g_{ik}}{\dd x_{r}}\, \xi^{r}\Add{,}
\Tag{(20')} \\
-\delta F_{ik}
&= F_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
&&+ F_{rk}\, \frac{\dd \xi^{r}}{\dd x_{i}}
&&+ \frac{\dd F_{\Typo{ir}{ik}}}{\dd x_{r}}\, \xi^{r}\Add{.}
\Tag{(20'')}
\end{alignat*}
And, on account of
\[
F_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
\quad\text{we have}\quad
\delta F_{ik}
= \frac{\dd (\delta \phi_{i})}{\dd x_{k}}
- \frac{\dd (\delta \phi_{k})}{\dd x_{i}}\Add{,}
\Tag{(21)}
\]
for since the former is an invariant relation, we get from it
\[
\bar{F}_{ik}(\bar{x})
= \frac{\dd \bar{\phi}_{i}(\bar{x})}{\dd \bar{x}_{k}}
- \frac{\dd \bar{\phi}_{k}(\bar{x})}{\dd \bar{x}_{i}},
\quad\text{and also }
\bar{F}_{ik}(x)
= \frac{\dd \bar{\phi}_{i}(x)}{\dd x_{k}}
- \frac{\dd \bar{\phi}_{k}(x)}{\dd x_{i}}\Add{.}
\]
\PageSep{235}
Substitution gives us
\[
-\delta \vL
= (\vT_{i}^{k} + \vH^{rk} F_{ri} + \vs^{k} \phi_{i}) \frac{\dd \xi}{\dd x_{k}}
+ (\tfrac{1}{2} \vT^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
+ \dots + ) \xi^{i}\Add{.}
\]
If we remove the derivatives of~$\xi^{i}$ by partial integration, and use
the abbreviation
\[
\vV_{i}^{k}
= \vT_{i}^{k} + F_{ir} \vH^{kr}
+ \phi_{i} \vs^{k} - \delta_{i}^{k} \vL\Add{,}
\]
we get a formula of the following form
\[
-\delta' \int_{\rX} \vL\, dx
= \int_{\rX} \frac{\dd (\vV_{i} \xi^{i})}{\dd x_{k}}\, dx
+ \int_{\rX} (\vt_{i} \xi^{i})\, dx = 0\Add{.}
\Tag{(22)}
\]
It follows from this that, as we know, by choosing the~$\xi^{i}$'s appropriately,
namely, so that they vanish outside a definite region,
which we here take to be~$\rX$, we must have, at every point,
\[
\vt_{i} = 0\Add{.}
\Tag{(23)}
\]
Accordingly, the first summand of~\Eq{(22)} is also equal to zero. The
identity which comes about in this way is valid for arbitrary
quantities~$\xi^{i}$ and for any finite region of integration~$\rX$. Hence,
since the integral of a continuous function taken over any and
every region can vanish only if the function itself $= 0$, we must
have
\[
\frac{\dd (\vV_{i}^{k} \xi^{i})}{\dd x_{k}}
= \vV_{i}^{k}\, \frac{\dd \xi^{i}}{\dd x_{k}}
+ \frac{\dd \vV_{i}^{k}}{\dd x_{k}}\, \xi^{i} = 0.
\]
Now, $\xi^{i}$~and $\dfrac{\dd \xi^{i}}{\dd x_{k}}$ may assume any values at one and the same
point. Consequently,
\[
\vV_{i}^{k} = 0\qquad
\left(\frac{\dd \vV_{i}^{k}}{\dd x_{k}} = 0\right).
\]
This gives us the desired result
\[
\vT_{i}^{k} = \vL \delta_{i}^{k} - F_{ir} \vH^{kr} - \phi_{i} \vs^{k}.
\]
These considerations simultaneously give us the theorems of conservation
of energy and of momentum, which we found by calculation
in §\,26; they are contained in equations~\Eq{(23)}. The change in the
\emph{Action} of the whole world for an infinitesimal deformation which
vanishes outside a finite region of the world is found to be
\[
\int \delta \vL\, dx
= \int \tfrac{1}{2} \vT^{ik}\, \delta g_{ik}\, dx
+ \int \delta_{0} \vL\, dx = 0\Add{.}
\Tag{(24)}
\]
In consequence of the equations~\Eq{(21)} and of \Emph{Hamilton's Principle},
namely
\[
\int \delta_{0} \vL\, dx = 0\Add{,}
\Tag{(25)}
\]
\PageSep{236}
which is here valid, the second part (in Maxwell's equations) disappears.
But the first part, as we have already calculated, is
\[
\Squeeze[0.95]{-\int \left(\vT_{i}^{k}\, \frac{\dd \xi^{i}}{\dd x_{k}}
+ \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta} \xi^{i}\right) dx
= \int \left(\frac{\dd \vT_{i}^{k}}{\dd x_{k}}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta}\right) \xi^{i}\, dx.}
\]
Thus, \Emph{as a result of the laws of the electromagnetic field, we
get the mechanical equations}
\[
\frac{\dd \vT_{i}^{k}}{\dd x_{k}}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vT^{\alpha\beta} = 0\Add{.}
\Tag{(26)}
\]
(On account of the presence of the additional term due to gravitation
\index{Einstein's Law of Gravitation}%
\index{General principle of relativity}%
\index{Gravitation!Einstein's Law of (general form)}%
these equations can no longer in the general theory of
relativity be fitly termed theorems of conservation. The question
\index{Relativity!principle of!(general)}%
whether proper theorems of conservation may actually be set up
will be discussed in §\,33.)
The Hamiltonian Principle which has been \Emph{supplemented by
\index{Hamilton's!principle!Maxwell@{(according to Maxwell and Lorentz)}}%
the \Typo{Action}{\emph{Action}} of the gravitational field}, namely
\[
\delta \int (\vL + \vG)\, dx = 0\Add{,}
\Tag{(27)}
\]
and in which the electromagnetic and the \Emph{gravitational} condition
(phase) of the field may be subjected independently of one another
to virtual infinitesimal variations gives rise to the gravitational
equations~\Eq{(15)} in addition to the electromagnetic laws. If we
apply the process above, which ended in~\Eq{(26)}, to~$\vG$ instead of to~$\vL$---here,
too, we have, for the variation~$\delta$ caused by a deformation
of the world-continuum which vanishes outside a finite region, that
%[** TN: Not displayed in the original]
\[
\displaystyle\delta \int \vG\, dx = \delta \int \tfrac{1}{2}R \sqrt{g}\, dx = 0
\]
---we arrive at \Emph{mathematical identities}
analogous to~\Eq{(26)}, namely
\[
\frac{\dd [\vG]_{i}^{k}}{\dd x_{k}}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} [\vG]^{\alpha\beta} = 0.
\]
The fact that $\vG$~contains the derivatives of the~$g_{ik}$'s as well as the
$g_{ik}$'s themselves is of no account. Accordingly, \emph{the mechanical
equations~\Eq{(26)} are just as much a consequence of the gravitational
equations~\Eq{(15)} as of the electromagnetic laws of the field}.
The wonderful relationships, which here reveal themselves,
may be formulated in the following way independently of the
question whether Mie's theory of electrodynamics is valid or not.
The phase (or condition) of a physical system is described relatively to
a co-ordinate system by means of certain variable space-time phase-quantities~$\phi$
(these were our $\phi_{i}$'s above). Besides these, we have
also to take account of the \Emph{metrical field} in which the system is
embedded and which is characterised by its potentials~$g_{ik}$. The
\PageSep{237}
uniformity underlying the phenomena occurring in the system is
expressed by an invariant integral $\Dint \vL\, dx$; in it, the scalar-density~$\vL$
is a function of the~$\phi$'s and of their derivatives of the first and
if need be, of the second order, and also a function of the~$g_{ik}$'s,
but the latter quantities alone and not their derivatives occur in~$\vL$.
We form the total differential of the function~$\vL$ by writing down
explicitly only that part which contains the differentials~$\delta g_{ik}$, namely,
\[
\delta \vL = \tfrac{1}{2} \vT^{ik} \delta g_{ik} + \delta_{0} \vL.
\]
$\vT_{i}^{k}$~is then the tensor-density of the \Emph{energy} (identical with \Emph{matter})
\index{Energy!(acts gravitationally)}%
associated with the physical state or phase of the system. The
determination of its components is thus reduced once and for all
to a determination of Hamilton's Function~$\vL$. \emph{The general theory
of relativity alone, which allows the process of variation to be applied
to the metrical structure of the world, leads to a true definition of
energy.} The phase-laws emerge from the ``partial'' principle of
action in which only the phase-quantities~$\phi$ are to be subjected to
variation; just as many equations arise from it as there are
quantities~$\phi$. The additional ten gravitational equations~\Eq{(15)} for
the ten potentials~$g_{ik}$ result if we enlarge the partial principle of
action to the total one~\Eq{(27)}, in which the~$g_{ik}$'s are also to be subjected
to variation. The \Emph{mechanical equations}~\Eq{(26)} are a consequence
of the phase-laws as well as of the gravitational laws;
they may, indeed, be termed the eliminant of the latter. Hence,
in the system of phase and gravitational laws, there are four
superfluous equations. The general solution must, in fact, contain
four arbitrary functions, since the equations, in virtue of their
invariant character, leave the co-ordinate system of the~$x_{i}$'s indeterminate;
hence, arbitrary continuous transformations of these
co-ordinates derived from \Emph{one} solution of the equations always
give rise to new solutions in their turn. (These solutions, however,
represent the same objective course of the world.) The old
subdivision into geometry, mechanics, and physics must be replaced
in Einstein's Theory by the separation into physical phases
and metrical or gravitational fields.
For the sake of completeness we shall once again revert to the
Hamiltonian Principle used in the theory of Lorentz and Maxwell.
Variation applied to the~$\phi_{i}$'s gives the electromagnetic laws, but
applied to the~$g_{ik}$'s the gravitational laws. Since the \emph{Action} is an
invariant, the infinitesimal change which an infinitesimal deformation
of the world-continuum calls up in it $= 0$; this deformation is
to affect the electromagnetic and the gravitational field as well as
the world-lines of the substance-elements. This change consists of
\PageSep{238}
three summands, namely, of the changes which are caused in turn
by the variation of the electromagnetic field, of the gravitational
field, and of the substance-paths. The first two parts are zero as
a consequence of the electromagnetic and the gravitational laws;
hence the third part also vanishes and we see that the mechanical
equations are a result of the two groups of laws mentioned just
above. Recapitulating our former calculations we may derive
this result by taking the following steps. From the gravitational
laws there follow~\Eq{(26)}, i.e.\
\[
\mu U_{i} + u_{i} M
= -\left\{\frac{\dd \vS_{i}^{k}}{\dd x_{k}}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vS^{\alpha\beta}\right\}\Add{,}
\Tag{(28)}
\]
in which $\vS_{i}^{k}$~is the tensor-density of the electromagnetic energy of
field, namely, of
\[
U_{i} = \frac{du_{i}}{ds}
- \tfrac{1}{2} \frac{\dd g_{\alpha\beta}}{\dd x_{i}} u^{\alpha} u^{\beta}\Add{,}
\]
and $M$~is the left-hand member of the equation of continuity for
matter, namely
\[
M = \frac{\dd (\mu u^{i})}{\dd x_{i}}.
\]
As a result of Maxwell's equations the right-hand member of~\Eq{(28)}
\[
= \vp_{i} = -F_{ik} \vs^{k}\qquad
(\vs^{i} = \rho u^{i}).
\]
If we then multiply~\Eq{(28)} by~$u^{i}$ and sum up with respect to~$i$, we
get $M = 0$; in this way we have arrived at the equation of continuity
for matter and also at the mechanical equations in their usual
form.
After having gained a full survey of how the gravitational laws
of Einstein are to be arranged into the scheme of the remaining
physical laws, we are still faced with the task of working out the
explicit expression for the~$[\vG]_{i}^{k}$'s (\textit{vide} \FNote{6}). The virtual change
\[
\delta \Gamma_{ik}^{r} = \delta \Chr{ik}{r} = \gamma_{ik}^{r}
\]
of the components of the affine relationship is, as we know (\Pageref{114}),
a tensor. If we use a geodetic co-ordinate system at a certain
point, then we get directly from the formula for~$R^{ik}$ (\Eq{(60)}, §\,17) that
\[
\delta R_{ik}
= \frac{\dd \gamma_{ik}^{r}}{\dd x_{r}} - \frac{\dd \gamma_{ir}^{r}}{\dd x_{k}}
\]
and
\[
g^{ik}\, \delta R_{ik}
= g^{ik}\, \frac{\dd \gamma_{ik}^{r}}{\dd x_{r}}
- g^{ir}\, \frac{\dd \gamma_{ik}^{k}}{\dd x_{r}}.
\]
If we set
\[
g^{ik} \gamma_{ik}^{r} - g^{ir} \gamma_{ik}^{k} = w^{r}
\]
\PageSep{239}
we get
\[
g^{ik}\, \delta R_{ik} = \frac{\dd w^{r}}{\dd x_{r}}\Add{,}
\]
or, for any arbitrary co-ordinate system,
\[
\delta R
= R_{ik}\, \delta g^{ik}
+ \frac{1}{\sqrt{g}}\, \frac{\dd (\sqrt{g} w^{r})}{\dd x_{r}}\Add{.}
\]
The divergence disappears in the integration and hence, since by
definition we are to have
\[
\delta \int R\sqrt{g}\, dx
= \int [\vG]^{ik}\, \delta g_{ik}\, dx
= -\int [\vG]_{ik}\, \delta g^{ik}\, dx
\]
and since the~$R_{ik}$'s are symmetrical in Riemann's space, we get
\begin{align*}
[\vG]_{ik}
&= \sqrt{g} (\tfrac{1}{2}g_{ik} R - R_{ik})
= \tfrac{1}{2} g_{ik} \vR - \vR_{ik}\Add{,} \\
[\vG]_{i}^{k}
&= \tfrac{1}{2} \delta_{i}^{k} \vR - \vR_{i}^{k}.
\end{align*}
Therefore the gravitational laws are
\[
\framebox{$\vR_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vR = \vT_{i}^{k}$}
\Tag{(29)}
\]
Here, of course (exactly as was done for the unit of charge in
electromagnetic equations), the unit of mass has been suitably
chosen. If we retain the units of the c.g.s.\ system, a universal
constant~$8\pi\kappa$ will have to be added as a factor to the right-hand side.
It might still appear doubtful now at the outset whether $\kappa$~is positive
or negative, and whether the right-hand side of equation~\Eq{(29)}
should not be of opposite sign. We shall find, however, in the
next paragraph that, in virtue of the fact that masses attract one
another and do not repel, $\kappa$~is actually positive.
It is of mathematical importance to notice that \Emph{the exact
gravitational laws are not linear}; although they are linear in
the derivatives of the field-components~$\dChr{ik}{r}$, they are not linear in
the field-components themselves. If we contract equations~\Eq{(29)},
that is, set $k = i$, and sum with respect to~$i$, we get $-\vR = \vT = \vT_{i}^{\Typo{l}{i}}$;
hence, in place\Typo{}{ of}~\Eq{(29)} we may also write
\[
\vR_{i}^{k} = \vT_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vT\Add{.}
\Tag{(30)}
\]
In the first paper in which Einstein set up the gravitational
equations without following on from Hamilton's Principle, the
term~$-\frac{1}{2} \delta_{i}^{k} \vT$ was missing on the right-hand side; he recognised
only later that it is required as a result of the energy-momentum-theorem
(\textit{vide} \FNote{7}). The whole series of relations here described
and which is subject to Hamilton's Principle, has become manifest
in further works by H.~A. Lorentz, Hilbert, Einstein, Klein,
and the author (\textit{vide} \FNote{8}).
\PageSep{240}
In the sequel we shall find it desirable to know the value of~$\vG$.
To convert
\[
\int R \sqrt{g}\, dx
\quad\text{into}\quad
2 \int \vG\, dx
\]
by means of partial integration (that is, by detaching a divergence),
we must set
\begin{alignat*}{2}
\sqrt{g} g^{ik}\, \frac{\dd}{\dd x_{r}} \Chr{ik}{r}
&= \frac{\dd}{\dd x_{r}} \left(\sqrt{g} g^{ik} \Chr{ik}{r}\right)
&&- \Chr{ik}{r} \frac{\dd}{\dd x_{r}}(\sqrt{g} g^{ik})\Add{,} \\
\sqrt{g} g^{ik}\, \frac{\dd}{\dd x_{k}} \Chr{ir}{r}
&= \frac{\dd}{\dd x_{k}} \left(\sqrt{g} g^{ik} \Chr{ir}{r}\right)
&&- \Chr{ir}{r} \frac{\dd}{\dd x_{k}}(\sqrt{g} g^{ik})\Add{.}
\end{alignat*}
Thus we get
\begin{multline*}% [** TN: Set on one line in the original]
2\vG = \Chr{is}{s} \frac{\dd}{\dd x_{k}} (\sqrt{g} g^{ik})
- \Chr{ik}{r} \frac{\dd}{\Typo{\dd xr}{\dd x_{r}}} (\sqrt{g} g^{ik}) \\
+ \left(\Chr{ik}{r} \Chr{rs}{s} - \Chr{ir}{s} \Chr{ks}{r}\right)
\sqrt{g} g^{ik}\Add{.}
\end{multline*}
By \Eq{(57')},~\Eq{(57'')} of §\,17, however, the first two terms on the right, if
we omit the factor~$\sqrt{g}$,
\begin{align*}
&= -\Chr{is}{s} \Chr{kr}{i} g^{kr}
+ 2\Chr{ik}{r} \Chr{rs}{i} g^{sk}
- \Chr{ik}{r} \Chr{rs}{s} g^{ik} \\
&= \left(-\Chr{rs}{s} \Chr{ik}{r}
+ 2 \Chr{sk}{r} \Chr{ri}{s}
- \Chr{ik}{r} \Chr{rs}{s}\right) g^{ik} \\
&= 2 g^{ik} \left(\Chr{ir}{s} \Chr{ks}{r} - \Chr{ik}{r} \Chr{rs}{s}\right)\Add{.}
\end{align*}
Hence we finally arrive at
\[
\frac{1}{\sqrt{g}} \vG
= \tfrac{1}{2} g^{ik} \left(\Chr{ir}{s} \Chr{ks}{r} - \Chr{ik}{r} \Chr{rs}{s}\right)\Add{.}
\Tag{(31)}
\]
This completes our development of the foundations of Einstein's
Theory of Gravitation. We must now inquire whether observation
confirms this theory which has been built up on purely speculative
grounds, and above all, whether the motions of the planets can be
explained just as well (or better) by it as by Newton's law of attraction.
§§\,29--32 treat of the solution of the gravitational equations.
\index{Gravitational!field}%
The discussion of the general theory will not be resumed till §\,33.\Pagelabel{240}
\Section{29.}{The Stationary Gravitational Field---Comparison with
Experiment}
\index{Static!gravitational field|(}%
\index{Stationary!field}%
To establish the relationship of Einstein's laws with the results
of observations of the planetary system, we shall first specialise
them for the case of a stationary gravitational field (\textit{vide} \FNote{9}).
The latter is characterised by the circumstance that, if we use
\PageSep{241}
appropriate co-ordinates, the world resolves into space and time, so
that for the metrical form
\[
ds^{2} = f^{2}\, dt^{2} - d\sigma^{2},\qquad
d\sigma^{2} = \sum_{i,k=1}^{3} \gamma_{ik}\, dx_{i}\, dx_{k}\Add{,}
\]
we get
\[
g_{00} = f^{2};\quad
g_{0i} = g_{i0} = 0;\quad
g_{ik} = -\gamma_{ik}\qquad
(i, k = 1, 2, 3)\Add{,}
\]
and also that the co-efficients $f$~and~$\gamma^{ik}$ occurring in it depend only
on the space-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, and not on the time $t = x_{0}$.
$d\sigma^{2}$~is a positive definite quadratic differential form which determines
the metrical nature of the space having co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$;
$f$~is obviously the velocity of light. The measure~$t$ of time is fully
determined (when the unit of time has been chosen) by the postulates
that have been set up, whereas the space co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$ are
fixed only to the extent of an arbitrary continuous transformation of
these co-ordinates among themselves. In the statical case, therefore,
the metrics of the world gives, besides the measure-determination of
the space, also a scalar field~$f$ in space.
If we denote the Christoffel $3$-indices symbol, relating to the
ternary form~$d\sigma^{2}$, by an appended~$*$, and if the index letters $i$,~$k$,~$l$
assume only the values $1, 2, 3$ in turn, then it easily follows from
definition that
\begin{gather*}
\Chr{ik}{l} = \Chr{ik}{l}^{*}\Add{,}\displaybreak[0] \\
\Chr{ik}{0} = 0,\qquad
\Chr{0i}{k} = 0,\qquad
\Chr{00}{0} = 0\Add{,}\displaybreak[0] \\
\Chr{i0}{0} = \frac{f_{i}}{f},\qquad
\Chr{00}{0} = f\!f^{i}.
\end{gather*}
In the above, $f_{i} = \dfrac{\dd f}{\dd x_{i}}$ are co-variant components of the three-dimensional
gradient, and $f^{i} = \gamma^{ik} f_{k}$ are the corresponding contra-variant
components, whereas $\sqrt{\gamma} f^{i} = \vf^{i}$ are the components of a contra-variant
vector-density in space. For the determinant~$\gamma$ of the~$\gamma_{ik}$'s
we have $\sqrt{g} = f\sqrt{\gamma}$. If we further set
\[
f_{ik} = \frac{\dd f_{i}}{\dd x_{k}} - \Chr{ik}{r}^{*} f_{r}
= \frac{\dd^{2} f}{\dd x_{i}\, \dd x_{k}} - \Chr{ik}{r}^{*} \frac{\dd f}{\dd x_{r}}
\]
(the summation letter~$r$ also assumes only the three values $1, 2, 3$),
and if we also set
\[
\Delta f = \frac{\dd \vf}{\dd x_{i}}\qquad
\Delta f = \sqrt{\gamma} · f_{i}^{i})\Add{,}
\]
we arrive by an easy calculation at the following relations between
the components $R_{ik}$~and $\Rho_{ik}$ of the curvature tensor of the second
\PageSep{242}
order which belongs to the quadratic groundform~$ds^{2}$ for~$d\sigma^{2}$,
respectively
\begin{align*}
R_{ik} &= \Rho_{ik} - \frac{f_{ik}}{f}\Add{,} \\
R_{i0} &= R_{0i} = 0\Add{,} \\
R_{00} &= f · \frac{\Delta f}{\sqrt{\gamma}}\qquad
(\vR_{0}^{0} = \Delta f).
\end{align*}
For statical matter which is non-coherent (i.e.\ of which the parts
do not act on one another by means of stresses), $\vT_{0}^{0} = \mu$ is the only
component of the energy-density tensor that is not zero; hence
$\vT = \mu$. Matter at rest produces a statical gravitational field.
Among the gravitational equations~\Eq{(30)} the only one that is of
% [** TN: Ordinal]
interest to us is the~$\Chg{\dbinom{0}{0}}{\binom{0}{0}}$th: it gives us
\[
\Delta f = \tfrac{1}{2} \mu
\Tag{(32)}
\]
or, if we insert the constant factor of proportionality~$8\pi\kappa$, we get
\[
\Delta f = 4\pi \kappa \mu\Add{.}
\Tag{(32')}
\]
If we assume that, for an appropriate choice of the space-co-ordinates
$x_{1}$,~$x_{2}$,~$x_{3}$, $ds^{2}$~differs only by an infinitesimal amount from
\[
c^{2}\, dt^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})
\Tag{(33)}
\]
---the masses producing the gravitational field must be infinitely
small if this is to be true---we get, by setting
\[
f = c + \frac{\Phi}{c}\Add{,}
\Tag{(34)}
\]
that
\[
\Delta \Phi
= \frac{\dd^{2} \Phi}{\dd x_{1}^{2}}
+ \frac{\dd^{2} \Phi}{\dd x_{2}^{2}}
+ \frac{\dd^{2} \Phi}{\dd x_{3}^{2}}
= 4\pi \kappa c\mu\Add{,}
\Tag{(10)}
\]
and $\mu$~is $c$-times the mass-density in the ordinary units. We find
that actually, according to all our geometric observations, this
assumption is very approximately true for the planetary system.
Since the masses of the planets are very small compared with
the mass of the sun which produces the field and is to be considered
at rest, we may treat the former as ``test-bodies'' that are embedded
in the gravitational field of the sun. The motion of each of them
is then given by a geodetic world-line in this statical gravitational
field, if we neglect the disturbances due to the influence of the
planets on one another. The motion thus satisfies the principle of
variation
\[
\delta \int ds = 0\Add{,}
\]
\PageSep{243}
the ends of the portion of world-line remaining fixed. For the case
of rest, this gives us
\[
\delta \int \sqrt{f^{2} - v^{2}}\, dt = 0\Add{,}
\]
in which
\[
v^{2} = \left(\frac{d\sigma}{dt}\right)^{2}
= \sum_{i,k=1}^{3} \gamma_{ik}\, \frac{dx_{i}}{dt}\, \frac{dx_{k}}{dt}
\]
is the square of the velocity. This is a principle of variation of the
same form as that of classical mechanics; the ``Lagrange Function''
in this case is
\[
L = \sqrt{f^{2} - v^{2}}.
\]
If we make the same approximation as just above and notice that
in an infinitely weak gravitational field the velocities that occur will
\index{Gravitational!constant}%
\index{Gravitational!potential}%
also be infinitely small (in comparison with~$c$), we get
\[
\sqrt{f^{2} - v^{2}}
= \sqrt{c^{2} - 2\Phi - v^{2}}
= c + \frac{1}{c}(\Phi - \tfrac{1}{2} v^{2})\Add{,}
\]
and since we may now set
\[
v^{2} = \sum_{i,k=1}^{3} \left(\frac{dx_{i}}{dt}\right)^{2}
= \sum_{i} \dot{x}_{i}^{2}\Add{,}
\]
we arrive at
\[
\delta \int \left\{\tfrac{1}{2} \sum_{i} \dot{x}_{i}^{2} - \Phi\right\} dt = 0\Add{;}
\]
that is, the planet of mass~$m$ moves according to the laws of
classical mechanics, if we assume that a force with the potential~$m\Phi$
acts in it. \Emph{In this way we have linked up the theory with
that of Newton}: $\Phi$~is the Newtonian potential that satisfies
Poisson's equation~\Eq{(10)}, and $\Kappa = c^{2}\kappa$ is the gravitational constant of
Newton. From the well-known numerical value of the Newtonian
constant~$\Kappa$, we get for~$8\pi\kappa$ the numerical value
\[
8\pi\kappa = \frac{8\pi\Kappa}{c^{2}} = 1\Chg{,}{.}87 · 10^{-27} \text{cm} · \text{gr}^{-1}.
\]
The deviation of the metrical groundform from that of Euclid~\Eq{(33)}
is thus considerable enough to make the geodetic world-lines differ
from rectilinear uniform motion by the amount actually shown by
planetary motion---although the geometry which is valid in space
and is founded on~$d\sigma^{2}$ differs only very little from Euclidean
geometry as far as the dimensions of the planetary system are concerned.
(The sum of the angles in a geodetic triangle of these
dimensions differs very very slightly from~$180°$.) The chief cause
\PageSep{244}
of this is that the radius of the earth's orbit amounts to about eight
light-minutes whereas the time of revolution of the world in its
orbit is a whole year!
We shall pursue the exact theory of the motion of a point-mass
and of light-rays in a statical gravitational field a little further (\textit{vide}
\FNote{10}). According to §\,17 the geodetic world-lines may be
characterised by the two principles of variation
\[
\Squeeze{\delta \int \sqrt{Q}\, ds = 0
\quad\text{or}\quad
\delta \int Q\, ds = 0,
\quad
\text{in which }
Q = g_{ik}\, \frac{dx_{i}}{ds}\, \frac{dx_{k}}{ds}\Add{.}}
\Tag{(35)}
\]
The second of these takes for granted that the parameter~$s$ has
been chosen suitably. The second alone is of account for the
``null-lines'' which satisfy the condition $Q = 0$ and depict the
progress of a light-signal. The variation must be performed in
such a way that the ends of the piece of world-line under consideration
remain unchanged. If we subject only $x_{0} = t$ to
variation, we get in the statical case
\[
\delta \int Q\, ds
= \left[2f^{2}\, \frac{dx_{0}}{ds}\, \delta x_{0}\right]
- 2 \int \frac{d}{ds} \left(f^{2}\, \frac{dx_{0}}{ds}\right) \delta x_{0}\, ds\Add{.}
\Tag{(36)}
\]
Thus we find that
\[
f^{2}\, \frac{dx_{0}}{ds} = \text{const.\quad holds.}
\]
If, for the present, we keep our attention fixed on the case of the
light-ray, we can, by choosing the unit of measure of the parameter~$s$
appropriately ($s$~is standardised by the principle of variation itself
except for an arbitrary unit of measure), make the constant which
occurs on the right equal to unity. If we now carry out the
variation more generally by varying the spatial path of the ray
whilst keeping the ends fixed but dropping the subsidiary condition
imposed by time, namely, that $\delta x_{0} = 0$ for the ends, then, as is
evident from~\Eq{(36)}, the principle becomes
\[
\delta \int Q\, ds = 2[\delta t] = 2\delta \int dt.
\]
If the path after variation is, in particular, traversed with the
velocity of light just as the original path, then for the varied world-line,
too, we have
\[
Q = 0,\qquad
d\sigma = f\,dt\Add{,}
\]
and we get
\[
\delta \int dt = \delta \int \frac{d\sigma}{f} = 0\Add{.}
\Tag{(37)}
\]
This equation fixes only the spatial position of the light-ray; it is
nothing other than \Emph{Fermat's principle of the shortest path}. In
\index{Fermat's Principle}%
\PageSep{245}
\index{Curvature!light@{of light rays in a gravitational field}}%
the last formulation time has been eliminated entirely; it is valid
for any arbitrary portion of the path of the light-ray if the latter
\index{Light!ray!(curved in gravitational field)}%
alters its position by an infinitely small amount, its ends being kept
fixed.
If, for a statical field of gravitation, we use any space-co-ordinates
$x_{1}$,~$x_{2}$,~$x_{3}$, we may construct a graphical representation of
a Euclidean space by representing the point whose co-ordinates are
$x_{1}$,~$x_{2}$,~$x_{3}$ by means of a point whose Cartesian co-ordinates are
$x_{1}$,~$x_{2}$,~$x_{3}$. If we mark the position of two stars $S_{1}$,~$S_{2}$ which are at
rest and also an observer~$B$, who is at rest, in this picture-space,
then the angle at which the stars appear to the observer is not
equal to the angle between the straight lines $BS_{1}$,~$BS_{2}$ connecting
the stars with the observer; we must connect~$B$ with $S_{1}$,~$S_{2}$ by
means of the curved lines of shortest path resulting from~\Eq{(37)} and
then, by means of an auxiliary construction, transform the angle
which these two lines make with one another at~$B$ from Euclidean
measure to that of Riemann determined by the metrical groundform~$d\sigma^{2}$
(cf.\ formula~\Eq{(15)}, §\,11). The angles which have been
calculated in this way are those which determine the actually
observed position of the stars to one another, and which are read
off on the divided circle of the observing instrument. Whereas
$B$,~$S_{1}$,~$S_{2}$ retain their positions in space, this angle~$S_{1}BS_{2}$ may
change, if great masses happen to get into proximity of the path of
the rays. It is in this sense that we may talk of \Emph{light-rays being
curved as a result of the gravitational field}. But the rays are
not, as we assumed in §\,12 to get at general results, geodetic lines
in space with the metrical groundform~$d\sigma^{2}$; they do not make the
integral $\Dint d\sigma$ but $\displaystyle\int \dfrac{d\sigma}{f}$ assume a limiting value. The bending of
%[** TN: [sic] "occur"]
light-rays occur, in particular, in the gravitational field of the sun.
If for our graphical representation we use co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$,
for which the Euclidean formula $d\sigma^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}$ holds
at infinity, then numerical calculation for the case of a light-ray
passing by close to the sun shows that it must be diverted from its
path to the extent of $1.74$~seconds (\textit{vide} §\,31). This entails a displacement
of the positions of the stars in the apparent immediate
neighbourhood of the sun, which should certainly be measurable.
These positions of the stars can be observed, of course, only during
a total eclipse of the sun. The stars which come into consideration
must be sufficiently bright, as numerous as possible, and sufficiently
close to the sun to lead to a measurable effect, and yet sufficiently
far removed to avoid being masked by the brilliance of the corona.
The most favourable day for such an observation is the 29th~May,
\PageSep{246}
and it was a piece of great good fortune that a total eclipse
of the sun occurred on the 29th~May, 1919. Two English
expeditions were dispatched to the zone in which the total
eclipse was observable, one to Sobral in North Brazil, the
other to the Island of Principe in the Gulf of Guinea, for the
express purpose of ascertaining the presence or absence of the
Einstein displacement. The effect was found to be present to the
amount predicted; the final results of the measurements were
$1.98'' ± 0.12''$ for Sobral, $1.61'' ± 0.30''$ for Principe (\textit{vide} \FNote{11}).
Another optical effect which should present itself, according to
\index{Displacement current!towards red due to presence of great masses}%
\index{Red, displacement towards the}%
Einstein's theory of gravitation, in the statical field and which,
under favourable conditions, may just be observable, arises from
the relationship\Pagelabel{246}
\[
ds = f\, dt
\]
holding between the cosmic time~$dt$ and the proper-time~$ds$ at a
\index{Time}%
fixed point in space. If two sodium atoms at rest are objectively
fully alike, then the events that give rise to the light-waves of the
$D$-line in each must have the same frequency, as measured in
\Emph{proper-time}. Hence, if $f$~has the values $f_{1}$,~$f_{2}$, respectively at the
points at which the atoms are situated, then between $f_{1}$,~$f_{2}$ and the
frequencies $\nu_{1}$,~$\nu_{2}$ in cosmic time, there will exist the relationship
\[
\frac{\nu_{1}}{f_{1}} = \frac{\nu_{2}}{f_{2}}.
\]
But the light-waves emitted by an atom will have, of course, the
same frequency, measured in \Emph{cosmic} time, at all points in space
(for, in a \Emph{static} metrical field, Maxwell's equations have a solution
in which time is represented by the factor~$e^{i\nu t}$, $\nu$~being an arbitrary
\Emph{constant} frequency). Consequently, if we compare the
sodium $D$-line produced in a spectroscope by the light sent from a
star of great mass with the same line sent by an earth-source into
the same spectroscope, there should be a slight displacement of the
former line towards the red as compared with the latter, since $f$~has
a slightly smaller value in the neighbourhood of great masses
than at a great distance from them. The ratio in which the
frequency is reduced, has according to our approximate formula~\Eq{(34)}
the value $1 - \dfrac{\kappa m_{0}}{r}$ at the distance~$r$ from a mass~$m_{0}$. At
the surface of the sun this amounts to a displacement of $.008$~Angströms
for a line in the blue corresponding to the wave-length
$4000~\text{Å}$. This effect lies just within the limits of observability.
Superimposed on this, there are the disturbances due to the Doppler
effect, the uncertainty of the means used for comparison on the
\PageSep{247}
earth, certain irregular fluctuations in the sun's lines the causes of
which have been explained only partly, and finally, the mutual
disturbances of the densely packed lines of the sun owing to the
overlapping of their intensities (which, under certain circumstances,
causes two lines to merge into one with a single maximum of intensity).
If all these factors are taken into consideration, the
observations that have so far been made, seem to confirm the displacement
towards the red to the amount stated (\textit{vide} \FNote{12}).
This question cannot, however, yet be considered as having been
definitely answered.
A third possibility of controlling the theory by means of experiment
\index{Perihelion, motion of Mercury's}%
is this. According to Einstein, Newton's theory of the
planets is only a first approximation. The question suggests itself
whether the divergence between Einstein's Theory and the latter
are sufficiently great to be detected by the means at our disposal.
It is clear that the chances for this are most favourable for the
planet Mercury which is nearest the sun. In actual fact, after
Einstein had carried the approximation a step further, and after
Schwarzschild (\textit{vide} \FNote{13}) had determined accurately the radially
symmetrical field of gravitation produced by a mass at rest and
also the path of a point-mass of infinitesimal mass, both found that
the \Emph{elliptical orbit of Mercury should undergo a slow rotation
in the same direction as the orbit is traversed} (over and above
the disturbances produced by the remaining planets), \Emph{amounting
to $43''$~per century}. Since the time of Leverrier an effect of this
magnitude has been known among the secular disturbances of
Mercury's perihelion, which could not be accounted for by the
usual causes of disturbance. Manifold hypotheses have been proposed
to remove this discrepancy between theory and observation
(\textit{vide} \FNote{14}). We shall revert to the rigorous solution given by
Schwarzschild in §\,31.
Thus we see that, however great is the revolution produced in
our ideas of space and time by Einstein's theory of gravitation, the
actual deviations from the old theory are exceedingly small in our
field of observation. Those which are measurable have been confirmed
up to now. The chief support of the theory is to be found
less in that lent by observation hitherto than in its inherent logical
consistency, in which it far transcends that of classical mechanics,
and also in the fact that it solves the perplexing problem of gravitation
and of the relativity of motion at one stroke in a manner
highly satisfying to our reason.
Using the same method as for the light-ray, we may set up
for the motion of a point-mass in a statical gravitational field a
\PageSep{248}
``minimum'' principle affecting only the path in space, corresponding
to Fermat's principle of the shortest path. If $s$~is the
parameter of proper-time, then\Typo{,}{}
\[
Q = 1,\quad\text{and}\quad
f^{2}\, \frac{dt}{ds} = \text{const.} = \frac{1}{E}
\Tag{(38)}
\]
is the energy-integral. We now apply the first of the two principles
of variation~\Eq{(35)} and generalise it as above by varying the spatial
path quite arbitrarily while keeping the ends, $x_{0} = t$, fixed. We get
\[
\delta \int \sqrt{Q}\, ds
= \left[\frac{1}{E}\, \delta t\right]
= \delta \int \frac{dt}{E}\Add{.}
\Tag{(39)}
\]
To eliminate the proper-time we divide the first of the equations~\Eq{(38)}
by the square of the second; the result is
\[
\frac{1}{f^{4}} \left\{f^{2} - \left(\frac{d\sigma}{dt}\right)^{2}\right\} = E^{2}\qquad
d\sigma = f^{2} \sqrt{U}\, dt\Add{,}
\Tag{(40)}
\]
in which
\[
U = \frac{1}{f^{2}} - E^{2}.
\]
\Eq{(40)}~is the law of velocity according to which the point-mass
traverses its path. If we perform the variation so that the varied
path is traversed according to the same law with the same constant~$E$,
it follows from~\Eq{(39)}\Typo{,}{} that
\[
\Squeeze[0.975]{\delta \int \frac{dt}{E}
= \delta \int \sqrt{f^{2} - \left(\frac{d\sigma}{dt}\right)^{2}}\, dt
= \delta \int Ef^{2}\, dt
\quad\text{i.e.}\
\delta \int f^{2} U\, dt = 0}
\]
or, finally, by expressing $dt$ in terms of the spatial element of arc~$d\sigma$,
and thus eliminating the time entirely, we get
\[
\delta \int \sqrt{U}\, d\sigma = 0.
\]
The path of the point-mass having been determined in this way,
we get as a relation giving the time of the motion in this path,
from~\Eq{(40)}, that
\[
dt = \frac{d\sigma}{f^{2} \sqrt{U}}.
\]
For $E = 0$, we again get the laws for the light-ray.
\index{Static!gravitational field|)}%
\Section{30.}{Gravitational Waves}
\index{Gravitational!waves|(}%
By assuming that the generating energy-field~$\vT_{i}^{k}$ is infinitely
weak, Einstein has succeeded in integrating the gravitational
equations generally (\textit{vide} \FNote{15}). The~$g_{ik}$'s will, under these
circumstances, if the co-ordinates are suitably chosen, differ from
\PageSep{249}
the~$\go_{ik}$'s by only infinitesimal amounts~$\gamma_{ik}$. We then regard the
world as ``Euclidean,'' having the metrical groundform
\[
\go_{ik}\, dx_{i}\, dx_{k}
\Tag{(41)}
\]
and the~$\gamma_{ik}$'s as the components of a symmetrical tensor-field of
the second order in this world. The operations that are to be performed
in the sequel will always be based on the metrical groundform~\Eq{(41)}.
For the present we are again dealing with the special
theory of relativity. We shall consider the co-ordinate system
which is chosen to be a ``normal'' one, so that $\go_{ik} = 0$ for $i \neq k$ and
\[
g_{00} = 1,\qquad
\go_{11} = \go_{22} = \go_{33} = -1.
\]
$x_{0}$~is the time, $x_{1}$,~$x_{2}$,~$x_{3}$ are Cartesian space-co-ordinates; the velocity
of light is taken equal to unity.
We introduce the quantities
\[
\psi_{i}^{k} = \gamma_{i}^{k} - \gamma \delta_{i}^{k}\Typo{,}{}
\qquad (\gamma = \tfrac{1}{2} \gamma_{i}^{i})\Add{,}
\]
and we next assert that we may without loss of generality set
\[
\frac{\dd \psi_{i}^{k}}{\dd x_{k}} = 0\Add{.}
\Tag{(42)}
\]
For, if this is not so initially, we may, by an infinitesimal change,
alter the co-ordinate system so that \Eq{(42)}~holds. The transformation
formulæ that lead to a new co-ordinate system~$\bar{x}$, namely,
\[
\Typo{x}{\bar{x}}_{i} = x_{i} + \xi(x_{0}\Com x_{1}\Com x_{2}\Com x_{3})
\]
contain the unknown functions~$\xi^{i}$, which are of the same order of
infinitesimals as the~$\gamma$'s. We get new co-efficients~$\bar{g}_{ik}$ for which,
according to earlier formulæ, we must have
\[
g_{ik}(x) - \bar{g}_{ik}(x)
= g_{ir}\, \frac{\dd \xi^{r}}{\dd \Typo{\xi_{k}}{x_{k}}}
+ g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}
+ \frac{\dd g_{ik}}{\dd x_{r}}\, \xi^{r}
\]
so that, here, we have
\[
\gamma_{ik}(x) - \bar{\gamma}_{ik}(x)
= \frac{\dd \xi_{i}}{\dd x_{k}} + \frac{\dd \xi_{k}}{\dd x_{i}},\qquad
\gamma(x) - \bar{\gamma}(x) = \frac{\dd \xi^{i}}{\dd x_{i}} = \Xi\Add{,}
\]
and we finally get
\[
\frac{\dd \gamma_{i}^{k}}{\dd x_{k}} - \frac{\dd \bar{\gamma}_{i}^{k}}{\dd x_{k}}
= \nabla \xi_{i} + \frac{\dd \Xi}{\dd x_{i}},\qquad
\frac{\dd \gamma}{\dd x_{i}} - \frac{\dd \bar{\gamma}}{\dd x_{i}}
= \frac{\dd \Xi}{\dd x_{i}}\Add{,}
\]
in which $\nabla$~denotes, for an arbitrary function, the differential
operator
\[
\nabla f = \frac{\dd}{\dd x_{i}} \left(\go_{ik}\, \frac{\dd f}{\dd x_{k}}\right)
= \frac{\dd^{2} f}{\dd x_{0}^{2}}
- \left(\frac{\dd^{2} f}{\dd x_{1}^{2}}
+ \frac{\dd^{2} f}{\dd x_{2}^{2}}
+ \frac{\dd^{2} f}{\dd x_{3}^{2}}\right).
\]
\PageSep{250}
The desired condition will therefore be fulfilled in the new
\index{Potential!retarded}%
\index{Retarded potential}%
co-ordinate system if the~$\xi^{i}$'s are determined from the equations
\[
\nabla \xi^{i} = \frac{\dd \psi_{i}^{k}}{\dd x_{k}}\Add{,}
\]
which may be solved by means of retarded potentials (cf.\ Chapter~III,
\Pageref{165}). If the linear Lorentz transformations are discarded,
the co-ordinate system is defined not only to the first order of
small quantities but also to the second. It is very remarkable
that such an invariant normalisation is possible.
We now calculate the components~$R_{ik}$ of curvature. As the
field-quantities $\dChr{ik}{r}$ are infinitesimal, we get, by confining ourselves
to terms of the first order
\[
R_{ik} = \frac{\dd}{\dd x_{r}} \Chr{ik}{r} - \frac{\dd}{\dd x_{k}} \Chr{ir}{r}.
\]
Now,
\[
\Chrsq{ik}{r}
= \tfrac{1}{2} \left(\frac{\dd \gamma_{ir}}{\dd x_{k}}
+ \frac{\dd \gamma_{kr}}{\dd x_{i}}
- \frac{\dd \gamma_{ik}}{\dd x_{r}}\right)\Add{,}
\]
hence
\[
\Chr{ik}{r}
= \tfrac{1}{2} \left(\frac{\dd \gamma_{i}^{r}}{\dd x_{k}}
+ \frac{\dd \gamma_{k}^{r}}{\dd x_{i}}
- \go_{rs}\, \frac{\dd \gamma_{ik}}{\dd x_{s}}\right).
\]
Taking into account equations~\Eq{(42)} or
\[
\frac{\dd \gamma_{i}^{k}}{\dd x_{k}} = \frac{\dd \gamma}{\dd x_{i}}\Add{,}
\]
we get
\[
\frac{\dd}{\dd x_{r}} \Chr{ik}{r}
= \frac{\dd^{2} \gamma}{\dd x_{i}\, \dd x_{k}}
- \tfrac{1}{2} \nabla \gamma_{ik}.
\]
In the same way we obtain
\[
\frac{\dd}{\dd x_{k}} \Chr{ir}{r}
= \frac{\dd^{2} \gamma}{\dd x_{i}\, \dd x_{k}}.
\]
The result is
\[
R_{ik} = -\tfrac{1}{2} \nabla \gamma_{ik}.
\]
Consequently, $R = -\nabla \gamma$ and
\[
R_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} R
= -\tfrac{1}{2} \nabla \psi_{i}^{k}.
\]
The gravitational equations are, however,
\[
\tfrac{1}{2} \nabla \psi_{i}^{k} = -T_{i}^{k}\Add{,}
\Tag{(43)}
\]
and may be directly integrated with the help of retarded potentials
(cf.\ \Pageref{165}). Using the same notation, we get
\[
\psi_{i}^{k} = -\int \frac{T_{i}^{k}(t - r)}{2\pi r}\, dV.
\]
\PageSep{251}
Accordingly, \emph{every change in the distribution of matter produces a
gravitational effect which is propagated in space with the velocity of
\index{Velocity!gravitation@{of propagation of gravitation}}%
light}. Oscillating masses produce gravitational waves. Nowhere in
the Nature accessible to us do mass-oscillations of sufficient power
occur to allow the resulting gravitational waves to be observed.
Equations~\Eq{(43)} correspond fully to the electromagnetic equations
\[
\nabla \phi^{i} = s^{i}
\]
and, just as the potentials~$\phi^{i}$ of the electric field had to satisfy
the secondary condition
\[
\frac{\dd \phi^{i}}{\dd x_{i}} = 0
\]
because the current~$s^{i}$ fulfils the condition
\[
\frac{\dd s^{i}}{\dd x_{i}} = 0\Add{,}
\]
so we had here to introduce the secondary conditions~\Eq{(42)} for the
system of gravitational potentials~$\psi_{i}^{k}$, because they hold for the
matter-tensor
\[
\frac{\dd T_{i}^{k}}{\dd x_{k}} = 0.
\]
\Emph{Plane gravitational waves} may exist: they are propagated
in space free from matter: we get them by making the same
supposition as in optics, i.e.\ by setting
\[
\psi_{i}^{k}
= a_{i}^{k} · e^{(\alpha_{0} x_{0} + \alpha_{1} x_{1} + \alpha_{2} x_{2} + \alpha_{3} x_{3})\sqrt{-1}}.
\]
The~$a_{i}^{k}$'s and the~$\alpha_{i}$'s are constants; the latter satisfy the condition
$\alpha_{i} \alpha^{i} = 0$. Moreover, $\alpha_{0} = \nu$ is the frequency of the vibration and
$\alpha_{1} x_{1} + \alpha_{2} x_{2} + \alpha_{3} x_{3} = \text{const.}$ are the planes of constant phase. The
differential equations $\nabla \psi_{i}^{k} = 0$ are satisfied identically. The
secondary conditions~\Eq{(42)} require that
\[
a_{i}^{k} \alpha_{k} = 0\Add{.}
\Tag{(44)}
\]
If the $x_{1}$-axis is the direction of propagation of the wave, we have
\index{Propagation!of gravitational disturbances}%
\[
\alpha_{2} = \alpha_{3} = 0,\qquad
-\alpha_{1} = \alpha_{0} = \nu\Add{,}
\]
and equations~\Eq{(44)} state that
\[
a_{i}^{0} = a_{i}^{1}
\quad\text{or}\quad
a_{0i} = -a_{1i}\Add{.}
\Tag{(45)}
\]
Accordingly, it is sufficient to specify the space part of the constant
symmetrical tensor~$a$, namely,
\[
\left\lVert\begin{array}{@{}ccc@{}}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} \\
\end{array}\right\rVert
\]
\PageSep{252}
since the~$a$'s with the index~$0$ are determined from these by~\Eq{(45)};
the space part, however, is subject to no limitation. In its turn it
splits up into the three summands in the direction of propagation
of the waves:
\[
\left\lVert\begin{array}{@{}ccc@{}}
a_{11} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{array}\right\rVert
+ \left\lVert\begin{array}{@{}ccc@{}}
0 & a_{12} & a_{13} \\
a_{21} & 0 & 0 \\
a_{31} & 0 & 0 \\
\end{array}\right\rVert
+ \left\lVert\begin{array}{@{}ccc@{}}
0 & 0 & 0 \\
0 & a_{22} & a_{23} \\
0 & a_{32} & a_{33} \\
\end{array}\right\rVert\Add{.}
\]
The tensor-vibration may hence be resolved into three independent
components: a longitudinal-longitudinal, a longitudinal-transverse,
and a transverse-transverse wave.
H.~Thirring has made two interesting applications of integration
based on the method of approximation used here for the
gravitational equations (\textit{vide} \FNote{16}). With its help he has investigated
the influence of the rotation of a large, heavy, hollow
sphere on the motion of point-masses situated near the centre of
the sphere. He discovered, as was to be expected, a force effect
of the same kind as centrifugal force. In addition to this a second
force appears which seeks to drag the body into the \Chg{æquatorial}{equatorial}
plane according to the same law as that according to which centrifugal
force seeks to drive it away from the axis. Secondly (in
conjunction with J.~Lense), he has studied the influence of the
rotation of a central body on its planets or moons, respectively. In
the case of the fifth moon of Jupiter, the disturbance caused attains
an amount that may make it possible to compare theory with
observation.
Now that we have considered in §§\,29,~30 the approximate
integration of the gravitational equations that occur if only linear
terms are taken into account, we shall next endeavour to arrive at
rigorous solutions: our attention will, however, be confined to
statical gravitation.
\Section[Rigorous Solution of the Problem of One Body]
{31.}{Rigorous Solution of the Problem of One Body\protect\footnotemark}
\footnotetext{\textit{Vide} \Chg{note~(17)}{\FNote{17}}.}
For a statical gravitational field we have
\index{Gravitational!waves|)}%
\index{Radial symmetry}%
\[
ds^{2}= f^{2}\, dx_{0}^{2} - d\sigma^{2}
\]
in which $d\sigma^{2}$~is a definitely positive quadratic form in the three-space
variables $x_{1}$,~$x_{2}$,~$x_{3}$; the velocity of light~$f$ is likewise dependent
only on these. The field is \Emph{radially symmetrical} if, for
a proper choice of the space-co-ordinates, $f$~and~$d\sigma^{2}$ are invariant
with respect to linear orthogonal transformations of these co-ordinates.
\PageSep{253}
If this is to be the case, $f$~must be a function of the
distance
\[
r = \sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}},
\]
from the centre, but $d\sigma^{2}$~must have the form
\[
\lambda(dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})
+ l(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})^{2}
\Tag{(46)}
\]
in which $\lambda$~and~$l$ are likewise functions of $r$~alone. Without disturbing
this normal form we may subject the space-co-ordinates to
a further transformation which consists in replacing $x_{1}$,~$x_{2}$,~$x_{3}$ by
$\tau x_{1}$,~$\tau x_{2}$,~$\tau x_{3}$, the factor of proportionality~$\tau$ being an arbitrary
function of the distance~$r$. By choosing $\lambda$~appropriately we may
clearly succeed in getting $\lambda = 1$; let us suppose this to have been
done. Then, using the notation of §\,29, we have
\[
\gamma_{ik} = -g_{ik} = \delta_{i}^{k} + l · x_{i} x_{k}
\qquad (i, k = 1, 2, 3).
\]
We shall next define this radially symmetrical field so that
it satisfies the homogeneous gravitational equations which hold
wherever there is no matter, that is, wherever the energy-density~$\vT_{i}^{k}$
vanishes. These equations are all included in the principle of
variation
\[
\delta \int \vG\, dx = 0.
\]
\Emph{The gravitational field}, which we are seeking, \Emph{is that which is
produced by statical masses which are distributed about
the centre with radial symmetry.} If the accent signify differentiation
with respect to~$r$, we get
\[
\frac{\dd \gamma_{ik}}{\dd x_{\alpha}}
= l' \frac{x_{\alpha}}{r} x_{i} x_{k}
+ l(\delta_{i}^{\alpha} x_{k} + \delta_{k}^{\alpha} x_{i})\Add{,}
\]
and hence
\[
-\Chrsq{ik}{\alpha}
= \tfrac{1}{2} \frac{x_{\alpha}}{r}\, l' x_{i} x_{k} + l \delta_{i}^{k} x_{\alpha}
\qquad (i, k, \alpha = 1, 2, 3).
\]
Since it follows from
\[
x_{\alpha} = \sum_{\beta=1}^{3} \gamma_{\alpha\beta} x^{\beta}
\]
that
\[
x_{\alpha} = \frac{1}{h^{2}} x_{\alpha}
\quad\text{and}\quad
h^{2} = 1 + lr^{2},
\]
as may be verified by direct substitution, we must have
\[
\Chr{ik}{\alpha}
= \tfrac{1}{2}\, \frac{x_{\alpha}}{r}\,
\frac{l' x_{i} x_{k} + 2lr\delta_{i}^{k})}{h^{2}}.
\]
\PageSep{254}
\index{Problem of one body}%
It is sufficient to carry out the calculation of~$\vG$ for the point
$x_{1} = r$, $x_{2} = 0$, $x_{3} = 0$. At this point, we get for the three-indices
symbols just calculated:
\[
\Chr{11}{1} = \frac{h'}{h}
\quad\text{and}\quad
\Chr{22}{1} = \Chr{33}{1} = \frac{lr}{h^{2}}\Add{,}
\]
whereas the remaining ones are equal to zero. Of the three-indices
symbols containing~$0$, we find by §\,29 that
\[
\Chr{10}{0} = \Chr{01}{0} = \frac{f'}{f}
\quad\text{and}\quad
\Chr{00}{1} = \frac{f\!f'}{h^{2}}\Add{,}
\]
whereas all the others $= 0$. Of the~$g_{ik}$'s all those situated in the
main diagonal ($i = k$) are equal, respectively, to
\[
f^{2},\quad
-h^{2},\quad
-1,\quad
-1
\]
whereas the lateral ones all vanish. Hence definition~\Eq{(31)} of~$\vG$
gives us
\begin{gather*}
-\frac{2}{\sqrt{g}} \vG = \\
\begin{array}{@{}r|l@{}}
\dfrac{1}{f^{2}}
& \dChr{00}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right) - 2\dChr{01}{0} \dChr{00}{1} \\
%
-\dfrac{1}{h^{2}}
& \dChr{11}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right) - \dChr{10}{0} \dChr{10}{0} - \dChr{11}{1} \dChr{11}{1} \\
-1 & \dChr{22}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right) \\
-1 & \dChr{33}{1} \left(\dChr{10}{0} + \dChr{11}{1}\right). \\
\end{array}
\end{gather*}
The terms in the first and second row taken together lead to
\[
\left(\Chr{11}{1} - \Chr{10}{0}\right)
\left(\frac{1}{f^{2}} \Chr{00}{1} - \frac{1}{h^{2}} \Chr{10}{0}\right).
\]
The second factor in this product, however, is equal to zero.
Since, by~\Eq{(57)} §\,17
\[
\sum_{i=0}^{3} \Chr{1i}{i} = \frac{\Delta'}{\Delta}
\qquad (\Delta = \sqrt{g} = hf)\Add{,}
\]
the sum of the terms in the third and fourth row is equal to
\[
-\frac{2lr}{h^{2}} · \frac{\Delta'}{\Delta}.
\]
If we wish to take the world-integral~$\vG$ over a fixed interval with
respect to the time~$x_{0}$, and over a shell enclosed by two spherical
surfaces with respect to space, then, since the element of integration
is
\[
dx = dx_{0} · d\Omega · r^{2}\, dr
\qquad (d\Omega = \text{solid angle}),
\]
\PageSep{255}
the equation of variation that is to be solved is
\[
\delta \int \vG r^{2}\, dr = 0.
\]
Hence, if we set
\[
\frac{lr^{3}}{h^{2}}
= \frac{lr^{3}}{1 + lr^{2}}
= \left(1 - \frac{1}{h^{2}}\right) r
= w\Add{,}
\]
we get
\[
\delta \int w \Delta'\, dr = 0
\]
in which $\Delta$~and~$w$ may be regarded as the two functions that may
be varied arbitrarily.
By varying~$w$, we get
\[
\Delta' = 0,\qquad
\Delta = \text{const.}
\]
and hence, if we choose the unit of time suitably
\[
\Delta = hf = 1.
\]
Partial integration gives
\[
\int w \Delta'\, dr = [w\Com \Delta] - \int \Delta w'\, dr.
\]
Hence, if we vary~$\Delta$, we arrive at
\[
w' = 0,\qquad
w = \text{const.} = 2m.
\]
Finally, from the definition of $w$~and~$\Delta = 1$, we get
\[
\framebox{$f^{2} = 1 - \dfrac{2m}{r}$,\qquad $h^{2} = \dfrac{1}{f^{2}}$}
\]
This completes the solution of the problem. The unit of time has
been chosen so that the velocity of light at infinity $ = 1$. For
distances~$r$, which are great compared with~$m$, the Newtonian
value of the potential holds in the sense that the quantity~$m_{0}$,
introduced by the equation $m = \kappa m_{0}$ occurs as the \Emph{field-producing
mass} in it; we call~$m$ the \Emph{gravitational radius} of the matter
\index{Gravitational!radius of a great mass}%
causing the disturbance of the field. Since $4\pi m$~is the flux of the
spatial vector-density~$\vf^{i}$ through an arbitrary sphere enclosing the
masses, we get, from~\Eq{(32')}, for discrete or non-coherent mass
\[
m_{0} = \int \mu\, dx_{1}\, dx_{2}\, dx_{3}.
\]
Since $f^{2}$~cannot become negative, it is clear from this that, if we use
the co-ordinates here introduced for the region of space devoid of
matter, $r$~must be~$> 2m$. Further light is shed on this by the
special case of a sphere of liquid which is to be discussed in §\,32,
and for which the gravitational field \emph{inside} the mass, too, will be
determined. We may apply the solution found to the gravitational
\PageSep{256}
field of the sum external to itself if we neglect the effect due to the
planets and the distant stars. The gravitational radius is about
$1.47$~kilometres for the sun's mass, and only $5$~millimetres for the
earth.
The motion of a planet (supposed infinitesimal in comparison
\index{Planetary motion}%
with the sun's mass) is represented by a geodetic world-line. Of
its four equations
\[
\frac{d^{2} x_{i}}{ds^{2}}
+ \Chr{\alpha\beta}{i} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0\Add{,}
\]
the one corresponding to the index $i = 0$ gives, for the statical
gravitational field, the energy-integral
\[
f^{2}\, \frac{dx_{0}}{ds} = \text{const.}
\]
as we saw above; or, since,
\[
\left(f\, \frac{dx_{0}}{ds}\right)^{2} = 1 + \left(\frac{d\sigma}{ds}\right)^{2}\Add{,}
\]
we get
\[
f^{2} \left[1 + \left(\frac{d\sigma}{ds}\right)^{2}\right] = \text{const.}
\]
In the case of a radially symmetrical field the equations corresponding
to the indices $i = 1, 2, 3$ give the proportion
\[
\frac{d^{2} x_{1}}{ds^{2}}
: \frac{d^{2} x_{2}}{ds^{2}}
: \frac{d^{2} x_{3}}{ds^{2}}
= x_{1} : x_{2} : x_{3}
\]
(this is readily seen from the three-indices symbols that are written
down). And from them, there results, in the ordinary way, the
three equations which express the Law of Areas
%[** TN: First two equations omitted in the original]
\[
%x_{2}\, \frac{dx_{3}}{ds} - x_{3}\, \frac{dx_{2}}{ds} = \text{const.},\qquad
%x_{3}\, \frac{dx_{1}}{ds} - x_{1}\, \frac{dx_{3}}{ds} = \text{const.},\qquad
\makebox[1.5in][c]{\dotfill,}\qquad
x_{1}\, \frac{dx_{2}}{ds} - x_{2}\, \frac{dx_{1}}{ds} = \text{const.}
\]
This theorem differs from the similar one derived in Newton's
Theory, in that the differentiations are made, not according to
cosmic time, but according to the proper-time~$s$ of the planet. On
account of the Law of Areas the motion takes place in a plane
that we may choose as our co-ordinate plane $x_{3} = 0$. If we
introduce polar co-ordinates into it, namely
\[
x_{1} = r\cos \phi,\qquad
x_{2} = r\sin \phi\Add{,}
\]
the integral of the area is
\[
r^{2}\, \frac{d\phi}{ds} = \text{const.} = b\Add{.}
\Tag{(47)}
\]
The energy-integral, however, since
\begin{gather*}
dx_{1}^{2} + dx_{2}^{2} = dr^{2} + r^{2}\, d\phi^{2},\qquad
x_{1}\, dx_{1} + x_{2}\, dx_{2} = r\, dr\Add{,} \\
d\sigma^{2} = (dr^{2} + r^{2}\, d\phi^{2}) + l(r\, dr)^{2}
= h^{2}\, dr^{2} + r^{2}\, d\phi^{2}\Add{,}
\end{gather*}
\PageSep{257}
becomes
\[
f^{2} \left\{1 + h^{2} \left(\frac{dr}{ds}\right)^{2}
+ r^{2} \left(\frac{d\phi}{ds}\right)^{2}
\right\} = \text{const.}
\]
\Typo{since}{Since} $fh = 1$, we get, by substituting for~$f^{2}$ its value, that
\[
-\frac{2m}{r} + \Typo{\left(\frac{dr^{2}}{ds}\right)}{\left(\frac{dr}{ds}\right)^{2}}
+ r(r - 2m)\left(\frac{d\phi}{ds}\right)^{2} = -E = \text{const.}
\Tag{(48)}
\]
Compared with the energy-equation of Newton's Theory this
equation differs from it only in having $r - 2m$ in place of~$r$ in the
last term of the left-hand side.
The succeeding steps are the same as those of Newton's Theory.
We substitute $\dfrac{d\phi}{ds}$ from~\Eq{(47)} into~\Eq{(48)}, getting
\[
\left(\frac{dr}{ds}\right)^{2}
= \frac{2m}{r} - E - \frac{b^{2} (r - 2m)}{r^{3}},
\]
or, using the reciprocal distance $\rho = \dfrac{1}{r}$ in place of~$r$,
\[
\left(\frac{d\rho}{\rho^{2}\, ds}\right)^{2}
= 2m\rho - E - b^{2} \rho^{2} (1 - 2m\rho).
\]
To arrive at the orbit of the planet we eliminate the proper-time
by dividing this equation by the square of~\Eq{(47)}, thus
\[
\left(\frac{d\rho}{d\phi}\right)^{2}
= \frac{2m}{b^{2}} \rho - \frac{E}{b^{2}} - \rho^{2} + 2m\rho^{3}.
\]
In Newton's Theory the last term on the right is absent. Taking
into account the numerical conditions that are presented in the case
of planets, we find that the polynomial of the third degree in~$\rho$ on
the right has three positive roots $\rho_{0} > \rho_{1} > \rho_{2}$ and hence
\[
= 2m(\rho_{0} - \rho) (\rho_{1} - \rho) (\rho - \rho_{2})\Add{;}
\]
$\rho$~assumes values ranging between $\rho_{1}$~and~$\rho_{2}$. The root~$\rho_{0}$ is very
great in comparison with the remaining two. As in Newton's
Theory, we set
\[
\frac{1}{\rho_{1}} = a(1 - e)\Add{,}\qquad
\frac{1}{\rho_{2}} = a(1 + e)\Add{,}
\]
and call $a$~the semi-major axis and $e$~the eccentricity. We then
get
\[
\rho_{1} + \rho_{2} = \frac{2}{a(1 - e^{2})}.
\]
If we compare the co-efficients of~$\rho^{2}$ with one another, we find that
\[
\rho_{0} + \rho_{1} + \rho_{2} = \frac{1}{2m}.
\]
$\phi$~is expressed in terms of~$\rho$ by an elliptic integral of the first kind
and hence, conversely, $\rho$~is an elliptic function of~$\phi$. The motion
\PageSep{258}
is of precisely the same type as that executed by the spherical
pendulum. To arrive at simple formulæ of approximation, we
make the same substitution as that used to determine the Kepler
orbit in the Newtonian Theory, namely
\[
\rho - \frac{\rho_{1} + \rho_{2}}{2} + \frac{\rho_{1} - \rho_{2}}{2}\cos\theta.
\]
Then
\[
\phi \Typo{-}{=} \int \frac{d\theta}
{\sqrt{2m \left(\rho_{0}
- \dfrac{\rho_{1} + \rho_{2}}{2}
- \dfrac{\rho_{1} - \rho_{2}}{2}\cos\theta
\right)}}\Add{.}
\Tag{(49)}
\]
The perihelion is characterised by the values $\theta = 0, 2\pi,~\dots$. The
increase of the azimuth~$\phi$ after a full revolution from perihelion to
perihelion is furnished by the above integral, taken between the
limits $0$ and~$2\pi$. With easily sufficient accuracy this increase may
be set
\[
= \frac{2\pi}{\sqrt{2m \left(\rho_{0} - \dfrac{\rho_{1} + \rho_{2}}{2}\right)}}\Add{.}
\]
We find, however, that
\[
\rho_{0} + \frac{\rho_{1} + \rho_{2}}{2}
= (\rho_{0} + \rho_{1} + \rho_{2}) - \tfrac{3}{2}(\rho_{1} + \rho_{2})
= \frac{1}{2m} - \frac{3}{a(1 - e^{2})}.
\]
Consequently the above increase (of azimuth)
\[
= \frac{2\pi}{\sqrt{1 - \dfrac{6m}{a(1 - e^{2})}}}
\sim 2\pi \left\{1 + \frac{3m}{a(1 - e^{2})}\right\}\Add{,}
\]
and \Emph{the advance of the perihelion per revolution}
\[
= \frac{6\pi m}{a(1 - e^{2})}.
\]
In addition, $m$, the gravitational radius of the sun may be expressed
according to Kepler's third law, in terms of the time of revolution~$T$
of the planet and the semi-major axis~$a$, thus
\[
m = \frac{4\pi^{2} a^{3}}{c^{2} T^{2}}.
\]
Using the most delicate means at their disposal, astronomers have
hitherto been able to establish the existence of this advance of the
perihelion only in the case of Mercury, the planet nearest the sun
(\textit{vide} \FNote{18}).
Formula~\Eq{(49)} also gives the deflection~$\alpha$ of the path of a ray of light.
If $\theta_{0} = \dfrac{\pi}{2} + \epsilon$ is the angle~$\theta$ for which $\rho = 0$, then the value of the
\PageSep{259}
integral, taken between $-\theta_{0}$ and $+\theta_{0} = \pi + \alpha$. Now in the
present case
\[
2m(\rho_{0} - \rho) (\rho_{1} - \rho) (\rho - \rho_{2})
= \frac{1}{b^{2}} - \rho^{2} + 2m \rho^{3}.
\]
The values of~$\rho$ fluctuate between $0$~and~$\rho_{2}$. Moreover, $\dfrac{1}{\rho_{1}} = r$ is the
nearest distance to which the light-ray approaches the centre of
mass~$O$, whilst $b$~is the distance of the two asymptotes of the light-ray
from~$O$ (for in the case of any curve, this distance is given by
the value of~$\dfrac{d\phi}{d\rho}$ for $\rho = 0$). Now,
\[
2m(\rho_{0} + \rho_{1} + \rho_{2}) = 1
\]
is accurately true. If $\dfrac{m}{b}$~is a small fraction, we get to a first
degree of approximation that
\begin{gather*}
m\rho_{1} = -m\rho_{2} = \frac{m}{b}\Add{,}\qquad
\frac{m}{2}(\rho_{1} + \rho_{2}) = \left(\frac{m}{b}\right)^{2}\Add{,}\qquad
\epsilon = \frac{m}{b}\Add{,} \\
\alpha = \int_{-\theta_{0}}^{\theta_{0}} (1 + \frac{m}{b}\cos\theta)\, d\theta - \pi
= 2\epsilon + \frac{2m}{b}
\quad\text{and hence}\quad
\framebox{$\alpha = \dfrac{4m}{b}$}
\end{gather*}
If we calculate the path of the light-ray according to Newton's
Theory, taking into account the gravitation of light, that is, considering
it as the path of a body that has the velocity~$c$ at infinity, then if we
set
\[
\frac{1}{b^{2}} + \frac{2m}{b^{2}}\, \rho - \rho^{2}
= (\rho_{1} - \rho) (\rho - \rho_{2})
\]
in which $\rho_{1} > 0$, $\rho_{2} < 0$ and set
\[
\cos\theta_{0} = -\frac{\rho_{1} + \rho_{2}}{\rho_{1} - \rho_{2}}\Add{,}
\]
we get
\[
\pi + \alpha = 2\theta_{0}\Add{,}\qquad
\alpha \sim \frac{2m}{b}.
\]
Thus Newton's law of attraction leads to a deflection which is only
half as great as that predicted by Einstein. The observations
made at Sobral and Principe decide the question definitely in
favour of Einstein (\textit{vide} \FNote{19}).
\Section{32.}{Additional Rigorous Solutions of the Statical Problem
of Gravitation}
In a Euclidean space with Cartesian co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, the
equation of a surface of revolution having as its axis of rotation the
$x_{3}$-axis is
\[
x_{3} = F(r),\qquad
r = \sqrt{x_{1}^{2} + x_{2}^{2}}.
\]
\PageSep{260}
On it, the square of the distance~$d\sigma$ between two infinitely near
points is
\begin{align*}
d\sigma^{2}
&= (dx_{1}^{2} + dx_{2}^{2}) + \bigl(F'(r)\bigr)^{2}\, dr^{2} \\
&= (dx_{1}^{2} + dx_{2}^{2}) + \left(\frac{F'(r)}{r}\right)^{2}\,
(x_{1}\, dx_{1} + x_{2}\, dx_{2})^{2}.%[** TN: Period before exponent in the original]
\end{align*}
In a radially symmetrical statical gravitational field we have for a
plane ($x_{3} = 0$) passing through the centre
\[
d\sigma^{2} = (dx_{1}^{2} + dx_{2}^{2}) + l(x_{1}\, dx_{1} + x_{2}\, dx_{2})^{2}
\]
in which
\[
l = \frac{h^{2} - 1}{r^{2}}
= \frac{2m}{r^{2}(r - 2m)}.
\]
The two formulæ are identical if we set
\[
F'(r) = \sqrt{\frac{2m}{r - 2m}}\Add{,}\qquad
F(r) = \sqrt{8m(r - 2m)}.
\]
\emph{The geometry which holds on this plane is therefore the same as that
which holds in Euclidean space on the surface of revolution of a
parabola}
\[
z = \sqrt{8m(r - 2m)}
\]
(\textit{vide} \FNote{20}).
A \Emph{charged sphere}, besides calling up a radially symmetrical
\index{Electron}%
\index{Sphere, charged}%
gravitational field, calls up a similar electrostatic field. Since both
fields influence one another mutually, they may be determined only
conjointly and simultaneously (\textit{vide} \FNote{21}). If we use the ordinary
units of the c.g.s.\ system (and not those of Heaviside which dispose
of the factor~$4\pi$ in another way and which we have generally used
in the foregoing) for electricity as well as for the other quantities,
then in the region devoid of masses and charges the integral becomes
\[
\int \left\{w \Delta' - \kappa\, \frac{\Phi'^{2} r^{2}}{\Delta}\right\} dr\Add{.}
\]
It assumes a stationary value for the condition of equilibrium. The
notation is the same as above, $\Phi$~denoting the electrostatic potential.
The square of the numerical value of the field is used as a basis for
the function of \Typo{Action}{\emph{Action}} of the electric field, in accordance with the
classical theory. Variation of~$w$ gives, just as in the case of no
charges,
\[
\Delta' = 0\Add{,}\qquad
\Delta = \text{const.} = c.
\]
But variation of~$\Phi$ leads to
\[
\frac{d}{dr} \left(\frac{r^{2} \Phi'}{\Delta}\right) = 0
\quad\text{and hence}\quad
\Phi = \frac{e_{0}}{r}.
\]
\PageSep{261}
For the electrostatic potential we therefore get the same formula as
when gravitation is disregarded. The constant~$e_{0}$ is the electric
charge which excites the field. If, finally, $\Delta$~be varied, we get
\[
w' - \kappa\, \frac{\Phi'^{2} r^{2}}{\Delta^{2}} = 0
\]
and hence
\[
w = 2m - \frac{\kappa}{c^{2}}\, \frac{e_{0}^{2}}{r},\qquad
\frac{1}{h^{2}} = \left(\frac{f}{c}\right)^{2}
= 1 - \frac{2\kappa m_{0}}{r} + \frac{\kappa}{c^{2}}\, \frac{e_{0}^{2}}{r^{2}}
\]
in which $m_{0}$~denotes the mass which produces the gravitational
field. In $f^{2}$ there occurs, as we see, in addition to the term
depending on the mass, an electrical term which decreases
more rapidly as $r$~increases. We call $m = \kappa m_{0}$ the gravitational
radius of the mass~$m_{0}$, and $\dfrac{\sqrt{\kappa}}{c} e_{0} = e$ the gravitational radius of
the charge~$e_{0}$. Our formula leads to \Emph{a view of the structure of
the electron which diverges essentially from the one commonly
accepted}. A finite radius has been attributed to the electron; this
has been found to be necessary, if one is to avoid coming to the
conclusion that the electrostatic field it produces has infinite total
energy, and hence an infinitely great inertial mass. If the inertial
mass of the electron is derived from its field-energy alone, then its
radius is of the order of magnitude
\[
a = \frac{e_{0}^{2}}{m_{0} c^{2}}.
\]
But in our formula a finite mass~$m_{0}$ (producing the gravitational
field) occurs quite independently of the smallness of the value of~$r$
for which the formula is regarded as valid; how are these results
to be reconciled? According to Faraday's view the charge enclosed
by a surface~$\Omega$ is nothing more than the flux of the electrical field
through~$\Omega$. Analogously to this it will be found in the next paragraph
that the true meaning of the conception of mass, both as field-producing
mass and as inertial or gravitational mass, is expressed
by a field-flux. If we are to regard the statical solution here given
as valid for all space, the flux of the electrical field through any
sphere is $4\pi e_{0}$ at the centre. On the other hand the mass which is
enclosed by a sphere of radius~$r$, assumes the value
\[
m_{0} - \tfrac{1}{2}\, \frac{e_{0}^{2}}{c^{2} r}
\]
which is dependent on the value of~$r$. The mass is consequently
distributed continuously. The density of mass coincides, of course,
with the density of energy. The ``initial level'' at the centre, from
which the mass is to be calculated, is not equal to~$0$ but to~$-\infty$.
\PageSep{262}
Therefore the mass~$m_{0}$ of the electron cannot be determined from
this level at all, but signifies the ``ultimate level'' at an infinitely
great distance. $a$~now signifies the radius of the sphere which
encloses the mass zero. Contrary to Mie's view \Emph{matter} now
appears \Emph{as a real singularity of the field}. In the general
theory of relativity, however, space is no longer assumed to be
Euclidean, and hence we are not compelled to ascribe to it the
relationships of Euclidean space. It is quite possible that it has
other limits besides infinity, and, in particular, that its relationships
are like those of a Euclidean space which contains punctures
(cf.\ §\,34). We may, therefore, claim for the ideas here developed---according
to which there is no connection between the total
mass of the electron and the potential of the field it produces, and
in which there is no longer a meaning in talking of a cohesive
pressure holding the electron together---equal rights as for those
of Mie. An unsatisfactory feature of the present theory is that the
field is to be entirely free of charge, whereas the mass ($=$~energy) is
to permeate the whole of the field with a density that diminishes
continuously.
It is to be noted that $a : e = e : m$ or, that $e = \sqrt{am}$. In the case
of the electron the quotient~$\dfrac{e}{m}$ is a number of the order of magnitude~$10^{20}$,
$\dfrac{a}{m}$~of the order~$10^{40}$; that is, the electric repulsion which two
electrons (separated by a great distance) exert upon one another is
$10^{40}$~times as great as that which they exert in virtue of gravitation.
The circumstance that in an electron an integral number of this
kind occurs which is of an order of magnitude varying greatly from
unity makes the thesis contained in Mie's Theory, namely, that all
pure figures determined from the measures of the electron must
be derivable as mathematical constants from the exact physical
laws, rather doubtful: on the other hand, we regard with equal
scepticism the belief that the structure of the world is founded on
certain pure figures of accidental numerical value.
The gravitational field that is present in the interior of \Emph{massive
bodies} is, according to Einstein's Theory, determined only when the
dynamical constitution of the bodies are fully known; since the
mechanical conditions are included in the gravitational equations,
the conditions of equilibrium are given for the statical case. The
simplest conditions that offer themselves for consideration are given
when we deal with bodies that are composed of a \Emph{homogeneous
incompressible fluid}. The energy-tensor of a fluid on which no
\index{Fluid, incompressible}%
volume forces are acting is given according to §\,25, by
\[
T_{ik} = \mu^{*} u^{i} u_{k} - pg_{ik}
\]
\PageSep{263}
in which the~$u_{i}$'s are co-variant components of the world-direction
of the matter, the scalar~$p$ denotes the pressure, and $\mu^{*}$~is determined
from the constant density~$\mu_{0}$ by means of the equation $\mu^{*} = \mu_{0} + p$. We introduce the quantities
\[
\mu^{*} u_{i} = v_{i}
\]
as independent variables, and set
\[
L = \frac{1}{\sqrt{g}}\, \vL
= \mu_{0} - \sqrt{v_{i} v^{i}}.
\]
Then, if we vary only the~$g^{ik}$'s, not the~$v_{i}$'s,
\[
d\vL = -\tfrac{1}{2} \vT_{ik}\, \delta g_{ik}.
\]
Consequently, by referring these equations to this kind of variation,
we may epitomise them in the formula
\[
\delta \int (\vL + \vG)\, dx = 0.
\]
It must carefully be noted, however, that, if the~$v_{i}$'s are varied
\index{Hydrodynamics}% [** TN: Hyphenated (but text usage inconsistent)]
\index{Hydrostatic pressure}%
\index{Pressure, on all sides!hydrostatic}%
as independent variables in this principle, it does \Emph{not} lead to the
correct \Chg{hydro-dynamical}{hydrodynamical} equations (instead, we should get $\dfrac{v^{i}}{\sqrt{v_{i} v^{i}}} = 0$,
which leads to nowhere). But these conservation theorems of energy
and momentum, are already included in the gravitational equations.
In the statical case, $v_{1} = v_{2} = v_{3} = 0$, and all quantities are independent
of the time. We set $v_{0} = v$ and apply the symbol of
variation~$\delta$ just as in §\,28 to denote a change that is produced by an
infinitesimal deformation (in this case a pure spatial deformation).
Then
\[
\delta\vL = \tfrac{1}{2} \vT^{ik}\, \delta g_{ik} - h\, \delta v\qquad
\left(h = \frac{\Delta}{f}\right)
\]
in which $\delta v$~denotes nothing more than the difference of~$v$ at two
points in space that are generated from one another as a result of
the displacement. By now arguing backwards from the conclusion
which gave us the energy-momentum theorem in §\,28, we infer from
this theorem, namely
\[
\int \vT^{ik}\, \delta g_{ik}\, dx = 0\Add{,}
\]
and from the equation
\[
\int \delta \vL\, dx = 0,
\]
which expresses the invariant character of the world-integral of~$\vL$,
that $\delta v = 0$. This signifies that, \Emph{in a connected space filled with
fluid, $v$~has a constant value}. The theorem of energy is true
\PageSep{264}
identically, and the law of momentum is expressed most simply by
this fact. A single mass of fluid in equilibrium will be radially
symmetrical in respect of the distribution of its mass and its field.
In this special case we must make the same assumption for~$ds^{2}$,
involving the three unknown functions $\lambda$,~$l$,~$f$, as at the beginning
of §\,31. If we start by setting $\lambda = 1$, we lose the equation which
is derived by varying~$\lambda$. A full substitute for it is clearly given by
the equation that asserts the invariance of the \emph{Action} during an
infinitesimal spatial displacement in radial directions, that is, the
theorem of $\text{momentum} : v = \text{const}$. The problem of variation that
has now to be solved is given by
\[
\delta \int \bigl\{\Delta' w + r^{2} \mu_{0} \Delta - r^{2} vh\bigr\}\, dr = 0
\]
in which $\Delta$~and~$h$ are to undergo variation, whereas
\[
w = \left(1 - \frac{1}{h^{2}}\right) r.
\]
Let us begin by varying~$\Delta$; we get
\[
w' - \mu_{0} r^{2} = 0
\quad\text{and}\quad
w = \frac{\mu_{0}}{3} r^{3}\Add{,}
\]
that is
\[
\framebox{$\dfrac{1}{h^{2}} = 1 - \dfrac{\mu_{0}}{3}\, r^{2}$}
\Tag{(50)}
\]
Let the spherical mass of fluid have a radius $r = r_{0}$. It is obvious
that $r_{0}$~must remain
\[
< a = \sqrt{\frac{3}{\mu_{0}}}.
\]
The energy and the mass are expressed in the rational units given
by the theory of gravitation. For a sphere of water, for example,
this upper limit of the radius works out to
\[
\sqrt{\frac{3}{8\pi \kappa}} = 4 · 10^{8} \text{ km.} = 22 \text{ light-minutes.}
\]
Outside the sphere our earlier formulæ are valid, in particular
\[
\frac{1}{h^{2}} = 1 - \frac{2m}{r},\qquad
\Delta = 1.
\]
The boundary conditions require that $h$~and~$f$ have continuous
values in passing over the spherical surface, and that the pressure~$p$
vanish at the surface. From the continuity of~$h$ we get for the
gravitational radius~$m$ of the sphere of fluid
\[
m = \frac{\mu_{0} r_{0}^{3}}{6}.
\]
\PageSep{265}
The inequality, which holds between $r_{0}$~and~$\mu_{0}$, shows that the
radius~$r_{0}$ must be greater than~$2m$. Hence, if we start from infinity,
then, before we get to the singular sphere $r = 2m$ mentioned
above, we reach the fluid, within which other laws hold. If we
now adopt the gramme as our unit, we must replace~$\mu_{0}$ by~$8\pi \kappa \mu_{0}$,
whereas $m = \kappa m_{0}$, if $m_{0}$~denotes the gravitating mass. We then
find that
\[
m_{0} = \mu_{0}\, \frac{4\pi r_{0}^{3}}{3}\Add{.}
\]
Since
\[
v = \mu^{*} f = \frac{\mu^{*} \Delta}{h}
\]
is a constant, and assumes the value~$\dfrac{\mu_{0}}{h_{0}}$ at the surface of the sphere,
in which $h_{0}$~denotes the value of~$h$ there as given by~\Eq{(50)}, we see
that in the whole interior
\[
v = (\mu_{0} + p) f = \frac{\mu_{0}}{h_{0}}\Add{.}
\Tag{(51)}
\]
Variation of~$h$ leads to
\[
-\frac{2\Delta'}{h^{3}} + rv = 0.
\]
Since it follows from~\Eq{(50)} that
\[
\frac{h'}{h^{3}} = \frac{\mu_{0}}{3} r\Add{,}
\]
we get immediately
\[
\Delta = \frac{3v}{2\mu_{0}} h + \text{const.}
\]
Further, if we use the value of the constant~$v$ given by~\Eq{(51)},
and calculate the value of the integration constant that occurs, by
using the boundary condition $\Delta = 1$ at the surface of the sphere,
then\Pagelabel{265}
\[
\Delta = \frac{3h - h_{0}}{2h_{0}},\qquad
\framebox{$f = \dfrac{3h - h_{0}}{2hh_{0}}$}
\]
Finally, we get from~\Eq{(51)}
\[
\framebox{$p = \mu_{0} · \dfrac{h_{0} - h}{3h - h_{0}}$}
\]
These results determine the metrical groundform of space
\[
d\sigma^{2}
= (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})
+ \frac{(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})}{a^{2} - r^{2}},
\Tag{(52)}
\]
the gravitational potential or the velocity of light~$f$, and the
pressure-field~$p$.
\PageSep{266}
If we introduce a superfluous co-ordinate
\[
x_{4} = \sqrt{a^{2} - r^{2}}
\]
into space, then
\[
x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = a^{2}
\Tag{(53)}
\]
and hence
\[
x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3} + x_{4}\, dx_{4} = 0\Add{.}
\]
\Eq{(52)}~then becomes
\[
d\sigma^{2} = \Typo{dx_{2}^{1}}{dx_{1}^{2}} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2}.
\]
\emph{In the whole interior of the fluid sphere spatial spherical geometry
\index{Geometry!spherical}%
\index{Spherical!geometry}%
is valid, namely, that which is true on the ``sphere''~\Eq{(53)} in four-dimensional
Euclidean space with Cartesian co-ordinates~$x_{i}$.} The
fluid covers a cap-shaped portion of the sphere. The pressure in
it is a linear fractional function of the ``vertical height,'' $z = x_{4}$ on
the sphere:
\[
\frac{p}{\mu_{0}} = \frac{z - z_{0}}{3z_{0} - z}.
\]
Further, it is shown by this formula that, since the pressure~$p$ may
not pass, on a sphere of latitude, $z = \text{const.}$, from positive to negative
values through infinity, $3z_{0}$~must be $> a$, and the upper limit~$a$
found above for the radius of the fluid sphere must be correspondingly
reduced to~$\dfrac{2a\sqrt{2}}{3}$.
These results for a sphere of fluid were first obtained by
Schwarzschild (\textit{vide} \FNote{22}). After the most important cases of
radially symmetrical statical gravitational fields had been solved,
the author succeeded in solving the more general problem of the
\Emph{cylindrically symmetrical statical field} (\textit{vide} \FNote{23}). We
shall here just mention briefly the simplest results of this investigation.
Let us consider first \Emph{uncharged masses} and a gravitational
field in space free from matter. It then follows from the gravitational
equations, if certain space-co-ordinates $r$,~$\theta$,~$z$ (so-called
canonical \Emph{cylindrical co-ordinates}) are used, that
\index{Canonical cylindrical co-ordinates}%
\[
ds^{2} = f^{2}\, dt^{2} - d\sigma^{2}\Add{,}\qquad
d\sigma^{2} = h(dr^{2} + dz^{2}) + \frac{r^{2}\, d\theta^{2}}{f^{2}}\Add{.}
\]
{\Loosen $\theta$~is an angle whose modulus is~$2\pi$; that is, corresponding to values
of~$\theta$ that differ by integral multiples of~$2\pi$ there is only one
point. On the axis of rotation $r = \Typo{o}{0}$. Also, $h$~and~$f$ are functions
of $r$~and~$z$. We shall plot real space in terms of a Euclidean space,
in which $r$,~$\theta$,~$z$ are cylindrical co-ordinates. The canonical co-ordinate
system is uniquely defined except for a displacement in
the direction of the axis of rotation $z' = z + \text{const}$. When
\PageSep{267}
$h = f = 1$, $d\sigma^{2}$~is identical with the metrical groundform of the
Euclidean picture-space (used for the plotting). The gravitational
problem may be solved just as easily on this theory as on that of
Newton, if the distribution of the matter is known in terms of
canonical co-ordinates. For if we transfer these masses into our
picture-space, that is, if we make the mass contained in a portion
of each space equal to the mass contained in the corresponding
portion of the picture-space, and if $\psi$~is then the Newtonian
potential of this mass-distribution in the Euclidean picture-space,
the simple formula}
\[
f = e^{\psi/c^{2}}
\Tag{(54)}
\]
holds. The second still unknown function~$h$ may also be determined
by the solution of an ordinary Poisson equation (referring to
the meridian plane $\theta = 0$). In the case of \Emph{charged bodies}, too,
the canonical co-ordinate system exists. If we assume that the
masses are negligible in comparison with the charges, that is, that
for an arbitrary portion of space the gravitational radius of the
electric charges contained in it is much greater than the gravitational
radius of the masses contained in it, and if $\phi$~denotes the
electrostatic potential (calculated according to the classical theory)
of the transposed charges in the canonical picture-space, then $f$~and
the electrostatic potential~$\Phi$ in real space are given by the formulæ
\[
\Phi = \frac{c}{\sqrt{\kappa}} \tan \left(\frac{\sqrt{\kappa}}{c} \phi\right)\Add{,}\qquad
f = \frac{1}{\cos \left(\dfrac{\sqrt{\kappa}}{c} \phi\right)}\Add{.}
\Tag{(54')}
\]
It is not quite easy to subordinate the radially symmetrical case to
this more general theory: it becomes necessary to carry out a rather
complicated transformation of the space-co-ordinates, into which
we shall not enter here.
Just as the laws of Mie's electrodynamics are non-linear, so
also \Emph{Einstein's laws of gravitation}. This non-linearity is not
perceptible in those measurements that are accessible to direct
observation, because, in them, the non-linear terms are quite
negligible in comparison with the linear ones. It is as a result of
this that the \Emph{principle of superposition} is found to be confirmed
by the interplay of forces in the visible world. Only, perhaps, for
the unusual occurrences within the atom, of which we have as yet
no clear picture, does this non-linearity come into consideration.
Non-linear differential equations involve, in comparison with linear
equations, particularly as regards singularities, extremely intricate,
unexpected, and, at the present, quite uncontrollable conditions.
The suggestion immediately arises that these two circumstances,
\PageSep{268}
the remarkable behaviour of non-linear differential equations and
the peculiarities of intra-atomic occurrences, are to be related to
one another. Equations \Eq{(54)}~and~\Eq{(54')} offer a beautiful and simple
example of how the principle of superposition becomes modified in
the strict theory of gravitation: the field-potentials $f$~and~$\Phi$ depend
in the one case on the exponential function of the quantity~$\psi$, and
in the other on a trigonometrical function of the quantity~$\phi$, these
quantities being those which satisfy the principle of superposition.
At the same time, however, these equations demonstrate clearly
that the non-linearity of the gravitational equations will be of no
\index{Gravitational!energy}%
assistance whatever for explaining the occurrences within the
atom or the constitution of the electron. For the differences
between $\phi$~and~$\Phi$ become appreciable only when $\dfrac{\sqrt{\kappa}}{c} \phi$ assumes
values that are comparable with~$1$. But even in the interior of the
electron this case arises only for spheres whose radius corresponds
to the order of gravitational radius
\[
e = \frac{\sqrt{\kappa}}{c} e_{0} \sim 10^{-33} \text{ cms.}
\]
for the charge~$e_{0}$ of the electron.
It is obvious that the statical differential equations of gravitation
cannot uniquely determine the solutions, but that boundary
conditions at infinity, or conditions of symmetry such as the
postulate of radial symmetry must be added. The solutions which
we found were those for which the metrical groundform converges,
at spatial infinity, to
\[
dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})\Add{,}
\]
the expression which is a characteristic of the special theory of
relativity.
A further series of elegant investigations into problems of
statical gravitation have been initiated by Levi-Civita (\textit{vide} \FNote{24}).
The Italian mathematicians have studied, besides the statical
case, also the ``stationary'' one, which is characterised by the
circumstance that all the~$g_{ik}$'s are independent of the time-co-ordinate~$x_{0}$,
whereas the ``lateral'' co-efficients $g_{01}$,~$g_{02}$,~$g_{03}$ need not
vanish (\textit{vide} \FNote{25}): an example of this is given by the field that
surrounds a body which is in stationary rotation.
\Section{33.}{Gravitational Energy. The Theorems of Conservation}
An \Emph{isolated system} sweeps out in the course of its history a
\index{World ($=$ space-time)!-canal}%
``world-canal''; we assume that outside this canal the stream-density
\PageSep{269}
$\vs^{i}$~vanishes (if not entirely, at least to such a degree that the
following argument retains its validity). It follows from the
equation of continuity
\[
\frac{\dd \vs^{i}}{\dd x_{i}} = 0
\Tag{(55)}
\]
that the flux of the vector-density~$\vs^{i}$ has the same value~$e$ through
every three-dimensional ``plane'' across the canal. To fix the sign
of~$e$, we shall agree to take for its direction that leading from the
past into the future. The invariant~$e$ is the \Emph{charge} of our system.
\index{Charge!(\emph{generally})}%
\index{Conservation, law of!electricity@{of electricity}}%
If the co-ordinate system fulfils the conditions that every ``plane''
$x_{0} = \text{const.}$ intersects the canal in a finite region and that these
planes, arranged according to increasing values of~$x_{0}$, follow one
another in the order, past $\to$~future, then we may calculate~$e$ by
means of the equation
\[
\int \vs^{0}\, dx_{1}\, dx_{2}\, dx_{3} = e
\]
in which the integration is taken over any arbitrary plane of the
family $x_{0} = \text{const}$. This integral $e = e(x_{0})$ is accordingly independent
of the ``time''~$x_{0}$, as is readily seen, too, from~\Eq{(55)} if we
integrate it with respect to the ``space-co-ordinates'' $x_{1}$,~$x_{2}$,~$x_{3}$. What
has been stated above is valid in virtue of the equation of continuity
alone; the idea of substance and the convention to which it
leads in Lorentz's Theory, namely, $\vs^{i} = \rho u^{i}$ do not come into
question in this case.
Does a similar \Emph{theorem of conservation} hold true for \Emph{energy
\index{Energy-momentum, tensor!(for the whole system, including gravitation)}%
\index{Energy-momentum, tensor!(in the general theory of relativity)}%
\index{Energy-momentum, tensor!(of the gravitational field)}%
and momentum}? This can certainly not be decided from the
equation~\Eq{(26)} of §\,28, since the latter contains the additional term
which is a characteristic of the theory of gravitation. \Emph{It is
possible}, however, to write this addition term, too, in the form of a
divergence. We choose a definite co-ordinate system and subject
the world-continuum to an infinitesimal \Emph{deformation} in the true
sense, that is, we choose constants for the deformation components~$\xi^{i}$
in §\,28. Then, of course, for any finite region~$\rX$
\[
\delta' \int_{\rX} \vG\, dx = 0
\]
(this is true for \Emph{every} function of the~$g_{ik}$'s and their derivatives: it
has nothing to do with properties of invariance; $\delta'$~denotes, as in
§\,28, the variation effected by the displacement). Hence, the displacement
gives us
\[
\int_{\rX} \frac{\dd (\vG \xi^{k})}{\dd x_{k}}\, dx
+ \int_{\rX} \delta \vG\, dx = 0.
\]
\PageSep{270}
If, as earlier, we set
\[
\delta \vG
= \tfrac{1}{2} \vG^{\alpha\beta}\, \delta g_{\alpha\beta}
+ \tfrac{1}{2} \vG^{\alpha\beta,k}\, \delta g_{\alpha\beta,k}\Add{,}
\Tag{(13)}
\]
then partial integration gives
\[
2 \int_{\rX} \delta \vG\, dx
= \int_{\rX} \frac{\dd (\vG^{\alpha\beta,k}\Typo{}{)}\, \delta g_{\alpha\beta,k}}{\dd x_{k}}
+\int_{\rX} [\vG]^{\alpha\beta}\, \delta g_{\alpha\beta}\, dx.
\]
Now, in this case, since the~$\xi$'s are constants,
\[
\delta g_{\alpha\beta} = -\frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, \xi^{i}.
\]
If we introduce the quantities
\[
\vG \delta_{i}^{k}
- \tfrac{1}{2} \vG^{\alpha\beta,k}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
= \vt_{i}^{k}
\]
then, by the preceding relation, we get the equation
\[
\int_{\rX} \left\{\frac{\dd \vt_{i}^{k}}{\dd x_{k}}
- \tfrac{1}{2} [\vG]^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\Typo{dx_{i}}{\dd x_{i}}}
\right\} \xi^{i}\, dx = 0.
\]
Since this holds for any arbitrary region~$\rX$, the integrand must be
equal to zero. In it the~$\xi^{i}$'s denote arbitrary constant numbers;
hence we get four identities:
\[
\tfrac{1}{2} [\vG]^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
= \frac{\dd \vt_{i}^{k}}{\dd x_{k}}.
\]
The left-hand side, by the gravitational equations,
\[
= -\tfrac{1}{2} \vT^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}
\]
and, accordingly, the mechanical equations~\Eq{(26)} become
\[
\frac{\dd \vU_{i}^{k}}{\dd x_{k}} = 0,\qquad
\text{where }
\vU_{i}^{k} = \vT_{i}^{k} + \vt_{i}^{k}\Add{.}
\Tag{(56)}
\]
It is thus shown that if we regard the~$\vt_{i}^{k}$'s, which are dependent
only on the potentials and the field-components of gravitation, as
the components of \Emph{the energy-density of the gravitational field},
we get pure divergence equations for \Emph{all} energy associated with
``physical state or phase'' and ``gravitation'' (\textit{vide} \FNote{26}).
And yet, physically, it seems devoid of sense to introduce the~$\vt_{i}^{k}$'s
as energy-components of the gravitational field, for these
quantities \Emph{neither form a tensor nor are they symmetrical}.
In actual fact, if we choose an appropriate co-ordinate system, we
may make all the~$\vt_{i}^{k}$'s at one point vanish; it is only necessary to
choose a geodetic co-ordinate system. And, on the other hand, if
we use a curvilinear co-ordinate system in a ``Euclidean'' world
totally devoid of gravitation, we get $\vt_{i}^{k}$'s that are all different from
\PageSep{271}
\index{Conservation, law of!electricity@{of electricity}}%
zero, although the existence of gravitational energy in this case
can hardly come into question. Hence, although the differential
relations~\Eq{(56)} have no real physical meaning, we can derive from
them, by \Emph{integrating over an isolated system}, an invariant
theorem of conservation (\textit{vide} \FNote{27}).
During motion an isolated system with its accompanying gravitational
field sweeps out a canal in the ``world''. Beyond the
canal, in the empty surroundings of the system, we shall assume
that the tensor-density~$\vT_{i}^{k}$ and the gravitational field vanish. We
may then use co-ordinates $x_{0}$~($= t$), $x_{1}$,~$x_{2}$,~$x_{3}$, such that the
metrical groundform assumes constant co-efficients outside the
canal, and in particular assumes the form
\[
dt^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}).
\]
Hence, outside the canal, the co-ordinates are fixed except for a
linear (Lorentz) transformation, and the~$\vt_{i}^{k}$'s vanish there. We
assume that each of the ``planes'' $t = \text{const.}$ has only a finite
portion of section in common with the canal. If we integrate the
equations~\Eq{(56)} with respect to $x_{1}$,~$x_{2}$,~$x_{3}$ over such a plane, we find
that the quantities
\[
J_{i} = \int \vU_{i}^{0}\, dx_{1}\, dx_{2}\, dx_{3}
\]
are independent of the time; that is $\dfrac{dJ_{i}}{dt} = 0$. We call~$J_{0}$ the
\Emph{energy}, and $J_{1}$,~$J_{2}$,~$J_{3}$ the \Emph{momentum co-ordinates} of the
system.
These quantities have a significance which is independent of
the co-ordinate system. We affirm, firstly, that they retain their
value if the co-ordinate system is changed anywhere \Emph{within the
canal}. Let $\bar{x}_{i}$ be the new co-ordinates, identical with the old ones
for the region outside the canal. We mark out two ``surfaces''
\[
x_{0} = \text{const.} = a
\quad\text{and}\quad
\bar{x}_{0} = \text{const.} = \bar{a}\qquad
(\bar{a} \neq a)
\]
which do not intersect in the canal (for this it suffices to
choose $a$~and~$\bar{a}$ sufficiently different from one another). We can
then construct a third co-ordinate system~$x_{i}^{*}$ which is identical
with the~$x_{i}$'s in the neighbourhood of the first surface, identical
with the~$\bar{x}_{i}$ in that of the second system, and is identical with both
outside the canal. If we give expression to the fact that the
energy-momentum components~$J_{i}^{*}$ in this system assume the same
values for $x_{0}^{*} = a$ and $x_{0} = \bar{a}$, then we get the result which we
enunciated, namely, $J_{i} = \bar{J}_{i}$.
\PageSep{272}
Consequently, the behaviour of the~$J_{i}$'s need be investigated
only in the case of \Emph{linear} transformations of the co-ordinates.
With respect to such, however, the conception of a tensor with
components that are constant (that is, independent of position) is
invariant. We make use of an arbitrary vector~$p^{i}$ of this type, and
form $\vU^{k} = \vU_{i}^{k} p^{i}$, and deduce from~\Eq{(56)} that
\[
\frac{\dd \vU^{k}}{\dd x_{k}} = 0.
\]
By applying the same reasoning as was used above in the case of
the electric current, it follows from this that
\[
\int \vU^{0}\, dx_{1}\, dx_{2}\, dx_{3} = J_{i} p^{i}
\]
is an invariant with respect to linear transformations. \Emph{Accordingly,
the~$J_{i}$'s are the components of a constant co-variant
vector in the ``Euclidean'' surroundings of the system}; this
energy-momentum vector is uniquely determined by the phase (or
state) of the physical system. The direction of this vector determines
generally the direction in which the canal traverses the
surrounding world (a purely descriptive datum that can be expressed
in an exact form accessible to mathematical analysis only
with great difficulty). The invariant
\[
\sqrt{J_{0}^{2} - J_{1}^{2} - J_{2}^{2} - J_{3}^{2}}
\]
is the \Emph{mass} of the system.
\index{Matter}%
In the statical case $J_{1} = J_{2} = J_{3} = 0$, whereas $J_{0}$~is equal to\Pagelabel{272}
the space-integral of $\vR_{0}^{0} - (\frac{1}{2} \vR - \vG)$. According to §\,29~and §\,28
(\Pageref[p.]{240}), respectively,
\begin{gather*}
\vR_{0}^{0} = \frac{\dd \vf^{i}}{\Typo{dx_{i}}{\dd x_{i}}},
\quad\text{and in general,} \\
\tfrac{1}{2} \vR - \vG
= \tfrac{1}{2} \frac{\dd}{\dd x_{i}} \sqrt{g}
\left(g^{\alpha\beta} \Chr{\alpha\beta}{i}
- g^{i\alpha} \Chr{\alpha\beta}{\beta}\right),
\end{gather*}
and hence, in the notation of §\,29~and §\,31, the mass~$J_{0}$ is equal to
the flux of the (spurious) spatial vector-density
\[
\vm_{i} = \tfrac{1}{2} f \sqrt{g}
\left(\gamma^{\alpha\beta} \Chr{\alpha\beta}{i}
- \gamma^{i\alpha} \Chr{\alpha\beta}{\beta}\right)\quad
(i\Com \alpha\Com \beta = 1, 2, 3)\Add{,}
\Tag{(57)}
\]
which has yet to be multiplied by~$\dfrac{1}{8\pi \kappa}$ if we use the ordinary
units. Since at a great distance from the system the solution of
the field laws, which was found in §\,31, is always valid, and for
which $\vm^{i}$~is a radial current of intensity
\[
\frac{1 - f^{2}}{8\pi \kappa r} = \frac{m_{0}}{4\pi r^{2}},
\]
\PageSep{273}
we get that \emph{the energy,~$J_{0}$, or the inertial mass of the system, is
equal to the mass~$m_{0}$, which is characteristic of the gravitational
field generated by the system} (\textit{vide} \FNote{28}). On the other hand it
is to be remarked parenthetically that the physics based on the
notion of substance leads to the space-integral of~$\mu/f$ for the value
\index{Substance}%
of the mass, whereas, in reality, for incoherent matter $J_{0} = m_{0} =$
the space-integral of~$\mu$; this is a definite indication of how radically
erroneous is the whole idea of substance.
\Section{34.}{Concerning the Inter-connection of the World
as~a Whole}
\index{Analysis situs@{\emph{Analysis situs}}}%
\index{Relationship!of the world}%
\index{World ($=$ space-time)!-law}%
The general theory of relativity leaves it quite undecided whether
the world-points may be represented by the values of four co-ordinates~$x_{i}$
in a singly reversible continuous manner or not. It
merely assumes that the \Emph{neighbourhood} of every world-point admits
of a singly reversible continuous representation in a region of the
four-dimensional ``number-space'' (whereby ``point of the four-dimensional
number-space'' is to signify any number-quadruple);
it makes no assumptions at the outset about the inter-connection
of the world. When, in the theory of surfaces, we start with a
parametric representation of the surface to be investigated, we are
referring only to a piece of the surface, not to the whole surface,
which in general can by no means be represented uniquely and
continuously on the Euclidean plane or by a plane region. Those
properties of surfaces that persist during all one-to-one continuous
transformations form the subject-matter of \emph{analysis situs} (the
analysis of position); connectivity, for example, is a property
of \Chg{analysis situs}{\emph{analysis situs}}. Every surface that is generated from the
sphere by continuous deformation does not, from the point of view
of \Chg{analysis situs}{\emph{analysis situs}}, differ from the sphere, but does differ from an
anchor-ring, for instance. For on the anchor-ring there exist closed
lines, which do not divide it into several regions, whereas such lines
are not to be found on the sphere. From the geometry which
is valid on a sphere, we derived ``spherical geometry'' (which,
following Riemann, we set up in contrast with the geometry of
Bolyai-Lobatschefsky) by identifying two diametrically opposite
points of the sphere. The resulting surface~$\vF$ is from the point of
view of \emph{analysis situs} likewise different from the sphere, in virtue
of which property it is called one-sided. If we imagine on a surface
a small wheel in continual rotation in the one direction to
be moved along this surface during the rotation, the centre of the
wheel describing a closed curve, then we should expect that when
the wheel has returned to its initial position it would rotate in the
\PageSep{274}
same direction as at the commencement of its motion. If this is the
case, then whatever curve the centre of the wheel may have described
on the surface, the latter is called \Emph{two-sided}; in the reverse
\index{Surface}%
\index{Two-sided surfaces}%
case, it is called \Emph{one-sided}. The existence of one-sided surfaces
\index{One-sided surfaces}%
was first pointed out by Möbius. The surface~$\vF$ mentioned above
is \Typo{two}{one}-sided, whereas the sphere is, of course, \Typo{one}{two}-sided. This is
obvious if the centre of the wheel be made to describe a great
circle; on the sphere the \Emph{whole} circle must be traversed if this
path is to be closed, whereas on~$\vF$ only the half need be covered.
Quite analogously to the case of two-dimensional manifolds, four-dimensional
ones may be endowed with diverse properties with
regard to \emph{analysis situs}. But in every four-dimensional manifold
the neighbourhood of a point may, of course, be represented in a
continuous manner by four co-ordinates in such a way that different
co-ordinate quadruples always correspond to different points of this
neighbourhood. The use of the four world-co-ordinates is to be
interpreted in just this way.
Every world-point is the origin of the double-cone of the active
future and the passive past. Whereas in the special theory of
relativity these two portions are separated by an intervening region,
it is certainly possible in the present case for the cone of the active
future to overlap with that of the passive past; so that, in principle,
it is possible to experience events now that will in part be an effect
of my future resolves and actions. Moreover, it is not impossible
for a world-line (in particular, that of my body), although it has a
time-like direction at every point, to return to the neighbourhood
of a point which it has already once passed through. The result
would be a spectral image of the world more fearful than anything
the weird fantasy of E.~T.~A. Hoffmann has ever conjured up. In
actual fact the very considerable fluctuations of the~$g_{ik}$'s that would
be necessary to produce this effect do not occur in the region of
world in which we live. Nevertheless there is a certain amount of
interest in speculating on these possibilities inasmuch as they shed
light on the philosophical problem of cosmic and phenomenal time.
Although paradoxes of this kind appear, nowhere do we find any real
contradiction to the facts directly presented to us in experience.
We saw in §\,26 that, apart from the consideration of gravitation,
the fundamental electrodynamic laws (of Mie) have a form such
as is demanded by the \Emph{principle of causality}. The time-derivatives
of the phase-quantities are expressed in terms of these
quantities themselves and their spatial differential co-efficients.
These facts persist when we introduce gravitation and thereby
increase the table of phase-quantities $\phi_{i}$,~$F_{ik}$, by the~$g_{ik}$'s and the~$\dChr{ik}{r}$'s.
\PageSep{275}
But on account of the general invariance of physical
laws we must formulate our statements so that, from the values of
the phase-quantities for one moment, all those assertions concerning
them, \Emph{which have an invariant character}, follow as a
consequence of physical laws; moreover, it must be noted that this
statement does not refer to the world as a whole but only to a
portion which can be represented by four co-ordinates. Following
Hilbert (\textit{vide} \FNote{29}) we proceed thus. In the neighbourhood of
the world-point~$O$ we introduce $4$~co-ordinates~$x_{i}$ such that, at $O$
itself,
\[
ds^{2} = dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}).
\]
In the three-dimensional space $x_{0} = 0$ surrounding~$O$ we may
mark off a region~$\vR$, such that, in it, $-ds^{2}$~remains definitely
positive. Through every point of this region we draw the geodetic
world-line which is orthogonal to that region, and which has a
time-like direction. These lines will cover singly a certain four-dimensional
neighbourhood of~$O$. We now introduce new
co-ordinates which will coincide with the previous ones in the
three-dimensional space~$\vR$, for we shall now assign the co-ordinates
$x_{0}$,~$x_{1}$, $x_{2}$,~$x_{3}$ to the point~$P$ at which we arrive, if we go from
the point $P_{0} = (x_{1}, x_{2}, x_{3})$ in~$\vR$ along the orthogonal geodetic
line passing through it, so far that the proper-time of the arc
traversed,~$P_{0}P$, is equal to~$x_{0}$. This system of co-ordinates was
introduced into the theory of surfaces by Gauss. Since $ds^{2} = dx_{0}^{2}$
on each of the geodetic lines, we must get identically for all four
co-ordinates in this co-ordinate system:
\[
g_{00} = 1\Add{.}
\Tag{(58)}
\]
{\Loosen Since the lines are orthogonal to the three-dimensional space
$x_{0} = 0$, we get for $x_{0} = 0$}
\[
g_{01} = g_{02} = g_{03} = 0\Add{.}
\Tag{(59)}
\]
Moreover, since the lines that are obtained when $x_{1}$,~$x_{2}$,~$x_{3}$ are kept
constant and $x_{0}$~is varied are geodetic, it follows (from the equation
of geodetic lines) that
\[
\Chr{00}{i} = 0
\qquad(i = 0, 1, 2, 3)\Add{,}
\]
and hence also that
\[
\Chrsq{00}{i} = 0.
\]
Taking \Eq{(58)} into consideration, we get from the latter
\[
\frac{\dd g_{0}}{\dd x_{0}} = 0
\qquad (i = 1, 2, 3)
\]
\PageSep{276}
and, on account of~\Eq{(59)}, we have consequently not only for $x_{0} = 0$
but also identically for the four co-ordinates that
\[
g_{0i} = 0
\qquad (i = 1, 2, 3).
\Tag{(60)}
\]
The following picture presents itself to us: a family of geodetic
\index{World ($=$ space-time)!-law}%
lines with time-like direction which covers a certain world-region
singly and completely (without gaps); also, a similar uni-parametric
family of three-dimensional spaces $x_{0} = \text{const}$. According
to~\Eq{(60)} these two families are everywhere orthogonal to one another,
and all portions of arc cut off from the geodetic lines by two of
the ``parallel'' spaces $x_{0} = \text{const.}$ have the same proper-time. If
we use this particular co-ordinate system, then
\[
\frac{\dd g_{ik}}{\dd x_{0}} = -2\Chr{ik}{0}
\qquad (i, k = 1, 2, 3)
\]
and the gravitational equations enable us to express the derivatives
\[
\frac{\dd}{\dd x_{0}} \Chr{ik}{0}
\qquad (i, k = 1, 2, 3)
\]
not only in terms of the~$\phi_{i}$'s and their derivatives, but also in terms
of the~$g_{ik}$'s, their derivatives (of the first and second order) with
respect to $x_{1}$,~$x_{2}$,~$x_{3}$, and the $\dChr{ik}{0}$'s~themselves.
%[** TN: Line break without indentation in the original]
Hence, by regarding the twelve quantities,
\[
g_{ik},\quad
\Chr{ik}{0}
\qquad (i, k = 1, 2, 3)
\]
together with the electromagnetic quantities, as the unknowns, we
arrive at the required result ($x_{0}$~playing the part of time). The
cone of the passive past starting from the point~$O'$ with a positive
$x_{0}$~co-ordinate will cut a certain portion~$\vR'$ out of~$\vR$, which, with
the sheet of the cone, will mark off a finite region of the world~$\vG$
(namely, a conical cap with its vertex at~$O'$). If our assertion that
the geodetic null-lines denote the initial points of all action is
rigorously true, then the values of the above twelve quantities as well
as the electromagnetic potentials~$\phi_{i}$ and the field-quantities~$F_{ik}$ in
the three-dimensional region of space~$\vR'$ determine fully the values
of the two latter quantities in the world-region~$\vG$. This has
hitherto not been proved. \emph{In any case, we see that the differential
equations of the field contain the physical laws of nature in their
complete form}, and that there cannot be a further limitation due
to boundary conditions at spatial infinity, for example.
Einstein, arguing from cosmological considerations of the inter-connection
of the world as a whole (\textit{vide} \FNote{30}) came to the conclusion
\PageSep{277}
that the world is finite in space. Just as in the Newtonian
theory of gravitation the law of contiguous action expressed in
Poisson's equation entails the Newtonian law of attraction only if
the condition that the gravitational potential vanishes at infinity is
superimposed, so Einstein in his theory seeks to supplement the
differential equations by introducing boundary conditions at spatial
infinity. To overcome the difficulty of formulating conditions of a
general invariant character, which are in agreement with astronomical
facts, he finds himself constrained to assume that the world
is closed with respect to space; for in this case the boundary conditions
are absent. In consequence of the above remarks the
author cannot admit the cogency of this deduction, since the differential
equations in themselves, without boundary conditions, contain
the physical laws of nature in an unabbreviated form excluding
every ambiguity. So much more weight is accordingly to be
attached to another consideration which arises from the question:
How does it come about that our stellar system with the relative
velocities of the stars, which are extraordinarily small in comparison
with that of light, persists and maintains itself and has not,
even ages ago, dispersed itself into infinite space? This system
presents exactly the same view as that which a molecule in a gas
in equilibrium offers to an observer of correspondingly small dimensions.
In a gas, too, the individual molecules are not at rest but
the small velocities, according to Maxwell's law of distribution,
occur much more often than the large ones, and the distribution of
the molecules over the volume of the gas is, on the average, uniform,
so that perceptible differences of density occur very seldom. If
this analogy is legitimate, we could interpret the state of the stellar
system and its gravitational field according to the same \Emph{statistical
principles} that tell us that an isolated volume of gas is almost
always in equilibrium. This would, however, be possible only if
the \Emph{uniform distribution of stars at rest in a static gravitational
field, as an ideal state of equilibrium}, is reconcilable
with the laws of gravitation. In a statical field of gravitation the
world-line of a point-mass at rest, that is, a line on which $x_{1}$,~$x_{2}$,~$x_{3}$
remain constant and $x_{0}$~alone varies, is a geodetic line if
\[
\Chr{00}{i} = 0,
\qquad (i = 1, 2, 3)\Add{,}
\]
and hence
\[
\Chrsq{00}{i} = 0\Add{,}\qquad
\frac{\dd g_{00}}{\dd x_{i}} = 0.
\]
Therefore, a distribution of mass at rest is possible only if
\[
\sqrt{g_{00}} = f = \text{const.} = 1.
\]
\PageSep{278}
The equation
\[
\Delta f = \tfrac{1}{2} \mu\qquad
(\mu = \text{density of mass})
\Tag{(32)}
\]
then shows, however, that the ideal state of equilibrium under consideration
\Emph{is incompatible} with the laws of gravitation, as hitherto
assumed.
In deriving the gravitational equations in §\,28, however, we
committed a sin of omission. $R$~is not the only invariant dependent
on the~$g_{ik}$'s and their first and second differential co-efficients,
and which is linear in the latter; for the most general invariant of
this description has the form $\alpha R + \beta$, in which $\alpha$~and $\beta$ are
numerical constants. Consequently we may generalise the laws of
gravitation by replacing~$R$ by~$R + \lambda$ (and $\vG$~by $\vG + \frac{1}{2} \lambda \sqrt{g}$), in
which $\lambda$~denotes a universal constant. If it is not equal to~$0$, as
we have hitherto assumed, we may take it equal to~$1$; by this
means not only has the unit of time been reduced by the principle
of relativity\Typo{,}{} to the unit of length, and the unit of mass by the law
of gravitation to the same unit, but the unit of length itself is fixed
absolutely. With these modifications the gravitational equations
for statical non-coherent matter ($\vT_{0}^{0} = \mu = \mu_{0} \sqrt{g}$, all other components
of the tensor-density~$\vT$ being equal to zero) give, if we use
the equation $f = 1$ and the notation of §\,29:
\[
\lambda = \mu_{0} \quad\text{[in place of~\Eq{(32)}]}
\]
and
\[
P_{ik} - \lambda \gamma_{ik} = 0
\qquad (i, k = 1, 2, 3)\Add{.}
\Tag{(61)}
\]
Hence this ideal state of equilibrium is possible under these circumstances
if the mass is distributed with the density~$\lambda$. The
space must then be homogeneous metrically; and indeed the equations~\Eq{(61)}
are then actually satisfied for a spherical space of radius
$a = \sqrt{2/\lambda}$. Thus, in space, we may introduce four co-ordinates,
connected by
\[
x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = a^{2},
\Tag{(62)}
\]
for which we get
\[
d\sigma^{2} = dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2}.
\]
\Emph{From this we conclude that space is closed and hence finite.}
\index{Finitude of space}%
If this were not the case, it would scarcely be possible to imagine
how a state of statistical equilibrium could come about. If the
world is closed, spatially, it becomes possible for an observer to see
several pictures of one and the same star. These depict the star at
epochs separated by enormous intervals of time (during which light
travels once entirely round the world). We have yet to inquire
whether the points of space correspond singly and reversibly to the
\PageSep{279}
\index{Analysis situs@{\emph{Analysis situs}}}%
value-quadruples~$x_{i}$ which satisfy the condition~\Eq{(62)}, or whether
two value-systems
\[
(x_{1}, x_{2}, x_{3}, x_{4})
\quad\text{and}\quad
(-x_{1}, -x_{2}, -x_{3}, -x_{4})
\]
correspond to the same point. From the point of view of \emph{analysis
situs} these two possibilities are different even if both spaces are
two-sided. According as the one or the other holds, the total mass
of the world in grammes would be
\[
\frac{\pi a}{2\kappa}
\quad\text{or}\quad
\frac{\pi a}{4\kappa},
\quad\text{respectively.}
\]
Thus our interpretation demands that the total mass that happens
to be present in the world bear a definite relation to the universal
constant $\lambda = \dfrac{2}{a^{2}}$ which occurs in the law of action; this obviously
makes great demands on our credulity.
The radially symmetrical solutions of the modified homogeneous
equations of gravitation that would correspond to a world empty of
mass are derivable by means of the principle of variation (\textit{vide} §\,31
for the notation)
\[
\delta \int (2w \Delta' + \lambda \Delta r^{2})\, dr = 0.
\]
The variation of~$w$ gives, as earlier, $\Delta = 1$. On the other hand,
variation of~$\Delta$ gives
\[
w' = \frac{\lambda}{2} r^{2}\Add{.}
\Tag{(63)}
\]
It we demand regularity at $r = 0$, it follows from~\Eq{(63)} that
\begin{gather*}
w = \frac{\lambda}{6} r^{3} \\
\text{and}\quad
\frac{1}{h^{2}} = f^{2} = 1 - \frac{\lambda}{6} r^{2}\Add{.}
\Tag{(64)}
\end{gather*}
The space may be represented congruently on a ``sphere''
\[
x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = 3a^{2}
\Tag{(65)}
\]
of radius $a\sqrt{3}$ in four-dimensional Euclidean space (whereby one
of the two poles on the sphere, whose first three co-ordinates, $x_{1}$,~$x_{2}$,~$x_{3}$
each $= 0$, corresponds to the centre in our case). The world is a
cylinder erected on this sphere in the direction of a fifth co-ordinate
axis~$t$. But since on the ``greatest sphere'' $x_{4} = 0$, which may be
designated as the equator or the space-horizon for that centre,
$f$~becomes zero, and hence the metrical groundform of the world
becomes singular, we see that the possibility of a stationary empty
world is contrary to the physical laws that are here regarded as
\PageSep{280}
valid. There must at least be masses at the horizon. The calculation
may be performed most readily if (merely to orient ourselves
on the question) we assume an incompressible fluid to be present
there. According to §\,32 the problem of variation that is to be
solved is (if we use the same notation and add the $\lambda$~term)
\[
\delta \int \left\{
\Delta' w + \left(\mu_{0} + \frac{\lambda}{2}\right) r^{2} \Delta - r^{2} vh
\right\} dr = 0.
\]
In comparison with the earlier expression we note that the only
change consists in the constant~$\mu_{0}$ being replaced by $\mu_{0} + \dfrac{\lambda}{2}$. As
earlier, it follows that
\begin{gather*}
w' - \left(\mu_{0} + \frac{\lambda}{2}\right) r^{2} = 0,\qquad
w = -2M + \frac{2\mu_{0} + \lambda}{6}\, r^{3}, \\
\frac{1}{h^{2}} = 1 + \frac{2M}{r} - \frac{2\mu_{0} + \lambda}{6}\, r^{2}\Add{.}
\Tag{(66)}
\end{gather*}
If the fluid is situated between the two meridians $x_{4} = \text{const.}$,
which have a radius~$r_{0}$ ($< a\sqrt{3}$), then continuity of argument with~\Eq{(64)}
demands that the constant
\[
M = \frac{\mu_{0}}{6}\, r_{0}^{3}.
\]
To the first order $\dfrac{1}{h^{2}}$~becomes equal to zero for a value $r = b$ between
$r_{0}$ and~$a\sqrt{3}$. Hence the space may still be represented
on the sphere~\Eq{(65)}, but this representation is no longer congruent
for the zone occupied by fluid. The equation for~$\Delta$
(\Pageref[p.]{265}) now yields a value of~$f$ that does not vanish at the
equator. The boundary condition of vanishing pressure gives a
transcendental relation between $\mu_{0}$~and~$r_{0}$, from which it follows
that, if the mass-horizon is to be taken arbitrarily small, then the
fluid that comes into question must have a correspondingly great
density, namely, such that the total mass does not become less than
a certain positive limit (\textit{vide} \FNote{31}).
The general solution of~\Eq{(63)} is
\[
\frac{1}{h^{2}} = f^{2} = 1 - \frac{2m}{r} - \frac{\lambda}{6}\, r^{2}\qquad
(m = \text{const.}).
\]
It corresponds to the case in which a spherical mass is situated
at the centre. The world can be empty of mass only in a zone
$r_{0} \leq r \leq r_{1}$, in which this~$f^{2}$ is positive; a mass-horizon is again
necessary. Similarly, if the central mass is charged electrically;
for in this case, too, $\Delta = 1$. In the expression for $\dfrac{1}{h^{2}} = f^{2}$ the
\PageSep{281}
electrical term~$+\dfrac{e^{2}}{r^{2}}$ has to be added, and the electrostatic potential
$= \dfrac{e}{r}$.
Perhaps in pursuing the above reflections we have yielded too
readily to the allurement of an imaginary flight into the region of
masslessness. Yet these considerations help to make clear what
the new views of space and time bring within the realm of \Emph{possibility}.
The assumption on which they are based is at any rate
the simplest on which it becomes explicable that, in the world as
actually presented to us, statical conditions obtain as a whole, so
far as the electromagnetic and the gravitational field is concerned,
and that just those solutions of the statical equations are valid
which vanish at infinity or, respectively, converge towards
Euclidean metrics. For on the sphere these equations will have
a unique solution (boundary conditions do not enter into the
question as they are replaced by the postulate of regularity over
the whole of the closed configuration). If we make the constant~$\lambda$
arbitrarily small, the spherical solution converges to that which
satisfies at infinity the boundary conditions mentioned for the infinite
world which results when we pass to the limit.
A metrically homogeneous world is obtained most simply if,
in a five-dimensional space with the metrical groundform $ds^{2} = -\Omega(dx)$,
($-\Omega$~denotes a non-degenerate quadratic form with constant
co-efficients), we examine the four-dimensional ``conic-section''
defined by the equation $\Omega(x) = \dfrac{6}{\lambda}$. Thus this basis gives us a
solution of the Einstein equations of gravitation, modified by the
$\lambda$~term, for the case of no mass. If, as must be the case, the resulting
metrical groundform of the world is to have one positive
and three negative dimensions, we must take for~$\Omega$ a form with
four positive dimensions and one negative, thus
\[
\Omega(x) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} - x_{5}^{2}.
\]
By means of a simple substitution this solution may easily be transformed
into the one found above for the statical case. For if we set
\[
x_{4} = z \cosh t,\qquad
x_{5} = z \sinh t\Add{,}
\]
we get
\[
x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + z^{2} = \frac{6}{\lambda},\qquad
-ds^{2} = (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dz^{2}) - z^{2}\, dt^{2}.
\]
These ``new'' $z$,~$t$ co-ordinates, however, enable only the ``wedge-shaped''
section $x_{4}^{2} - x_{5}^{2} > 0$ to be represented. At the ``edge'' of
the wedge (at which $x_{4} = 0$ simultaneously with $x_{5} = 0$), $t$~becomes
\PageSep{282}
indeterminate. This edge, which appears as a two-dimensional
configuration in the original co-ordinates is, therefore, three-dimensional
in the new co-ordinates; it is the cylinder erected in the
direction of the $t$-axis over the equator $z = 0$ of the sphere~\Eq{(65)}.
The question arises whether it is the first or the second co-ordinate
system that serves to represent the whole world in a regular
manner. In the former case the world would not be static as a
whole, and the absence of matter in it would be in agreement with
physical laws; de~Sitter argues from this assumption (\textit{vide} \FNote{32}).
In the latter case we have a static world that cannot exist without
a mass-horizon; this assumption, which we have treated more
fully, is favoured by Einstein.
\Section[The Metrical Structure of the World as the Origin of Electromagnetic Phenomena]
{35.}{The Metrical Structure of the World as the Origin of
Electromagnetic Phenomena\protect\footnotemark}
\index{Electromagnetic field!origin@{(origin in the metrics of the world)}}%
\index{Field action of electricity!forces (contrasted with inertial forces)}%
\index{Force!(field force andinertial force)}%
\index{Metrics or metrical structure!(general)}%
\footnotetext{\textit{Vide} \FNote{33}.}
We now aim at a final synthesis. To be able to characterise
the physical state of the world at a certain point of it by means of
numbers we must not only refer the neighbourhood of this point
to a co-ordinate system but we must also fix on certain units of
measure. We wish to achieve just as fundamental a point of view
\index{Measure!relativity of}%
with regard to this second circumstance as is secured for the first
one, namely, the arbitrariness of the co-ordinate system, by the
Einstein Theory that was described in the preceding paragraph.
This idea, when applied to geometry and the conception of distance
(in Chapter~II) after the step from Euclidean to Riemann geometry
had been taken, effected the final entrance into the realm of infinitesimal
geometry. Removing every vestige of ideas of ``action at
a distance,'' let us assume that world-geometry is of this kind; we
then find that the metrical structure of the world, besides being
dependent on the quadratic form~\Eq{(1)}, is also dependent on a linear
differential form~$\phi_{i}\, dx_{i}$.
Just as the step which led from the special to the general theory
of relativity, so this extension affects immediately only the world-geometrical
\index{Relativity!of motion}%
foundation of physics. Newtonian mechanics, as also
the special theory of relativity, assumed that uniform translation is
a unique state of motion of a set of vector axes, and hence that the
position of the axes at one moment determines their position in
all other moments. But this is incompatible with the intuitive
principle of the \Emph{relativity of motion}. This principle could be
satisfied, if facts are not to be violated drastically, only by maintaining
the conception of \Emph{infinitesimal} parallel displacement of a
vector set of axes; but we found ourselves obliged to regard the
\PageSep{283}
affine relationship, which determines this displacement, as something
physically real that depends physically on the states of
matter (``\Emph{guiding field}''). The properties of \emph{gravitation} known
\index{Field action of electricity!guiding@{(``guiding'' or gravitational)}}%
from experience, particularly the equality of inertial and gravitational
mass, teach us, finally, that gravitation is already contained
in the guiding field besides inertia. And thus the general theory of
relativity gained a significance which extended beyond its original
\index{Relativity!of magnitude}%
important bearing on \Emph{world-geometry} to a significance which is
specifically \emph{physical}. The same certainty that characterises the
relativity of motion accompanies the principle of the \Emph{relativity of
magnitude}. We must not let our courage fail in maintaining this
principle, according to which the size of a body at one moment does
not determine its size at another, in spite of the existence of rigid
bodies.\footnote
{It must be recalled in this connection that the spatial direction-picture
which a point-eye with a given world-line receives at every moment from a
given region of the world, depends only on the ratios of the~$g_{ik}$'s, inasmuch as
this is true of the geodetic null-lines which are the determining factors in the
propagation of light.}
But, unless we are to come into violent conflict with
fundamental facts, this principle cannot be maintained without
retaining the conception of \emph{infinitesimal} congruent transformation;
that is, we shall have to assign to the world besides its \emph{measure-determination}
at every point also a \emph{metrical relationship}. Now
this is not to be regarded as revealing a ``geometrical'' property
which belongs to the world as a form of phenomena, but as being a
phase-field having physical reality. Hence, as the fact of the
propagation of action and of the existence of rigid bodies leads us
to found the affine relationship on the \emph{metrical} character of the
world which lies a grade lower, it immediately suggests itself to us,
not only to identify the co-efficients of the quadratic groundform
$g_{ik}\, dx_{i}\, dx_{k}$ with the potentials of the gravitational field, but also to
identify \Emph{the co-efficients of the linear groundform~$\phi_{i}\, dx_{i}$ with
the electromagnetic potentials}. The electromagnetic field and
the electromagnetic forces are then derived from the metrical
structure of the world or the \emph{metrics}, as we may call it. No other
truly essential actions of forces are, however, known to us besides
those of gravitation and electromagnetic actions; for all the others
statistical physics presents some reasonable argument which traces
them back to the above two by the method of mean values. We
thus arrive at the inference: \Emph{The world is a $(3 + 1)$-dimensional
metrical manifold; all physical field-phenomena are expressions
of the metrics of the world.} (Whereas the old view
was that the four-dimensional metrical continuum is the scene of
\PageSep{284}
physical phenomena; the physical essentialities themselves are,
however, things that exist ``in'' this world, and we must accept
them in type and number in the form in which experience gives us
cognition of them: nothing further is to be ``comprehended'' of
them.) We shall use the phrase ``state of the world-æther'' as
synonymous with the word ``metrical structure,'' in order to call
attention to the character of reality appertaining to metrical structure;
but we must beware of letting this expression tempt us to
form misleading pictures. In this terminology the fundamental
theorem of infinitesimal geometry states that the guiding field,
and hence also gravitation, is determined by the state of the
æther. The antithesis of ``physical state'' and ``gravitation''
which was enunciated in §\,28 and was expressed in very clear
terms by the division of Hamilton's Function into two parts, is
overcome in the new view, which is uniform and logical in itself.
Descartes' dream of a purely geometrical physics seems to be
attaining fulfilment in a manner of which he could certainly have
had no presentiment. The quantities of intensity are sharply
distinguished from those of magnitude.
The linear groundform~$\phi_{i}\, dx_{i}$ is determined except for an additive
total differential, but the tensor of distance-curvature
\[
f_{ik} = \frac{\dd \phi_{i}}{\dd x_{k}} - \frac{\dd \phi_{k}}{\dd x_{i}}
\]
which is derived from it, is free of arbitrariness. According to
Maxwell's Theory the same result obtains for the electromagnetic
potential. The electromagnetic field-tensor, which we denoted
earlier by~$F_{ik}$, is now to be identified with the distance-curvature~$f_{ik}$.
If our view of the nature of electricity is true, then the first
system of Maxwell's equations
\[
\frac{\dd f_{ik}}{\dd x_{l}}
+ \frac{\dd f_{kl}}{\dd x_{i}}
+ \frac{\dd f_{li}}{\dd x_{k}} = 0
\Tag{(67)}
\]
is an intrinsic law, the validity of which is wholly independent of
whatever physical laws govern the series of values that the physical
phase-quantities actually run through. In a four-dimensional
metrical manifold the simplest integral invariant that exists at all is
\[
\int \vl\, dx = \tfrac{1}{4} \int f_{ik} \vf^{ik}\, dx
\Tag{(68)}
\]
and it is just this one, in the form of \emph{Action}, on which Maxwell's
\index{Action@\emph{Action}!quantum of}%
Theory is founded! We have accordingly a good right to claim that
the whole fund of experience which is crystallised in Maxwell's
Theory weighs in favour of the world-metrical nature of electricity.
And since it is impossible to construct an integral invariant at all
of such a simple structure in manifolds of more or less than four
\PageSep{285}
dimensions the new point of view does not only lead to a deeper
understanding of Maxwell's Theory but the fact that the world is
\index{Maxwell's!theory!(derived from the world's metrics)}%
four-dimensional, which has hitherto always been accepted as merely
``accidental,'' becomes intelligible through it. In the linear groundform
$\phi_{i}\, dx_{i}$ there is an arbitrary factor in the form of an additive
total differential, but there is not a factor of proportionality; the
quantity \emph{Action} is a pure number. But this is only as it should be,
\index{Action@\emph{Action}!quantum of}%
\index{Quantum Theory}%
if the theory is to be in agreement with that atomistic structure of
the world which, according to the most recent results (Quantum
Theory), carries the greatest weight.
The \Emph{statical case} occurs when the co-ordinate system and
the calibration may be chosen so that the linear groundform
becomes equal to~$\phi\, dx_{0}$ and the quadratic groundform becomes
equal to
\[
f^{2}\, dx_{0}^{2} - d\sigma^{2}\Add{,}
\]
whereby $\phi$~and~$f$ are not dependent on the time~$x_{0}$, but only on the
space-co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$, whilst $d\sigma^{2}$~is a definitely positive quadratic
differential form in the three space-variables. This particular
form of the groundform (if we disregard quite particular cases) remains
unaffected by a transformation of co-ordinates and a re-calibration
only if $x_{0}$~undergoes a linear transformation of its own, and if the
space-co-ordinates are likewise transformed only among themselves,
whilst the calibration ratio must be a constant. Hence, in the
statical case, we have a three-dimensional Riemann space with
the groundform~$d\sigma^{2}$ and two scalar fields in it: the electrostatic
potential~$\phi$, and the gravitational potential or the velocity of light~$f$.
The length-unit and the time-unit (centimetre, second) are to be
chosen as arbitrary units; $d\sigma^{2}$~has dimensions~$\text{cm}^{2}$, $f$~has dimensions
$\text{cm} · \text{sec}^{-1}$, and $\phi$~has~$\text{sec}^{-1}$. Thus, as far as one may speak of a
space at all in the general theory of relativity (namely, in the statical
case), it appears as a \Emph{Riemann} space, and not as one of the more
general type, in which the transference of distances is found to be
non-integrable.
We have the case of the special theory of relativity again, if the
co-ordinates and the calibration may be chosen so that
\[
ds^{2} = dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2}).
\]
If $x_{i}$,~$\bar{x}_{i}$ denote two co-ordinate systems for which this normal form
for~$ds^{2}$ may be obtained, then the transition from $x_{i}$ to~$\bar{x}_{i}$ is a conformal
transformation, that is, we find
\[
dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})\Add{,}
\]
except for a factor of proportionality, is equal to
\[
d\bar{x}_{0}^{2} - (d\bar{x}_{1}^{2} + d\bar{x}_{2}^{2} + d\bar{x}_{3}^{2}).
\]
\PageSep{286}
\index{Co-ordinates, curvilinear!hexaspherical@{(hexaspherical)}}%
\index{Hexaspherical co-ordinates}%
The conformal transformations of the four-dimensional Minkowski
world coincide with spherical transformations (\textit{vide} \FNote{34}), that
\index{Spherical!transformations}%
is, with those transformations which convert every ``sphere'' of the
world again into a sphere. A sphere is represented by a linear
homogeneous equation between the homogeneous ``hexaspherical''
co-ordinates
\[
u_{0} : u_{1} : u_{2} : u_{3} : u_{4} : u_{5}
= x_{0} : x_{1} : x_{2} : x_{3} : \frac{(x\Com x) + 1}{2} : \frac{(x\Com x) - 1}{2}\Add{,}
\]
where
\[
(x\Com x) = x_{0}^{2} - (x_{1}^{2} + x_{2}^{2} + x_{3}^{2}).
\]
They are bound by the condition
\[
u_{0}^{2} - u_{1}^{2} - u_{2}^{2} \Typo{}{-} u_{3}^{2} - u_{4}^{2} + u_{5}^{2} = 0.
\]
The spherical transformations therefore express themselves as those
linear homogeneous transformations of the~$u_{i}$'s which leave this
condition, as expressed in the equation, invariant. Maxwell's
\index{Maxwell's!density of action}%
equations of the æther, in the form in which they hold in the
special theory of relativity, are therefore invariant not only with
respect to the $10$-parameter group of the linear Lorentz transformations
but also indeed with respect to the more comprehensive
$15$-parameter group of spherical transformations (\textit{vide} \FNote{35}).
To test whether the new hypothesis about the nature of the
electromagnetic field is able to account for phenomena, we must
work out its implications. We choose as our initial physical law a
Hamilton principle which states that the change in the \emph{Action}
$\Dint \vW\, dx$ for every infinitely small variation of the metrical structure
of the world that vanishes outside a finite region is zero. The
\emph{Action} is an invariant, and hence $\vW$~is a scalar-density (in the true
sense) which is derived from the metrical structure. Mie, Hilbert,
and Einstein assumed the \emph{Action} to be an invariant with respect to
transformations of the co-ordinates. We have here to add the
further limitation that it must also be invariant with respect to the
process of re-calibration, in which $\phi_{i}$,~$g_{ik}$ are replaced by
\[
\phi_{i} - \frac{1}{\lambda}\, \frac{\dd \lambda}{\dd x_{i}}
\quad\text{and}\quad
\lambda g_{ik},
\quad\text{respectively,}
\Tag{(69)}
\]
in which $\lambda$~is an arbitrary positive function of position. We assume
that $\vW$~is an expression of the second order, that is, built up, on the
one hand, of the~$g_{ik}$'s and their derivatives of the first and second
order, on the other hand, of the~$\phi_{i}$'s and their derivatives of the first
order. The simplest example is given by Maxwell's \emph{density of action~$\vl$}.
But we shall here carry out a general investigation without binding
ourselves to any particular form of~$\vW$ at the beginning. According
to Klein's method, used in §\,28 (and which will only now be applied
\PageSep{287}
with full effect), we shall here deduce certain mathematical identities,
which are valid for every scalar-density~$\vW$ which has its origin
in the metrical structure.
I\@. If we assign to the quantities $\phi_{i}$,~$g_{ik}$, which describe the
metrical structure relative to a system of reference, infinitely small
increments $\delta \phi_{i}$,~$\delta g_{ik}$, and if $\rX$~denote a finite region of the world,
then the effect of partial integration is to separate the integral of
the corresponding change~$\delta \vW$ in~$\vW$ over the region~$\rX$ into two
parts: \Inum{(\ia)}~a divergence integral and \Inum{(\ib)}~an integral whose integrand
is only a linear combination of $\delta \phi_{i}$ and~$\delta g_{ik}$, thus
\[
\int_{\rX} \delta \vW\, dx
= \int_{\rX} \frac{\dd (\delta \vv^{k})}{\dd x_{k}}\, dx
+ \int_{\rX} (\vw^{i}\, \delta \phi_{i} + \tfrac{1}{2} \vW^{ik}\, \delta g_{ik})\, dx
\Tag{(70)}
\]
whereby $\vW^{ki} = \vW^{ik}$.
The~$\vw^{i}$'s are components of a contra-variant vector-density, but
the~$\vW_{i}^{k}$'s are the components of a mixed tensor-density of the second
order (in the true sense). The~$\delta \vv^{k}$'s are linear combinations of
\[
\delta \phi_{\alpha},\qquad
\delta g_{\alpha\beta}\quad\text{and}\quad \delta g_{\alpha\beta,i}\qquad
\left[\delta g_{\alpha\beta,i} = \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\right].
\]
We indicate this by the formula
\[
\delta \vv^{k}
= (k\Com \alpha)\, \delta \phi_{\alpha}
+ (k\Com \alpha\Com \beta)\, \delta g_{\alpha\beta}
+ (k\Com i\Com \alpha\Com \beta)\, \delta g_{\alpha\beta,i}.
\]
The~$\delta \vv^{k}$'s are defined uniquely by equation~\Eq{(70)} only if the
normalising condition that the co-efficients $(k\Com i\Com \alpha\Com \beta)$ be symmetrical
in the indices $k$ and~$i$ is added. In the normalisation the~$\delta \vv^{k}$'s are
components of a vector-density (in the true sense), if the~$\delta \phi_{i}$'s are
regarded as the components of a co-variant vector of weight zero
and the~$\delta g_{ik}$'s as the components of a tensor of weight unity.
(There is, of course, no objection to applying another normalisation
in place of this one, provided that it is invariant in the same sense.)
First of all, we express that $\Dint_{\rX} \vW\, dx$ is a calibration invariant,
that is, that it does not alter when the calibration of the world is
altered infinitesimally. If the calibration ratio between the altered
and the original calibration is $\lambda = 1 + \pi$, $\pi$~is an infinitesimal scalar-field
which characterises the event and which may be assigned
arbitrarily. As a result of this process, the fundamental quantities
assume, according to~\Eq{(69)}, the following increments:
\[
\delta g_{ik} = \pi g_{ik},\qquad
\delta \phi_{i} = -\frac{\dd \pi}{\dd x_{i}}\Add{.}
\Tag{(71)}
\]
\PageSep{288}
If we substitute these values in~$\delta \vv^{k}$, let the following expressions
result:
\[
\vs^{k}(\pi) = \pi · \vs^{k} + \frac{\dd \pi}{\dd x_{\alpha}} · \vh^{k\alpha}\Add{.}
\Tag{(72)}
\]
They are the components of a vector-density which depends on the
scalar-field~$\pi$ in a linear-differential manner. It further follows
from this, that, since the~$\dfrac{\dd \pi}{\dd x_{\alpha}}$'s are the components of a co-variant
vector-field which is derived from the scalar-field, $\vs^{k}$~is a vector-density,
and $\vh^{k\alpha}$~is a contra-variant tensor-density of the second
order. The variation~\Eq{(70)} of the integral of \Typo{Action}{\emph{Action}} must vanish on
account of its calibration invariance; that is, we have
\[
\int_{\rX} \frac{\dd \vs^{k}(\pi)}{\dd x_{k}}\, dx
+ \int_{\rX} \left( -\vw^{i}\, \frac{\dd \pi}{\dd x_{i}} + \tfrac{1}{2} \vW_{i}^{i} \pi\right) dx = 0.
\]
If we transform the first term of the second integral by means of
partial integration, we may write, instead of the preceding equation,
\[
\int_{\rX} \frac{\dd \bigl(\vs^{k}(\pi) - \pi \vw^{k}\bigr)}{\dd x_{k}}\, dx
+ \int_{\rX} \pi\left(\frac{\dd \vw^{i}}{\dd x_{i}} + \tfrac{1}{2} \vW_{i}^{i}\right) dx = 0\Add{.}
\Tag{(73)}
\]
This immediately gives the identity
\[
\frac{\dd \vw^{i}}{\dd x_{i}} + \tfrac{1}{2} \vW_{i}^{i} = 0
\Tag{(74)}
\]
in the manner familiar in the calculus of variations. If the
function of position on the left were different from~$0$ at a point~$x_{i}$,
say positive, then it would be possible to mark off a neighbourhood~$\rX$
of this point so small that this function would be positive at every
point within~$\rX$. If we choose this region for~$\rX$ in~\Eq{(73)}, but choose
for~$\pi$ a function which vanishes for points outside~$\rX$ but is $> 0$
throughout~$\rX$, then the first integral vanishes, but the second is
found to be positive---which contradicts equation~\Eq{(73)}. Now that
this has been ascertained, we see that \Eq{(73)}~gives
\[
\int_{\rX} \frac{\dd \bigl(\vs^{k}(\pi) - \pi \vw^{k}\bigr)}{\dd x_{k}}\, dx = 0.
\]
For a given scalar-field~$\pi$ it holds for every finite region~$\rX$, and
consequently we must have
\[
\frac{\dd \bigl(\vs^{k}(\pi) - \pi \vw^{k}\bigr)}{\dd x_{k}} = 0\Add{.}
\Tag{(75)}
\]
If we substitute~\Eq{(72)} in this, and observe that, for a particular
\PageSep{289}
point, arbitrary values may be assigned to $\pi$, $\dfrac{\dd \pi}{\dd x}$, $\dfrac{\dd^{2} \pi}{\dd x_{i}\, \dd x_{k}}$, then this
single formula resolves into the identities:
\[
\frac{\dd \vs^{k}}{\dd x_{k}} = \frac{\dd \vw^{k}}{\dd x_{k}};\qquad
\vs^{i} + \frac{\dd \vh^{\alpha i}}{\dd x_{\alpha}} = \vw^{i};\qquad
\vh^{\alpha\beta} + \vh^{\beta\alpha} = 0\Add{.}
\Tag{(75_{1,2,3})}
\]
According to the third identity, $\vh^{ik}$~is a linear tensor-density of the
second order. In view of the skew-symmetry of~$\vh$ the first is a
result of the second, since
\[
\frac{\dd^{2} \vh^{\alpha\beta}}{\dd x_{\alpha}\, \dd x_{\beta}} = 0.
\]
II\@. We subject the world-continuum to an infinitesimal deformation,
in which each point undergoes a displacement whose
components are~$\xi^{i}$; let the metrical structure accompany the
deformation without being changed. Let $\delta$ signify the change
occasioned by the deformation in a quantity, if we remain at the
same space-time point, $\delta'$~the change in the same quantity if we
share in the displacement of the space-time point. Then, by \Eq{(20)},
\Eq{(21')}, \Eq{(71)}
\[
\left.
\begin{aligned}
-\delta \phi_{i}
&= \left(\phi_{r}\, \frac{\dd \xi^{r}}{\dd x_{i}}
\phantom{{}+ g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}}
\; + \frac{\dd \phi_{i}}{\dd x_{r}}\, \xi^{r}\right) + \frac{\dd \pi}{\dd x_{i}}\Add{,} \\
-\delta g_{ik}
&= \left(g_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
+ g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}
+ \frac{\dd g_{ik}}{\dd x_{r}}\, \xi^{r}\right) - \pi g_{ik}\Add{,}
\end{aligned}
\right\}
\Tag{(76)}
\]
in which $\pi$~denotes an infinitesimal scalar-field that has still been
left arbitrary by our conventions. The invariance of the \emph{Action}
with respect to transformation of co-ordinates and change of
calibration is expressed in the formula which relates to this
variation:
\[
\delta' \int_{\rX} \vW\, dx
= \int_{\rX} \left\{\frac{\dd(\vW \xi^{k})}{\dd x_{k}} + \delta \vW\right\} dx = 0\Add{.}
\Tag{(77)}
\]
If we wish to express the invariance with respect to the co-ordinates
alone we must make $\pi = 0$; but the resulting formulæ
of variation~\Eq{(76)} have not then an invariant character. This convention,
in fact, signifies that the deformation is to make the two
groundforms vary in such a way that the measure~$l$ of a line-element
remains unchanged, that is, $\delta' l = 0$. This equation does
not, however, express the process of congruent transference of a
distance, but indicates that
\[
\delta' l = -l(\phi_{i}\, \delta' x_{i}) = -l(\phi_{i} \xi^{i}).
\]
Accordingly, in~\Eq{(76)} we must choose~$\pi$ not equal to zero but equal
to~$-(\phi_{i} \xi^{i})$ if we are to arrive at invariant formulæ, namely,
\PageSep{290}
\index{Mechanics!fundamental law of!derived@{(derived from field laws)}}%
\[
\left.
\begin{aligned}
-\delta \phi_{i} &= f_{ir} \xi^{r}\Add{,} \\
-\delta g_{ik}
&= \left(g_{ir}\, \frac{\dd \xi^{r}}{\dd x_{k}}
+ g_{kr}\, \frac{\dd \xi^{r}}{\dd x_{i}}\right)
+ \left(\frac{\dd g_{ik}}{\dd x_{r}} + g_{ik} \phi_{r}\right) \xi^{r}\Add{.}
\end{aligned}
\right\}
\Tag{(78)}
\]
The change in the two groundforms which it represents is one
that makes \emph{the metrical structure appear carried along unchanged
by the deformation and every line-element to be transferred congruently}.
The invariant character is easily recognised analytically,
too; particularly in the case of the second equation~\Eq{(78)}, if we
introduce the mixed tensor
\[
\frac{\dd \xi^{i}}{\dd x_{k}} + \Gamma_{kr}^{i} \xi^{r} = \xi_{k}^{i}.
\]
The equation then becomes
\[
-\delta g_{ik} = \xi_{ik} + \xi_{ki}.
\]
Now that the calibration invariance has been applied in~\Inum{I}, we may
in the case of~\Eq{(76)} restrict ourselves to the choice of~$\pi$, which
was discussed just above, and which we found to be alone possible
from the point of view of invariance.
For the variation~\Eq{(78)} let
\[
\vW \xi^{k} + \delta \vv^{k} = \vS^{k}(\xi).
\]
$\vS^{k}(\xi)$~is a vector-density which depends in a linear differential
manner on the arbitrary vector-field~$\xi^{i}$. We write in an explicit
form
\[
\vS^{k}(\xi)
= \vS_{i}^{k} \xi^{i}
+ \Bar{\vH}_{i}^{k\alpha}\, \frac{\dd \xi^{i}}{\dd x_{\alpha}}
+ \tfrac{1}{2} \vH_{i}^{k\alpha\beta}\, \frac{\dd^{2} \xi^{i}}{\dd x_{\alpha}\, \dd x_{\beta}}
\]
(the last co-efficient is, of course, symmetrical in the indices $\alpha$,~$\beta$).
The fact that $\vS^{k}(\xi)$~is a vector-density dependent on the vector-field~$\xi^{i}$
expresses most simply and most fully the character of invariance
possessed by the co-efficients which occur in the expression
for~$\vS^{k}(\xi)$; in particular, it follows from this that the~$\vS_{i}^{k}$'s are not
components of a mixed tensor-density of the second order: we call
them the components of a ``pseudo-tensor-density''. If we insert
in~\Eq{(77)} the expressions \Eq{(70)}~and~\Eq{(78)}, we get an integral, whose
integrand is
\[
\frac{\dd \vS^{k}(\xi)}{\dd x_{k}}
- \xi^{i} \left\{f_{ki} \vw^{k}
+ \tfrac{1}{2}\left(\frac{\dd g_{\alpha\beta}}{\dd x_{i}}
+ g_{\alpha\beta} \phi_{i}\right) \vW^{\alpha\beta}
\right\}
\vW_{i}^{k}\, \frac{\dd \xi^{i}}{\dd x_{k}}.
\]
On account of
\[
\frac{\dd g_{\alpha\beta}}{\dd x_{i}} + g_{\alpha\beta} \phi_{i}
= \Gamma_{\alpha,\beta i} + \Gamma_{\beta,\alpha i}
\]
and of the symmetry of~$\vW^{\alpha\beta}$ we find
\[
\tfrac{1}{2} \left(\frac{\dd g_{\alpha\beta}}{\dd x_{i}} + g_{\alpha\beta} \phi_{i}\right) \vW^{\alpha\beta}
= \Gamma_{\alpha,\beta i} \vW^{\alpha\beta}
= \Gamma_{\beta i}^{\alpha} \vW_{\alpha}^{\beta}.
\]
\PageSep{291}
\index{Einstein's Law of Gravitation!(in its modified form)}%
\index{Energy-momentum, tensor!(of the electromagnetic field)}%
\index{Gravitation!Einstein's Law of (modified form)}%
If we apply partial integration to the last member of the integrand,
we get
\[
\int_{\rX} \frac{\dd\bigl(\vS^{k}(\xi) - \vW_{i}^{k} \xi^{i}\bigr)}{\dd x_{k}}\, dx
+ \int_{\rX} [\dots]_{i} \xi^{i}\, dx = 0.
\]
According to the method of inference used above we get from this
the identities:
\[
[\dots]_{i},\quad\text{that is, }
\left(\frac{\dd \vW_{i}^{k}}{\dd x_{k}} - \Gamma_{\beta}^{\alpha} \vW_{\alpha}^{\beta}\right) + f_{ik} \vw^{k} = 0
\Tag{(79)}
\]
and
\[
\frac{\dd\bigl(\vS^{k}(\xi) - \vW_{i}^{k} \xi^{i}\bigr)}{\dd x_{k}} = 0\Add{.}
\Tag{(80)}
\]
The latter resolves into the following four identities:
\[
\Squeeze{\left.
\begin{gathered}
\frac{\dd \vS_{i}^{k}}{\Typo{\dd x^{k}}{\dd x_{k}}}
= \frac{\dd \vW_{i}^{k}}{\dd x_{k}}; \\
(\Bar{\vH}_{i}^{\alpha\beta} + \Bar{\vH}_{i}^{\beta\alpha})
+ \frac{\dd \vH_{i}^{\gamma\alpha\beta}}{\dd x_{\gamma}} = 0;
\end{gathered}\quad
\begin{gathered}
\vS_{i}^{k} + \frac{\dd \Bar{\vH}_{i}^{\alpha k}}{\dd x_{\alpha}} = \vW_{i}^{k}\Add{;} \\
\vphantom{\dfrac{\dd x}{\dd x}}\vH_{i}^{\alpha\beta\gamma}
+ \vH_{i}^{\beta\gamma\alpha}
+ \vH_{i}^{\gamma\alpha\beta} = 0\Add{.}
\end{gathered}
\right\}}
\Tag{(80_{1,2,3,4})}
\]
If from~\Eq{({}_{4})} we replace in~\Eq{({}_{3})}
\[
\Bar{\vH}_{i}^{\gamma\alpha\beta}\quad\text{by}\quad
- \vH_{i}^{\alpha\beta\gamma} - \vH_{i}^{\beta\alpha\gamma}\Add{,}
\]
we get that
\[
\Bar{\vH}_{i}^{\alpha\beta} - \frac{\dd \vH_{i}^{\alpha\beta\gamma}}{\dd x_{\gamma}}
= \vH_{i}^{\alpha\beta}
\]
is skew-symmetrical in the indices $\alpha$,~$\beta$. If we introduce~$\vH_{i}^{\alpha\beta}$ in
place of~$\Bar{\vH}_{i}^{\alpha\beta}$ we see that \Eq{({}_{3})}~and~\Eq{({}_{4})} are merely statements regarding
symmetry, but \Eq{({}_{2})}~becomes
\[
\vS_{i}^{k} + \frac{\dd \vH_{i}^{\alpha k}}{\dd x_{\alpha}}
+ \frac{\dd^{2} \vH_{i}^{\alpha\beta k}}{\dd x_{\alpha}\, \dd x_{\beta}}
= \vW_{i}^{k}\Add{.}
\Tag{(81)}
\]
\Eq{({}_{1})}~follows from this because, on account of the conditions of
symmetry
\[
\frac{\dd^{2} \vH_{i}^{\alpha\beta}}{\dd x_{\alpha}\, \dd x_{\beta}} = 0,
\quad\text{we get}\quad
\frac{\dd^{3} \vH_{i}^{\alpha\beta\gamma}}
{\dd x_{\alpha}\, \dd x_{\beta}\, \dd x_{\gamma}} = 0\Add{.}
\]
%[** TN: Heading italicized in the original; boldface elsewhere]
\Par{Example.}---In the case of Maxwell's Action-density we have, as
\index{Density!based@{(based on the notion of substance)}}%
is immediately obvious
\[
\delta \vv^{k} = \vf^{ik}\, \delta \phi_{i}.
\]
Consequently
\[
\vs^{i} = 0,\
\vh^{ik} = \vf^{ik};\
\vS_{i}^{k} = \vl \delta_{i}^{k} - f^{i\alpha} \vf^{k\alpha},
\quad\text{and the quantities $\vH = 0$.}
\]
\PageSep{292}
Hence our identities lead to
\begin{gather*}
\vw^{i} = \frac{\dd \vf^{\alpha i}}{\dd x_{\alpha}}\qquad
\frac{\dd \vw^{i}}{\dd x_{i}} = 0,\qquad
\vW_{i}^{i} = 0\Add{,} \\
\vW_{i}^{k} = \vS_{i}^{k}\qquad
\left(\frac{\dd \vS_{i}^{k}}{\dd x_{k}} - \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}}\, \vS^{\alpha\beta}\right)
+ f_{i\alpha}\, \frac{\dd \vf^{\beta\alpha}}{\dd x_{\beta}} = 0.
\end{gather*}
We arrived at the last two formulæ by calculation earlier, the
former on \Pageref{230}, the latter on \Pageref{167}; the latter was found
to express the desired connection between Maxwell's tensor-density~$\vS_{i}^{k}$
of the field-energy and the ponderomotive force.
\Par{Field Laws and Theorems of Conservation.}---If, in~\Eq{(70)}, we
\index{Conservation, law of!energy@{of energy and momentum}}%
\index{Energy-momentum, tensor!(in physical events)}%
take for~$\delta$ an arbitrary variation which vanishes outside a finite
region, and for~$\rX$ we take the whole world or a region such that,
outside it, $\delta = 0$, we get
\[
\int \delta \vW\, dx
= \int (\vw^{i}\, \delta \phi^{i} + \tfrac{1}{2} \vW^{ik}\, \delta g_{ik})\, dx.
\]
If $\Dint \vW\, dx$ is the \emph{Action}, we see from this that the following invariant
laws are contained in Hamilton's Principle:
\index{Hamilton's!principle!general@{(in the general theory of relativity)}}%
\[
\vw^{i} = 0\Add{,}\qquad \vW_{i}^{k} = 0.
\]
Of these, we have to call the former the electromagnetic laws,
the latter the gravitational laws. Between the left-hand sides of
these equations there are five identities, which have been stated
in \Eq{(74)}~and~\Eq{(79)}. Thus there are among the field-equations five
superfluous ones corresponding to the transition (dependent on
five arbitrary functions) from one system of reference to another.
According to~\Eq{(75_{2})} the electromagnetic laws have the following
form:
\[
\frac{\dd \vh^{ik}}{\dd x_{k}} = \vs^{i}
\quad\text{[and~\Eq{(67)}]}
\Tag{(82)}
\]
in full agreement with Maxwell's Theory; $\vs^{i}$~is the density of the
$4$-current, and the linear tensor-density of the second order~$\vh^{ik}$
is the electromagnetic density of field. Without specialising
the \emph{Action} at all we can read off the whole structure of
Maxwell's Theory from the calibration invariance alone. The
particular form of Hamilton's function~$\vW$ affects only the formulæ
which state that current and field-density are determined by the
phase-quantities $\phi_{i}$,~$g_{ik}$ of the æther. In the case of Maxwell's
Theory in the restricted sense ($\vW = \vl$), which is valid only in
empty space, we get $\vh^{ik} = \vf^{ik}$, $\vs^{i} = 0$, which is as it should be.
Just as the~$\vs^{i}$'s constitute the density of the $4$-current, so the
scheme of~$\vS_{i}^{k}$'s is to be interpreted as the pseudo-tensor-density of
\PageSep{293}
\index{Mechanics!fundamental law of!derived@{(derived from field laws)}}%
the energy. In the simplest case, $\vW = \vl$, this explanation becomes
identical with that of Maxwell. According to \Eq{(75_{1})}~and~\Eq{(80_{1})} \Emph{the
theorems of conservation
\[
\frac{\dd \vs^{i}}{\dd x_{i}} = 0,\qquad
\frac{\dd \vS_{i}^{k}}{\dd x_{k}} = 0
\]
are generally valid}; and, indeed, they follow in two ways from
the field laws. For $\dfrac{\dd \vs^{i}}{\dd x_{i}}$~is not only identically equal to~$\dfrac{\dd \Typo{\vw}{\vw^{i}}}{\dd x_{i}}$, but also
to $-\frac{1}{2} \vW_{i}^{i}$, and $\dfrac{\dd \vS_{i}^{k}}{\dd x_{k}}$~is not only identically equal to~$\dfrac{\dd \vW_{i}^{k}}{\dd x_{k}}$, but also
to $\Gamma_{i\beta}^{\alpha} \vW_{\alpha}^{\beta} - f_{ik} \vw^{k}$. The form of the gravitational equations is given
by~\Eq{(81)}. The field laws and their accompanying laws of conservation
may, by \Eq{(75)}~and~\Eq{(80)}, be summarised conveniently in the two
equations
\[
\frac{\dd \vs^{i}(\pi)}{\dd x_{i}} = 0,\qquad
\frac{\dd \vS^{i}(\xi)}{\dd x_{i}} = 0.
\]
Attention has already been directed above to the intimate connection
between the laws of conservation of the energy-momentum
and the co-ordinate-invariance. To these four laws there is to be
added the law of conservation of electricity, and, corresponding to
it, there must, logically, be a property of invariance which will introduce
a fifth arbitrary function; the calibration-invariance here
appears as such. Earlier we derived the law of conservation of
energy-momentum from the co-ordinate-invariance only owing to
the fact that Hamilton's function consists of two parts, the \emph{\Typo{action}{Action}}-function
of the gravitational field and that of the ``physical phase'';
each part had to be treated differently, and the component results had
to be combined appropriately (§\,33). If those quantities, which are
derived from $\vW \xi^{k} + \delta \vv^{k}$ by taking the variation of the fundamental
quantities from~\Eq{(76)} for the case $\pi = 0$, instead of from~\Eq{(78)}, are
distinguished by a prefixed asterisk, then, in consequence of the
co-ordinate-invariance, the ``theorems of conservation'' $\dfrac{\dd {}^{*}\vS_{i}^{k}}{\dd x_{k}} = 0$
are generally valid. But the ${}^{*}\vS_{i}^{k}$'s are not the energy-momentum
components of the \Chg{two-fold}{twofold} action-function which have been used
as a basis since §\,28. For the gravitational component ($\vW = \vG$)
we defined the energy by means of~${}^{*}\vS_{i}^{k}$ (§\,33), but for the electromagnetic
component ($\vW = \vL$, §\,28) we introduced~$\vW_{i}^{k}$ as the
energy components. This second component~$\vL$ contains only the
$g_{ik}$'s~themselves, not their derivatives; for a quantity of this kind we
have, by~\Eq{(80_{2})}, $\vW_{i}^{k} = \vS_{i}^{k}$. Hence (\Emph{if we use the transformations
\PageSep{294}
which the fundamental quantities undergo during an infinitesimal
alteration of the calibration}), we can adapt the
two different definitions of energy to one another although we
cannot reconcile them entirely. These discrepancies are removed
only here since it is the new theory which first furnishes us with
an explanation of the current~$\vs^{i}$, of the electromagnetic density of
field~$\vh^{ik}$, and of the \Emph{energy}~$\vS_{i}^{k}$, which is no longer bound by the
assumption that the \emph{Action} is composed of two parts, of which the
one does not contain the~$\phi_{i}$'s and their derivatives, and the other
does not contain the derivatives of the~$g^{ik}$'s. The virtual deformation
of the world-continuum which leads to the definition of~$\vS_{i}^{k}$
must, accordingly, carry along the metrical structure and the
line-elements ``unchanged'' in \Emph{our} sense and not in that of
\Emph{Einstein}. The laws of conservation of the~$\vs^{i}$'s and the~$\Typo{\vS_{i}}{\vS_{i}^{k}}$'s are
then likewise not bound by an assumption concerning the composition
of the \emph{Action}. Thus, after the total energy had been introduced
in §\,33, we have once again passed beyond the stand taken
in §\,28 to a point of view which gives a more compact survey
of the whole. What is done by Einstein's theory of gravitation
with respect to the equality of inertial and gravitational matter,
namely, that it recognises their identity as necessary but not as a
consequence of an undiscovered law of physical nature, is accomplished
by the present theory with respect to the facts that find
expression in the structure of Maxwell's equations and the laws of
conservation. Just as is the case in §\,33 in which we integrate over
the cross-section of a canal of the system, so we find here that, as
a result of the laws of conservation, if the $\vs^{i}$'s~and $\vS_{i}^{k}$'s vanish
outside the canal, the system has a constant charge~$e$ and a constant
\index{Charge!(\emph{generally})}%
\index{Electrical!charge!flux@{(as a flux of force)}}%
energy-momentum~$J$. Both may be represented, by Maxwell's
equations~\Eq{(82)} and the gravitational equations~\Eq{(81)}, as the
flux of a certain spatial field through a surface~$\Omega$ that encloses the
system. If we regard this representation as a definition, the integral
theorems of conservation hold, even if the field has a real
singularity within the canal of the system. To prove this, let us
replace this field within the canal in any arbitrary way (preserving,
of course, a continuous connection with the region outside it) by a
regular field, and let us define the~$\vs^{i}$'s and the~$\vS_{i}^{k}$'s by the equations
\Eq{(82)},~\Eq{(81)} (in which the right-hand sides are to be replaced by
zero) in terms of the quantities $\vh$~and~$\vH$ belonging to the altered
field. The integrals of these fictitious quantities $\vs^{0}$~and~$\vS_{i}^{0}$, which
are to be taken over the cross-section of the canal (the interior of~$\Omega$),
are constant; on the other hand, they coincide with the fluxes
\PageSep{295}
mentioned above over the surface~$\Omega$, since on~$\Omega$ the imagined field
coincides with the real one.
\Section{36.}{Application of the Simplest Principle of Action. The
Fundamental Equations of Mechanics}
We have now to show that if we uphold our new theory it is
possible to make an assumption about~$\vW$ which, as far as the
results that have been confirmed in experience are concerned,
agrees with Einstein's Theory. The simplest assumption\footnote
{\textit{Vide} \FNote{36}.}
for
purposes of calculation (I do not insist that it is realised in
nature) is:
\[
\vW = -\tfrac{1}{4} F^{2} \sqrt{g} + \alpha \vl\Add{.}
\Tag{(83)}
\]
The quantity \emph{Action} is thus to be composed of the volume, measured
in terms of the radius of curvature of the world as unit of length
(cf.~\Eq{(62)}, §\,17) and of Maxwell's action of the electromagnetic field;
the positive constant~$\alpha$ is a pure number. It follows that
\[
\delta \vW = -\tfrac{1}{2} F \delta(F \sqrt{g}) + \tfrac{1}{4} F^{2} \delta\sqrt{g} + \alpha\, \delta \vl.
\]
We assume that $-F$~is positive; the calibration may then be uniquely
determined by the postulate $F = -1$; thus
\[
\delta \vW = \text{the variation of $\tfrac{1}{2} F \sqrt{g} + \tfrac{1}{4} \sqrt{g} + \alpha \vl$.}
\]
If we use the formula~\Eq{(61)}, §\,17 for~$F$, and omit the divergence
\[
\delta \frac{(\dd \sqrt{g} \phi^{i})}{\dd x_{i}}
\]
which vanishes when we integrate over the world, and if, by means
of partial integration, we convert the world-integral of $\delta(\frac{1}{2} R \sqrt{g})$
into the integral of~$\delta \vG$ (§\,28), then our principle of action takes the
form
\[
\delta \int \vV\, dx = 0,
\text{ and we get }
\vV = \vG + \alpha \vl + \tfrac{1}{4} \sqrt{g} \bigl\{1 - 3(\phi_{i} \phi^{i})\bigr\}\Add{.}
\Tag{(84)}
\]
This normalisation denotes that we are measuring with cosmic
measuring rods. If, in addition, we choose the co-ordinates~$x_{i}$ so
that points of the world whose co-ordinates differ by amounts of
the order of magnitude~$1$, are separated by cosmic distances, then
we may assume that the~$g_{ik}$'s and the~$\phi_{i}$'s are of the order of magnitude~$1$.
(It is, of course, a fact that the potentials vary perceptibly
by amounts that are extraordinarily small in comparison with cosmic
distances.) By means of the substitution $x_{i} = \epsilon x_{i}'$ we introduce
co-ordinates of the order of magnitude in general use (that is having
dimensions comparable with those of the human body); $\epsilon$~is a very
small constant. The~$g_{ik}$'s do not change during this transformation,
\PageSep{296}
if we simultaneously perform the re-calibration which multiplies~$ds^{2}$
by~$\dfrac{1}{\epsilon^{2}}$. In the new system of reference we then have
\[
g_{ik}' = g_{ik},\qquad
\phi_{i}' = \phi_{i};\qquad
F' = -\epsilon^{2}.
\]
$\dfrac{1}{\epsilon}$~is accordingly, in our ordinary measures, the radius of curvature
of the world. If $g_{ik}$,~$\phi_{i}$ retain their old significance, but if we take
$x_{i}$~to represent the co-ordinates previously denoted by~$x_{i}'$, and if
$\Gamma_{ik}^{r}$~are the components of the affine relationship corresponding to
these co-ordinates, then
\begin{gather*}
\vV = (\vG + \alpha \vl)
+ \frac{\epsilon^{2}}{4} \sqrt{g} \bigl\{1 - 3(\phi_{i} \phi^{i})\bigr\}, \\
\Gamma_{ik}^{r} = \Chr{ik}{r}
+ \tfrac{1}{2} \epsilon^{2} (\delta_{i}^{r} \phi_{k} + \delta_{k}^{r} \phi_{i} - g_{ik} \phi^{r}).
\end{gather*}
\emph{Thus, by neglecting the exceedingly small cosmological terms, we
arrive exactly at the classical Maxwell-Einstein theory of electricity
and gravitation.} To make the expression correspond exactly with
that of §\,34 we must set $\dfrac{\epsilon^{2}}{2} = \lambda$. Hence our theory necessarily
gives us Einstein's cosmological term $\dfrac{1}{2} \lambda \sqrt{g}$. The uniform distribution
of electrically neutral matter at rest over the whole of
(spherical) space is thus a state of equilibrium which is compatible
with our law. But, whereas in Einstein's Theory (cf.~§\,34) there
must be a pre-established harmony between the universal physical
constant~$\lambda$ that occurs in it, and the total mass of the earth (because
each of these quantities in themselves already determine the curvature
of the world), here (where $\lambda$~\Emph{denotes} merely the curvature),
we have that the mass present in the world \Emph{determines} the
curvature. It seems to the author that just this is what makes
Einstein's cosmology physically possible. In the case in which a
physical field is present, Einstein's cosmological term must be
supplemented by the further term $-\dfrac{3}{2} \lambda \sqrt{g} (\phi_{i} \phi^{i})$; and in the components~$\Gamma_{ik}^{r}$
of the gravitational field, too, a cosmological term that
is dependent on the electromagnetic potentials occurs. Our theory
is founded on a definite unit of electricity; let it be~$e$ in ordinary
electrostatic units. Since, in~\Eq{(84)}, if we use these units, $\dfrac{2\kappa}{c^{2}}$~occurs
in place of~$\alpha$, we have
\[
\frac{2e^{2} \kappa}{c^{2}} = \frac{\alpha}{-F},\qquad
\frac{e \sqrt{\kappa}}{c} = \frac{1}{\epsilon} \sqrt{\frac{\alpha}{2}}:
\]
\PageSep{297}
our unit is that quantity of electricity whose gravitational radius is
$\sqrt{\dfrac{\Typo{a}{\alpha}}{2}}$~times the radius of curvature of the world. It is, therefore,
like the quantum of action~$\vl$, of cosmic dimensions. The cosmological
factor which Einstein added to his theory later is part of
ours from the very beginning.
Variation of the~$\phi_{i}$'s gives us Maxwell's equations\Typo{.}{}
\[
\frac{\dd \vf^{ik}}{\dd x_{k}} = \vs^{i}
\]
and, in this case, we have simply
\[
\vs^{i} = -\frac{3\lambda}{\alpha}\, \phi_{i} \sqrt{g}.
\]
Just as according to Maxwell the æther is the seat of energy and
mass so we obtain here an electric charge (plus current) diffused
thinly throughout the world. \Typo{Variatio}{Variation} of the~$g_{ik}$'s gives the gravitational
equations
\[
\vR_{i}^{k} - \frac{\vR + \lambda \sqrt{g}}{2}\, \delta_{i}^{k} = \alpha \vT_{i}^{k}
\Tag{(85)}
\]
where
\[
\vT_{i}^{k}
= \bigl\{\vl + \tfrac{1}{2}(\phi_{r} \vs^{r})\bigr\} \delta_{i}^{k}
- f_{ir}\vf^{kr}
= \phi_{i} \vs^{k}.
\]
The conservation of electricity is expressed in the divergence
equation
\[
\frac{\dd (\sqrt{g} \phi^{i})}{\dd x_{i}} = 0\Add{.}
\Tag{(86)}
\]
This follows, on the one hand, from Maxwell's equations, but must,
on the other hand, be derivable from the gravitational equations
according to our general results. We actually find, by contracting
the latter equations with respect to~$i\Com k$, that
\[
R + 2\lambda = \tfrac{3}{2} (\phi_{i} \phi^{i})\Add{,}
\]
and this in conjunction with $-F = 2\lambda$ again gives~\Eq{(86)}. We get
for the pseudo-tensor-density of the energy-momentum, as is to
be expected
\[
\vS_{i}^{k}
= \alpha \vT_{i}^{k}
+ \left\{\vG + \tfrac{1}{2}\lambda \sqrt{g} \delta_{i}^{k}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}\right\}.
\]
From the equation $\delta' \Dint \vV\, dx = 0$ for a variation~$\delta'$ which is produced
by the displacement in the true sense [from formula~\Eq{(76)} with $\xi^{i} = \text{const.}$,
$\pi = 0$], we get
\[
\frac{\dd ({}^{*} \vS_{i}^{k} \xi^{i})}{\dd x_{k}} = 0\Add{,}
\Tag{(87)}
\]
\PageSep{298}
where
\[
{}^{*}\vS_{i}^{k}
= \vV \delta_{i}^{k}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}
+ \alpha \frac{\dd \phi}{\dd x_{i}} \vf^{kr}.
\]
To obtain the conservation theorems, we must, according to our
earlier remarks, write Maxwell's equations in the form
\[
\frac{\dd \left(\pi \vs^{i} + \dfrac{\dd \pi}{\dd x_{k}} \vf^{ik}\right)}{\dd x_{i}} = 0\Add{,}
\]
then set $\pi = -(\phi_{i} \xi^{i})$, and, after multiplying the resulting equation
by~$\alpha$, add it to~\Eq{(87)}. We then get, in fact,
\[
\frac{\dd (\vS_{i}^{k} \xi^{i})}{\dd x_{k}} = 0.
\]
The following terms occur in~$\vS_{i}^{k}$: the Maxwell energy-density of
the electromagnetic field
\[
\vl \delta_{i}^{k} - f_{ir} \vf^{kr},
\]
the gravitational energy
\[
\vG \delta_{i}^{k}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}\Add{,}
\]
and the supplementary cosmological terms
\[
\tfrac{1}{2}(\lambda \sqrt{g} + \phi_{r} \vs^{r}) \delta_{i}^{k}
- \phi_{i} \vs^{k}.
\]
The statical world is by its own nature calibrated. The question
arises whether $F = \text{const.}$ for this calibration. The answer is in the
affirmative. For if we re-calibrate the statical world in accordance
with the postulate $F = -1$ and distinguish the resulting quantities
by a horizontal bar, we get
\begin{gather*}
\bar{\phi}_{i} = -\frac{F_{i}}{F},
\quad\text{where we set }
F_{i} = \frac{\dd F}{\dd x_{i}}\quad (i = 1, 2, 3)\Add{,} \\
\bar{g}_{ik} = -F g_{ik},
\quad\text{that is, }
\bar{g}^{ik} = -\frac{g^{ik}}{F},\qquad
\sqrt{g} = F^{2} \sqrt{g}\Add{,}
\end{gather*}
and equation~\Eq{(86)} gives
\[
\sum_{i=1}^{3} \frac{\dd \vF^{i}}{\dd x_{i}} = 0\qquad
(\vF^{i} = \sqrt{g} F^{i})\Add{.}
\]
From this, however, it follows that $F = \text{const}$.
From the fact that a further electrical term becomes added to
Einstein's cosmological term, the existence of a material particle
becomes possible without a mass horizon becoming necessary. The
particle is necessarily charged electrically. If, in order to determine
\PageSep{299}
the radially symmetrical solutions for the statical case, we
again use the old terms of §\,31, and take~$\phi$ to mean the electrostatic
potential, then the integral whose variation must vanish, is
\[
\int \vV r^{2}\, dr
= \int \left\{
w \Delta' - \frac{\alpha r^{2} \phi'^{2}}{2\Delta}
+ \frac{\lambda r^{2}}{2} \left(\Delta - \frac{3h^{2} \phi^{2}}{2\Delta}\right)
\right\} dr
\]
(the accent denotes differentiation with respect to~$r$). Variation of
$w$,~$\Delta$, and~$\phi$, respectively, leads to the equations
\begin{gather*}
\Delta \Delta' = \frac{3\lambda}{4} h^{4} \phi^{2} r\Add{,} \\
w' = \frac{\lambda r^{2}}{2}\left(1 + \tfrac{3}{2}\, \frac{h^{2} \phi^{2}}{\Delta^{2}}\right)
+ \frac{\alpha}{2}\, \frac{r^{2} \phi'^{2}}{\Delta^{2}}\Add{,} \\
\left(\frac{r^{2} \phi'}{\Delta}\right)'
= \frac{3}{2\alpha}\, \frac{h^{2} r^{2} \phi}{\Delta}.
\end{gather*}
As a result of the normalisations that have been performed, the
spatial co-ordinate system is fixed except for a Euclidean rotation,
and hence $h^{2}$~is uniquely determined. In $f$~and~$\phi$, as a result of the
free choice of the unit of time, a common constant factor remains
arbitrary (a circumstance that may be used to reduce the order of
the problem by~$1$). If the equator of the space is reached when
$r = r_{0}$, then the quantities that occur as functions of $z = \sqrt{r_{0}^{2} - r^{2}}$
must exhibit the following behaviour for $z = 0$: $f$~and~$\phi$ are regular,
and $f \neq 0$; $h^{2}$~is infinite to the second order, $\Delta$~to the first order.
The differential equations themselves show that the development of~$h^{2} z^{2}$
according to powers of~$z$ begins with the term~$h_{0}^{2}$, where
\[
h_{0}^{2} = \frac{2r_{0}^{2}}{\lambda r_{0}^{2} - 2}
\]
---this proves, incidentally, that $\lambda$~must be positive (the curvature~$F$
negative) and that $r_{0}^{2} > \dfrac{2}{\lambda}$---whereas for the initial values \Typo{of}{} $f_{0}$,~$\Typo{\phi}{\phi_{0}}$,
of $f$~and~$\phi$ we have
\[
f_{0}^{2} = \frac{3\lambda}{4} h_{0}^{2} \phi_{0}^{2}.
\]
% [** TN: [sic] "diametral"]
If diametral points are to be identified, $\phi$~must be an even function
of~$z$, and the solution is uniquely determined by the initial values
for $z = 0$, which satisfy the given conditions (\textit{vide} \FNote{37}). It
cannot remain regular in the whole region $0 \leq r \leq r_{0}$, but must, if
we let $r$~decrease from~$r_{0}$, have a singularity at least ultimately
when $r = 0$. For otherwise it would follow, by multiplying the
differential equation of~$\phi$ by~$\phi$, and integrating from $0$ to~$r_{0}$, that
\[
\int_{0}^{r_{0}} \frac{r^{2}}{\Delta}
\left(\phi'^{2} + \frac{3}{2\alpha} h^{2} \phi^{2}\right) dr = 0.
\]
\PageSep{300}
Matter is accordingly a true singularity of the field. The fact
that the phase-quantities vary appreciably in regions whose
linear dimensions are very small in comparison with~$\dfrac{1}{\sqrt{l}}$ may
be explained, perhaps, by the circumstance that a value must be
taken for~$r_{0}^{2}$ which is enormously great in comparison with~$\dfrac{1}{\lambda}$. The
fact that all elementary particles of matter have the same charge
and the same mass seems to be due to the circumstance that
they are all embedded in the same world (of the same radius~$r_{0}$);
this agrees with the idea developed in §\,32, according to which the
charge and the mass are determined from infinity.
In conclusion, we shall set up the mechanical equations that
govern the motion of a material particle. In actual fact we have
not yet derived these equations in a form which is admissible from
the point of view of the general theory of relativity; we shall now
endeavour to make good this omission. We shall also take this
opportunity of carrying out the intention stated in §\,32, that is, to show
that in general the inertial mass is the flux of the gravitational field
through a surface which encloses the particle, even when the
matter has to be regarded as a singularity which limits the field
and lies, so to speak, outside it. In doing this we are, of course,
debarred from using a substance which is in motion; the hypotheses
corresponding to the latter idea, namely (§\,27):
\[
dm\, ds = \mu\, dx,\qquad
\vT_{i}^{k} = \mu u_{i} u^{k}
\]
are quite impossible here, as they contradict the postulated properties
of invariance. For, according to the former equation, $\mu$~is a scalar-density
of weight~$\frac{1}{2}$, and, according to the latter, one of weight~$0$,
since $\vT_{i}^{k}$~is a tensor-density in the true sense. And we see that
these initial conditions are impossible in the new theory for the
same reason as in Einstein's Theory, namely, because they lead to a
false value for the mass, as was mentioned at the end of §\,33. This
is obviously intimately connected with the circumstance that the
integral $\Dint dm\, ds$ has now no meaning at all, and hence cannot be
introduced as ``substance-action of gravitation''. We took the first
\index{Substance-action of electricity and gravitation!mass@{($=$~mass)}}%
step towards giving a real proof of the mechanical equations in §\,33.
There we considered the special case in which the body is completely
isolated, and no external forces act on it.
From this we see at once that we must start from the laws of
conservation
\[
\frac{\dd \vS_{i}^{k}}{\dd x_{k}} = 0
\Tag{(89)}
\]
\PageSep{301}
which hold for the \Emph{total energy}. Let a volume~$\Omega$, whose dimensions
\index{Energy!(total energy of a system)}%
are great compared with the actual essential nucleus of the
particle, but small compared with those dimensions of the external
field which alter appreciably, be marked off around the material
particle. In the course of the motion $\Omega$~describes a canal in the
world, in the interior of which the current filament of the material
particle flows along. Let the co-ordinate system consisting of the
``time-co-ordinate'' $x_{0} = t$ and the ``space-co-ordinates'' $x_{1}$,~$x_{2}$,~$x_{3}$,
be such that the spaces $x_{0} = \text{const.}$ intersect the canal (the cross-section
is the volume~$\Omega$ mentioned above). The integrals
\[
\int_{\Omega} \vS_{i}^{0}\, dx_{1}\, dx_{2}\, dx_{3} = J_{i}\Add{,}
\]
which are to be taken in a space $x_{0} = \text{const.}$ over~$\Omega$, and which
are functions of the time alone, represent the energy ($i = 0$) and
the momentum ($i = 1, 2, 3$) of the material particle. If we integrate
the equation~\Eq{(89)} in the space $x_{0} = \text{const.}$ over~$\Omega$, the first
member ($k = 0$) gives the time-derivative~$\dfrac{dJ_{i}}{dt}$; the integral sum
over the three last terms, however, becomes transformed by Gauss'
Theorem into an integral~$K_{i}$ which is to be taken over the surface
of~$\Omega$. In this way we arrive at the mechanical equations
\[
\frac{dJ_{i}}{dt} = K_{i}\Add{.}
\Tag{(90)}
\]
On the left side we have the components of the ``inertial force,''
\index{Inertial force}%
and on the right the components of the external ``field-force''.
Not only the field-force but also the four-dimensional momentum~$J_{i}$
may be represented, in accordance with a remark at the end of
§\,35, as a flux through the surface of~$\Omega$. If the interior of the canal
encloses a real singularity of the field the momentum must, indeed,
be defined in the above manner, and then the device of the
``fictitious field,'' used at the end of §\,35, leads to the mechanical
equations proved above. \emph{It is of fundamental importance to notice
that in them only such quantities are brought into relationship with
one another as are determined by the course of the field outside the
particle \emph{(on the surface of~$\Omega$)}, and have nothing to do with the
singular states or phases in its interior.} The antithesis of kinetic
and potential which receives expression in the fundamental law of
mechanics does not, indeed, depend actually on the separation of
energy-momentum into one part belonging to the external field
and another belonging to the particle (as we pictured it in §\,25), but
rather on this juxtaposition, conditioned by the resolution into space
\PageSep{302}
and time, of the first and the three last members of the divergence
equations which make up the laws of conservation, that is, on the
circumstance that the singularity canals of the material particles
have an infinite extension in only \Emph{one} dimension, but are very
limited in \Emph{three} other dimensions. This stand was taken most
definitely by Mie in the third part of his epoch-making \Title{Foundations
of a Theory of Matter}, which deals with ``Force and Inertia''
(\textit{vide} \FNote{38}). Our next object is to work out the full consequences
of this view for the principle of action adopted in this chapter.
To do this, it is necessary to ascertain exactly the meaning of
the electromagnetic and the gravitational equations. If we discuss
Maxwell's equations first, we may disregard gravitation entirely
and take the point of view presented by the special theory of relativity.
We should be reverting to the notion of substance if we
were to interpret the Maxwell-Lorentz equation
\[
\frac{\dd f^{ik}}{\dd x_{k}} = \rho u^{i}
\]
so literally as to apply it to the volume-elements of an electron.
Its true meaning is rather this: Outside the $\Omega$-canal, the homogeneous
equations
\[
\frac{\dd f^{ik}}{\dd x_{k}} = 0
\Tag{(91)}
\]
hold. %[** TN: "hold" set in the display in the original]
The only statical radially symmetrical solution~$\bar{f}^{ik}$ of~\Eq{(91)} is that
derived from the potential~$\dfrac{e}{r}$; it gives the flux~$e$ (and not~$0$, as it
would be in the case of a solution of~\Eq{(91)} which is free from singularities)
of the electric field through an envelope~$\Omega$ enclosing the
particle. On account of the linearity of equations~\Eq{(91)}, these properties
are not lost when an arbitrary solution~$f_{ik}$ of equations~\Eq{(91)},
free from singularities, is added to~$\bar{f}_{ik}$; such a one is given by $f_{ik} = \text{const}$.
\Emph{The field which surrounds the moving electron must
be of the type:} $f_{ik} + \bar{f}_{ik}$, if we introduce at the moment under
consideration a co-ordinate system in which the electron is at rest.
This assumption concerning the constitution of the field outside~$\Omega$
is, of course, justified only when we are dealing with quasi-stationary
motion, that is, when the world-line of the particle
deviates by a sufficiently small amount from a straight line. The
term~$\rho u^{i}$ in Lorentz's equation is to express the general effect of the
charge-singularities for a region that contains many electrons.
But it is clear that this assumption comes into question only for
\Emph{quasi-stationary motion}. Nothing at all can be asserted about
what happens during rapid acceleration. The opinion which is so
\PageSep{303}
generally current among physicists nowadays, that, according to
classical electrodynamics, a greatly accelerated particle emits radiation,
seems to the author quite unfounded. It is justified only if
Lorentz's equations are interpreted in the too literal fashion repudiated
above, and if, also, it is assumed that the constitution of
the electron is not modified by the acceleration. \Emph{Bohr's Theory
of the Atom} has led to the idea that there are individual stationary
\index{Atom, Bohr's}%
\index{Bohr's model of the atom}%
\index{Stationary!orbits in the atom}%
orbits for the electrons circulating in the atom, and that they may
move permanently in these orbits without emitting radiations; only
when an electron jumps from one stationary orbit to another is the
energy that is lost by the atom emitted as electromagnetic energy of
vibration (\textit{vide} \FNote{39}). If matter is to be regarded as a boundary-singularity
of the field, our field-equations make assertions only
about \Emph{the possible states of the field}, and \Emph{not about the conditioning
of the states of the field by the matter}. This gap is
filled by the \Emph{Quantum Theory} in a manner of which the underlying
\index{Quantum Theory}%
principle is not yet fully grasped. The above assumption
about the singular component~$\bar{f}$ of the field surrounding the particle
is, in our opinion, true for a quasi-stationary electron. We may,
of course, work out other assumptions. If, for example, the particle
is a radiating atom, the~$\bar{f}^{ik}$'s will have to be represented as the field
of an oscillating Hertzian dipole. (This is a possible state of the
field which is caused by matter in a manner which, according to
Bohr, is quite different from that imagined by Hertz.)
As far as gravitation is concerned, we shall for the present
adopt the point of view of the original Einstein Theory. In it the
(homogeneous) gravitational equations have (according to §\,31) a
statical radially symmetrical solution, which depends \Emph{on a single
constant~$m$, the mass}. The flux of a gravitational field through
\index{Mass!producing@{(producing a gravitational field)}}%
a sufficiently great sphere described about the centre is not equal to~$0$,
as it should be if the solution were free from singularities, but
equal to~$m$. We assume that this solution is characteristic of the
moving particle in the following sense: We consider the values
traversed by the~$g_{ik}$'s outside the canal to be extended over the
canal, by supposing the narrow deep furrow, which the path of the
material particle cuts out in the metrical picture of the world,
to be smoothed out, and by treating the stream-filament of the
particle as a line in this smoothed-out metrical field. Let $d$s be
the corresponding proper-time differential. For a point of the
stream-filament we may introduce a (``normal'') co-ordinate
system such that, at that point,
\[
ds^{2} = dx_{0}^{2} - (dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2})\Add{,}
\]
\PageSep{304}
the derivatives $\dfrac{\dd g_{\alpha\beta}}{\dd x_{i}}$ vanish, and the direction of the stream-filament
is given by
\[
dx_{0} : dx_{1} : dx_{2} : dx_{3} = 1 : 0 : 0 : 0.
\]
In terms of these co-ordinates the field is to be expressed by the
above-mentioned statical solution (only, of course, in a certain
neighbourhood of the world-point under consideration, from which
the canal of the particle is to be cut out). If we regard the normal
co-ordinates~$x_{i}$ as Cartesian co-ordinates in a four-dimensional
Euclidean space, then the picture of the world-line of the particle
becomes a definite curve in the Euclidean space. Our assumption
is, of course, admissible again only if the motion is quasi-stationary,
that is, if this picture-curve is only slightly curved at the point
under consideration. (The transformation of the homogeneous
gravitational equations into non-homogeneous ones, on the right
side of which the tensor $\mu u_{i} u_{k}$ appears, takes account of the singularities,
due to the presence of masses, by fusing them into a continuum;
this assumption is legitimate only in the quasi-stationary
case.)
To return to the derivation of the mechanical equations! We
shall use, once and for all, the calibration normalised by $F = \text{const.}$,
and we shall neglect the cosmological terms outside the canal. The
influence of the charge of the electron on the gravitational field is, as
we know from §\,32, to be neglected in comparison with the influence
of the mass, provided the distance from the particle is sufficiently
great. Consequently, if we base our calculations on the normal co-ordinate
system, we may assume the gravitational field to be that
mentioned above. The determination of the electromagnetic field is
then, as in the gravitational case, a linear problem; it is to have the
form $f_{ik} + \bar{f}_{ik}$ mentioned above (with $f_{ik} = \text{const.}$ on the surface of~$\Omega$).
But this assumption is compatible with the field-laws only if
$e = \text{const}$. To prove this, we shall deduce from a fictitious field
that fills the canal regularly and that links up with the really
existing field outside, that
\[
\frac{\dd \vf^{ik}}{\dd x_{k}} = \vs^{i},\qquad
\int_{\Omega} \vs^{0}\, dx_{1}\, dx_{2}\, dx_{3} = e^{*}
\]
in any arbitrary co-ordinate system; $e^{*}$~is independent of the choice
of the fictitious field, inasmuch as it may be represented as a field-flux
through the surface of~$\Omega$. Since (if we neglect the cosmological
terms) the~$\vs^{i}$'s on this surface vanish, the equation of definition gives
us, if $\dfrac{\dd \vs^{i}}{\dd x_{i}} = 0$ is integrated, $\dfrac{de^{*}}{dt} = 0$; moreover, the arguments set
\PageSep{305}
out in §\,33 show that $e^{*}$~is independent of the co-ordinate system
chosen. If we use the normal co-ordinate system at one point, the
representation of~$e^{*}$ as a field-flux shows that $e^{*} = e$.
Passing on from the charge to the momentum, we must notice
\index{Mass!flux@{(as a flux of force)}}%
at once that, with regard to the representation of the energy-momentum
components by means of field-fluxes, we may not refer
to the general theory of §\,35, because, by applying the process of
partial integration to arrive at~\Eq{(84)}, we sacrificed the co-ordinate
invariance of our \emph{Action}. Hence we must proceed as follows. With
the help of the fictitious field which bridges the canal regularly, we
define~$\alpha \vS_{i}^{k}$ by means of
\[
(\vR_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vR)
+ \left(\vG \delta_{i}^{k}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} \vG^{\alpha\beta,k}\right).
\]
The equation
\[
\frac{\dd \vS_{i}^{k}}{\dd x_{k}} = 0
\Tag{(92)}
\]
is an identity for it. By integrating~\Eq{(92)} we get~\Eq{(90)}, whereby
\[
J_{i} = \int_{\Omega} \vS_{i}^{0}\, dx_{1}\, dx_{2}\, dx_{3}.
\]
$K_{i}$~expresses itself as the field-flux through the surface~$\Omega$. In these
expressions the fictitious field may be replaced by the real one, and,
moreover, in accordance with the gravitational equations, we may
replace
\[
\frac{1}{\alpha} (\vR_{i}^{k} - \tfrac{1}{2} \delta_{i}^{k} \vR)
\quad\text{by}\quad
\vl \delta_{i}^{k} - f_{ir} \vf^{kr}.
\]
If we use the normal co-ordinate system the part due to the gravitational
energy drops out; for its components depend not only
linearly but also quadratically on the (vanishing) derivatives~$\dfrac{\dd g_{\alpha\beta}}{\dd x_{i}}$.
We are, therefore, left with only the electromagnetic part, which is
to be calculated along the lines of Maxwell. Since the components
of Maxwell's energy-density depend quadratically on the field $f + \bar{f}$,
each of them is composed of three terms in accordance with the
formula
\[
(f + \bar{f})^{2} = f^{2} + \Typo{2\Bar{f\!f}}{2f\! \bar{f}} + \bar{f}^{2}.
\]
In the case of each, the first term contributes nothing, since the
flux of a constant vector through a closed surface is~$0$. The last
term is to be neglected since it contains the weak field~$\bar{f}$ as a square;
the middle term alone remains. But this gives us
\[
K_{i} = ef_{0i}\Add{.}
\]
\PageSep{306}
Concerning the momentum-quantities we see (in the same way as
in §\,33, by using identities~\Eq{(92)} and treating the cross-section of the
stream-filament as infinitely small in comparison with the external
field) \Inum{(1)}~that, for co-ordinate transformations that are to be regarded
as linear in the cross-section of the canal, the~$J_{i}$'s are the co-variant
components of a vector which is independent of the co-ordinate
system; and \Inum{(2)}~that if we alter the fictitious field occupying the
canal (in §\,33 we were concerned, not with this, but with a charge
of the co-ordinate system in the canal) the quantities~$J_{i}$ retain their
values. In the normal co-ordinate system, however, for which the
gravitational field that surrounds the particle has the form calculated
in §\,31, we find that, since the fictitious field may be chosen as a
statical one, according to \Pageref{272}: $J_{1} = J_{2} = J_{3} = 0$, and $J_{0} = $~the
flux of a spatial vector-density through the surface of~$\Omega$, and hence~$= m$.
On account of the property of co-variance possessed by~$J_{i}$,
we find that not only at the point of the canal under consideration,
but also just before it and just after it
\[
J_{i} = mu_{i}\qquad
\left(u^{i} = \frac{dx_{i}}{ds}\right).
\]
Hence the equations of motion of our particle expressed in the
normal co-ordinate system are
\[
\frac{d(mu_{i})}{dt} = ef_{0i}\Add{.}
\Tag{(93)}
\]
The $0$th~of these equations gives us: $\dfrac{dm}{dt} = 0$; thus the field equations
require that the mass be constant. But in any arbitrary co-ordinate
\index{Mass!producing@{(producing a gravitational field)}}%
system we have:
\[
\frac{d(mu_{i})}{ds}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} m u^{\alpha} u^{\beta}
= e · f_{ki} u^{k}\Add{.}
\Tag{(94)}
\]
For the relations~\Eq{(94)} are invariant with respect to co-ordinate
transformations, and agree with~\Eq{(93)} in the case of the normal co-ordinate
system. \emph{Hence, according to the field-laws, a necessary
condition for a singularity canal, which is to fit into the remaining
part of the field, and in the immediate neighbourhood of which the
field has the required structure, is that the quantities $e$~and~$m$ that
characterise the singularity at each point of the canal remain constant
along the canal, but that the world-direction of the canal
satisfy the equations}
\[
\frac{du_{i}}{ds}
- \tfrac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{i}} u^{\alpha} u^{\beta}
= \frac{e}{m} · f_{ki} u^{k}.
\]
In the light of these considerations, it seems to the author that
the opinion expressed in §\,25 stating that mass and field-energy are
\PageSep{307}
identical is a premature inference, and the whole of Mie's view of
matter assumes a fantastic, unreal complexion. It was, of course,
a natural result of the special theory of relativity that we should
come to this conclusion. It is only when we arrive at the general
theory that we find it possible to represent the mass as a field-flux,
and to ascribe to the world relationships such as obtain in
Einstein's \emph{Cylindrical World} (§\,34), when there are cut out of
it canals of circular cross-section which stretch to infinity in both
directions. This view of~$m$ states not only that inertial and
gravitational masses are identical in nature, but also that mass as
the \Emph{point of attack} of the metrical field is identical in nature with
mass as the \Emph{generator} of the metrical field. That which is
physically important in the statement that energy has inertia still
persists in spite of this. For example, a radiating particle loses
inertial mass of exactly the same amount as the electromagnetic
energy that it emits. (In this example Einstein first recognised the
intimate relationship between energy and inertia.) This may be
proved simply and rigorously from our present point of view.
Moreover, the new standpoint in no wise signifies a relapse to the
old idea of substance, but it deprives of meaning the problem of
the cohesive pressure that holds the charge of the electron together.
With about the same reasonableness as is possessed by
Einstein's Theory we may conclude from our results that a \Emph{clock}
in quasi-stationary motion indicates the proper time~$\Dint ds$ which
corresponds to the normalisation $F = \text{const}$.\footnote
{The invariant quadratic form $F · ds^{2}$ is very far from being distinguished
from all other forms of the type $E · ds^{2}$ ($E$~being a scalar of weight~$-1$) as is
the $ds^{2}$ of Einstein's Theory, which does not contain the derivatives of the
potentials at all. For this reason the inference made in our calculation of the
\Emph{displacement towards the infra-red} (\Pageref[p.]{246}), that similar atoms radiate
the same frequency measured in the proper time~$ds$ corresponding to the
normalisation $F = \text{const.}$, is by no means as convincing as in the theory of
Einstein: it loses its validity altogether if a principle of action other than that
here discussed holds.}
If during the motion
of a clock (e.g.\ an atom) with infinitely small period, the world-distance
traversed by it during a period were to be transferred
congruently from period to period in the sense of our world-geometry,
then two clocks which set out from the same world-point~$A$
\index{Clocks}%
with the same period, that is, which traverse congruent world-distances
in~$A$ during their first period will have, in general,
different periods when they meet at a later world-point~$B$. The
orbital motion of the electrons in the atom can, therefore, certainly
not take place in the way described, independently of their previous
\PageSep{308}
histories, since the atoms emit spectral lines of definite frequencies.
Neither does a measuring rod at rest in a statical field undergo a
congruent transference; for the measure $l = d\sigma^{2}$ of a measuring
rod at rest does not alter, whereas for a congruent transference it
would have to satisfy the equation $\dfrac{dl}{dt} = -l · \phi$. What is the
source of this discrepancy between the conception of congruent
transference and the behaviour of measuring rods, clocks, and
atoms? We may distinguish two modes of determining a quantity
\index{Adjustment@{\emph{Adjustment} and \emph{persistence}}}%
\index{Persistence@{\emph{Persistence}}}%
in nature, namely, that of \Emph{persistence} and that of \Emph{adjustment}.
This difference is illustrated in the following example. We may
prescribe to the axis of a rotating top any arbitrary direction in
space; but once this arbitrary initial direction has been fixed the
direction of the axis of the top when left to itself is determined from
it for all time by a \Emph{tendency of persistence} which is active from
one moment to another; at each instant the axis experiences an
infinitesimal parallel displacement. Diametrically opposed to this
is the case of a magnet needle in the magnetic field. Its direction
is determined at every moment, independently of the state of the
system at other moments, by the fact that the system, in virtue of
its constitution, \Emph{adjusts} itself to the field in which it is embedded.
There is no \textit{a~priori} ground for supposing a pure transference,
following the tendency of persistence, to be integrable. But even
if this be the case, as, for example, for rotations of the top in
Euclidean space, nevertheless two tops which set out from the
same point with axes in the same position, and which meet after
the lapse of a great length of time, will manifest any arbitrary
deviations in the positions of the axes, since they can never be
fully removed from all influences. Thus although, for example,
Maxwell's equations for the charge~$e$ of an electron make necessary
the equation of conservation $\dfrac{de}{dt} = 0$, this does not explain why an
electron itself after an arbitrarily long time still has the same
charge, and why this charge is the same for all electrons. This
circumstance shows that the charge is determined not by persistence
but by adjustment: there can be only \Emph{one} state of
equilibrium of negative electricity, to which the corpuscle adjusts
itself afresh at every moment. The same reason enables us to draw
the same conclusion for the spectral lines of the atoms, for what
is common to atoms emitting equal frequencies is their constitution
and not the equality of their frequencies at some moment when
they were together far back in time. In the same way, obviously,
the length of a measuring rod is determined by adjustment; for it
\PageSep{309}
would be impossible to give to \Emph{this} rod at \Emph{this} point of the field
any length, say two or three times as great as the one that it
now has, in the way that I can prescribe its direction arbitrarily.
The world-curvature makes it theoretically possible to determine a
length by adjustment. In consequence of its constitution the rod
assumes a length which has such and such a value in relation to
the radius of curvature of the world. (Perhaps the time of rotation
of a top gives us an example of a time-length that is determined by
persistence; if what we assumed above is true for direction then at
each moment of the motion of the top the rotation vector would
experience a parallel displacement.) We may briefly summarise as
follows: The affine and metrical relationship is an \textit{a~priori} datum
telling us how vectors and lengths alter, \Emph{if they happen to follow
the tendency of persistence}. But to what extent this is the case
in nature, and in what proportion persistence and adjustment
modify one another, can be found only by starting from the
physical laws that hold, i.e.\ from the principle of action.
The subject of the above discussion is the principle of action,
compatible with the new axiom of calibration invariance, which
most nearly approaches the Maxwell-Einstein theory. We have
seen that it accounts equally well for all the phenomena which are
explained by the latter theory and, indeed, that it has decided
advantages so far as the deeper problems, such as the cosmological
problems and that of matter are concerned. Nevertheless, I doubt
whether the Hamiltonian function~\Eq{(83)} corresponds to reality.
We may certainly assume that $\vW$~has the form~$W \sqrt{g}$, in which $W$~is
an invariant of weight~$-2$ formed in a perfectly rational manner
from the components of curvature. Only \Emph{four} of these invariants
may be set up, from which every other may be built up linearly by
means of numerical co-efficients (\textit{vide} \FNote{40}). One of these is
Maxwell's:
\[
l = \tfrac{1}{4} f_{ik} f^{ik}\Add{;}
\Tag{(95)}
\]
another is the~$F^{2}$ used just above. But curvature is by its nature
a linear matrix-tensor of the second order: $\sfF_{ik}\, dx_{i}\, \delta x_{k}$. According
to the same law by which~\Eq{(95)}, the square of the numerical value,
is produced from the distance-curvature~$f_{ik}$ we may form
\[
\tfrac{1}{4} \sfF_{ik} \sfF^{ik}
\Tag{(96)}
\]
from the total curvature. The multiplication is in this case to be interpreted
as a composition of matrices; \Eq{(96)}~is therefore itself again
a matrix. But its trace~$L$ is a scalar of weight~$-2$. The two
quantities $L$~and~$l$ seem to be invariant and of the kind sought, and
they can be formed most naturally from the curvature; invariants
\PageSep{310}
of this natural and simple type, indeed, exist only in a four-dimensional
world at all. It seems more probable that $W$~is a linear
combination of $L$~and~$l$. Maxwell's equations become then as
above: (when the calibration has been normalised by $F = \text{const.}$)
$\vs^{i} = $~a constant multiple of~$\sqrt{g} \phi^{i}$, and $\vh^{ik} = \vf^{ik}$. The gravitational
laws in the statical case here, too, agree to a first approximation
with Newton's laws. Calculations by Pauli (\textit{vide} \FNote{41}) have
indeed disclosed that the field determined in §\,31 is not only a
rigorous solution of Einstein's equations, but also of those favoured
here, so that the amount by which the perihelion of Mercury's
orbit advances and the amount of the deflection of light rays owing
to the proximity of the sun at least do not conflict with these
equations. But in the question of the mechanical equations and
of the relationship holding between the results obtained by
measuring-rods and clocks on the one hand and the quadratic
form on the other, the connecting link with the old theory seems
to be lost; here we may expect to meet with new results.
\Emph{One} serious objection may be raised against the theory in its
present state: it does not account for the \Emph{inequality of positive
and negative electricity} (\textit{vide} \FNote{42}). There seem to be two
ways out of this difficulty. Either we must introduce into the law
of action a square root or some other irrationality; in the discussion
on Mie's theory, it was mentioned how the desired inequality could
be caused in this way, but it was also pointed out what obstacles
lie in the way of such an irrational \emph{Action}. Or, secondly, there is
the following view which seems to the author to give a truer statement
of reality. We have here occupied ourselves only with the
\Emph{field} which satisfies certain generally invariant functional laws.
It is quite a different matter to inquire into the \Emph{excitation} or \Emph{cause}
of the field-phases that appear to be possible according to these
laws; it directs our attention to the reality lying beyond the field.
Thus in the æther there may exist convergent as well as divergent
electromagnetic waves; but only the latter event can be brought
about by an atom, situated at the centre, which emits energy owing
to the jump of an electron from one orbit to another in accordance
with Bohr's hypothesis. This example shows (what is immediately
obvious from other considerations) that the idea of causation (in
\Chg{contradistinction}{contra-distinction} to functional relation) is intimately connected
with the \Emph{unique direction of progress characteristic of Time},
namely \Emph{Past~$\to$ Future}. This oneness of sense in Time exists
beyond doubt---it is, indeed, the most fundamental fact of our perception
of Time---but \textit{a~priori} reasons exclude it from playing a part
in physics of the field, But we saw above (§\,33) that the sign, too,
\PageSep{311}
\index{Density!electricity@{(of electricity and matter)}}%
of an isolated system is fully determined, as soon as a definite sense
of flow, Past~$\to$ Future, has been prescribed to the world-canal
swept out by the system. This connects the inequality of positive
and negative electricity with the inequality of Past and Future;
but the roots of this problem are not in the field, but lie outside it.
Examples of such regularities of structure that concern, not the
field, but the causes of the field-phases are instanced: by the
existence of cylindrically shaped boundaries of the field: by our
assumptions above concerning the constitution of the field in their
immediate neighbourhood: lastly, and above all, by the facts of
the quantum theory. But the way in which these regularities
have hitherto been formulated are, of course, merely provisional in
character. Nevertheless, it seems that the \Emph{theory of statistics}
plays a part in it which is fundamentally necessary. We must
here state in unmistakable language that physics at its present
stage can in no wise be regarded as lending support to the belief
that there is a causality of physical nature which is founded on
rigorously exact laws. The extended field, ``æther,'' is merely the
\index{Aether@{Æther}!(in a generalised sense)}%
\emph{transmitter} of effects and is, of itself, powerless; it plays a part
that is in no wise different from that which space with its rigid
Euclidean metrical structure plays, according to the old view; but
now the rigid motionless character has become transformed into
one which gently yields and adapts itself. But freedom of action
in the world is no more restricted by the rigorous laws of field
physics than it is by the validity of the laws of Euclidean geometry
according to the usual view.
If Mie's view were correct, we could recognise the field as objective
reality, and physics would no longer be far from the goal
of giving so complete a grasp of the nature of the physical world,
of matter, and of natural forces, that logical necessity would extract
from this insight the unique laws that underlie the occurrence of
physical events. For the present, however, we must reject these
bold hopes. The laws of the metrical field deal less with reality
itself than with the shadow-like extended medium that serves as a
link between material things, and with the formal constitution of
this medium that gives it the power of transmitting effects. \Emph{Statistical
physics}, through the quantum theory, has already reached
a deeper stratum of reality than is accessible to field physics; but
the problem of matter is still wrapt in deepest gloom. But even
if we recognise the limited range of field physics, we must gratefully
acknowledge the insight to which it has helped us. Whoever
looks back over the ground that has been traversed, leading from
the Euclidean metrical structure to the mobile metrical field which
\PageSep{312}
depends on matter, and which includes the field phenomena of
gravitation and electromagnetism; whoever endeavours to get a
complete survey of what could be represented only successively
and fitted into an articulate manifold, must be overwhelmed by a
feeling of freedom won---the mind has cast off the fetters which
have held it captive. He must feel transfused with the conviction
that reason is not only a human, a too human, makeshift in the
struggle for existence, but that, in spite of all disappointments and
errors, it is yet able to follow the intelligence which has planned
the world, and that the consciousness of each one of us is the
centre at which the One Light and Life of Truth comprehends
itself in Phenomena. Our ears have caught a few of the fundamental
chords from that harmony of the spheres of which Pythagoras
and Kepler once dreamed.
\PageSep{313}
\BackMatter
%[** TN: Smaller type in the original]
\Appendix{I}{(Pp.\ \PageNo{179} and \PageNo{229})}
To distinguish ``normal'' co-ordinate systems among all others in the
\index{Co-ordinate systems!normal}%
\index{Normal calibration of Riemann's space!system of co-ordinates}%
special theory of relativity, and to determine the metrical groundform in
the general theory, we may dispense with not only rigid bodies but also
with clocks.
In the \emph{special} theory of relativity the postulate that, for the transformation
corresponding to the co-ordinates~$x_{i}$ of a piece of the world to
an Euclidean ``picture'' space, the world-lines of points moving freely
under no forces are to become \Emph{straight} lines (Galilei's and Newton's
Principle of Inertia), fixes this picture space \Emph{except for an affine
transformation}. For the theorem, that affine transformations of a portion
\Figure{15}
of space are the only
continuous ones which
transform straight lines
into straight lines, holds.
This is immediately evident
if, in Möbius' mesh
construction (\Fig{12}),
we replace infinity by a
straight line intersecting
our portion of space
(\Fig{15}). The phenomenon
of light propagation
then fixes \Emph{infinity}
and the \Emph{metrical structure}
in our four-dimensional
projective space;
for its (three dimensional) ``plane at infinity''~$E$ is characterised by the
property that the light-cones are projections, taken from different world-points,
of one and the same two-dimensional conic section situated in~$E$.
In the \emph{general} theory of relativity these deductions are best expressed
in the following form. The four-dimensional Riemann space,
which Einstein imagines the world to be, is a particular case of general
metrical space (§\,16). If we adopt this view we may say that the phenomenon
of light propagation determines the \Emph{quadratic} groundform~$ds^{2}$,
whereas the \Emph{linear} one remains unrestricted. Two different choices of
the linear groundform which differ by $d\phi = \phi_{i}\, dx_{i}$ correspond to two
different values of the affine relationship. Their difference is, according
to formula~\Typo{49}{\Eq{(49)}}, §\,16, given by
\[
[\Gamma_{\alpha\beta}^{i}]
= \tfrac{1}{2} (\delta_{\alpha}^{i} \phi_{\beta}
+ \delta_{\beta}^{i} \phi_{\alpha}
- g_{\alpha\beta} \phi^{i})\Add{.}
\]
\PageSep{314}
The difference between the two vectors that are derived from a world-vector~$u^{i}$
at the world-point~$O$ by means of an infinitesimal parallel
displacement of~$u^{i}$ in its own direction (by the same amount $dx_{i} = \epsilon · u^{i}$), is
therefore $\epsilon$~times
\[
u^{i} (\phi_{\alpha} u^{\alpha}) - \tfrac{1}{2} \phi^{i}\Add{,}
\Tag{(*)}
\]
whereby we assume $g_{\alpha\beta} u^{\alpha} u^{\beta} = 1$. If the geodetic lines passing through~$O$
in the direction of the vector~$u^{i}$ coincide for the two fields, then the
above two vectors derived from~$u^{i}$ by parallel displacement must be
coincident in direction; the vector~\Eq{(*)}, and hence~$\phi^{i}$, must have the same
direction as the vector~$u^{i}$. If this agreement holds for \Emph{two} geodetic lines
passing through~$O$ in different directions, we get $\phi^{i} = 0$. Hence if we
know the world-lines of two point-masses passing through~$O$ and moving
only under the influence of the guiding field, then the linear groundform,
as well as the quadratic groundform, is uniquely determined at~$O$.
\PageSep{315}
\Appendix{II}{(\Pageref[Page]{232})}
\emph{Proof of the Theorem that, in Riemann's space, $R$~is the sole invariant
that contains the derivatives of the~$g_{ik}$'s only to the second order, and those
of the second order only linearly.}
According to hypothesis, the invariant~$J$ is built up of the derivatives
of the second order:
\[
g_{ik,rs} = \frac{\dd^{2} g_{ik}}{\dd x_{r}\, \dd x_{s}}\Add{;}
\]
thus
\[
J = \sum \lambda_{ik,rs} g_{ik,rs} + \lambda.
\]
The $\lambda$'s denote expressions in the~$g_{ik}$'s and their first derivatives; they
satisfy the conditions of symmetry:
\[
\lambda_{ki,rs} = \lambda_{ik,rs},\qquad
\lambda_{ik,sr} = \lambda_{ik,rs}.
\]
At the point~$O$ at which we are considering the invariant, we introduce an
orthogonal geodetic co-ordinate system, so that, at that point, we have
\[
g_{ik} = \delta_{i}^{k},\qquad
\frac{\dd g_{ik}}{\dd x_{r}} = 0.
\]
The $\lambda$'s become \Emph{absolute constants}, if these values are inserted. The
unique character of the co-ordinate system is not affected by:
(1) linear orthogonal transformations;
(2) a transformation of the type
\[
x_{i} = x_{i}' + \frac{1}{6} \alpha_{krs}^{i} x_{k}' x_{r}' x_{s}'
\]
which contains no quadratic terms; the co-efficients~$\alpha$ are symmetrical in
$k$,~$r$, and~$s$, but are otherwise arbitrary.
Let us therefore consider in a Euclidean-Cartesian space (in which
arbitrary orthogonal linear transformations are allowable) the biquadratic
form dependent on two vectors $x = (x_{i})$, $y = (y_{i})$, namely
\[
G = g_{ik,rs} x_{i} x_{k} y_{r} y_{s}
\]
with arbitrary co-efficients~$g_{ik,rs}$ that are symmetrical in $i$~and~$k$, as also in
$r$~and~$s$; then
\[
\lambda_{ik,rs} g_{ik,rs}
\Tag{(1)}
\]
\PageSep{316}
must be an invariant of this form. Moreover, since as a result of the
%[** TN: Refers to item number, not equation number]
transformation~\Inum{(2)} above, the derivatives~$g_{ik,rs}$ transform themselves,
as may easily be calculated, according to the equation
\[
%[** TN: Display-style fraction in the original]
g_{ik,rs}' = g_{ik,rs} + \tfrac{1}{2}(\alpha_{krs}^{i} + \alpha_{irs}^{k})\Add{,}
\]
we must have
\[
\lambda_{ik,rs} \alpha_{krs}^{i} = 0
\Tag{(2)}
\]
for every system of numbers~$\alpha$ symmetrical in the three indices $k$,~$r$,~$s$.
Let us operate further in the Euclidean-Cartesian space; $(x\Com y)$~is to
signify the scalar product $x_{1} y_{1} + x_{2} y_{2} + \dots \Add{+} x_{n} y_{n}$. It will suffice to use
for~$G$ a form of the type
\[
G = (a\Com x)^{2} (b\Com y)^{2}
\]
in which $a$~and $b$ denote arbitrary vectors. If we now again write $x$~and~$y$
for $a$~and~$b$, then \Eq{(1)}~expresses the postulate that
\[
\Lambda = \Lambda_{x} = \sum \lambda_{ik,rs} x_{i} x_{k} y_{r} y_{s}
\Tag{(1^{*})}
\]
is an orthogonal invariant of the two vectors $x$,~$y$. In~\Eq{(2)} it is sufficient
to choose
\[
\alpha_{krs}^{i} = x_{i} · y_{k} y_{r} y_{s}
\]
and then this postulate signifies that the form which is derived from~$\Lambda_{x}$
by converting an~$x$ into a~$y$, namely,
\[
\Lambda_{y} = \sum \lambda_{ik,rs} x_{i} y_{k} y_{r} y_{s}
\Tag{(2^{*})}
\]
vanishes identically. (It is got from~$\Lambda_{x}$ by forming first the symmetrical
bilinear form~$\Lambda_{x\Com x'}$ in $x$,~$x'$ (it is related quadratically to~$y$), which, if the
series of variables~$x'$ be identified with~$x$, resolves into~$\Lambda_{x}$, and by then
replacing $x'$ by~$y$.) I now assert that it follows from~\Eq{(1^{*})} that $\Lambda$~is of the
form
\[
\Lambda = \alpha(x\Com x) (y\Com y) - \beta(x\Com y)^{2}
\textTag{(I)}
\]
and from~\Eq{(2^{*})} that
\[
\alpha = \beta\Add{.}
\textTag{(II)}
\]
This will be the complete result, for then we shall have
\[
J = \alpha(g_{ii,kk} - g_{ik,\Typo{+}{}ik}) + \lambda
\]
or since, in an orthogonal geodetic co-ordinate system, the Riemann
scalar of curvature is
\[
R = g_{ik,ik} - g_{ii,kk}
\]
we shall get
\[
J = -\alpha R + \lambda\Add{.}
\Tag{(*)}
\]
Proof of~\textEq{I}: We may introduce a Cartesian co-ordinate system such that
% [** TN: Ordinal]
$x$~coincides with the first co-ordinate axis, and $y$~with the $(1, 2)$th co-ordinate
plane, thus;
\begin{gather*}
x = (x_{1}, 0, 0, \dots\Add{,} 0),\qquad
y = (y_{1}, y_{2}, 0, \dots\Add{,} 0)\Add{,} \\
\Lambda = x_{1}^{2} (ay_{1}^{2} + 2b y_{1} y_{2} + cy_{2}^{2})\Add{,}
\end{gather*}
\PageSep{317}
whereby the sense of the second co-ordinate axis may yet be chosen
arbitrarily. Since $\Lambda$~may not depend on this choice, we must have $b = 0$,
therefore
\[
\Lambda = cx_{1}^{2} (y_{1}^{2} + y_{2}^{2}) + (a - c)(x_{1} y_{1})^{2}
= c(x\Com x)(y\Com y) + (a - c)(x\Com y)^{2}.
\]
Proof of~\textEq{II}: From the $\Lambda = \Lambda_{x}$ which are given under~\textEq{I}, we derive the
forms
\begin{align*}
\Lambda_{x\Com x'} &= \alpha(x\Com x') (y\Com y) - \beta(x\Com y) (x'\Com y)\Add{,} \\
\Lambda_{y} &= (\alpha - \beta)(x\Com y) (y\Com y).
\end{align*}
If $\Lambda_{y}$~is to vanish then $\alpha$~must equal~$\beta$.
We have tacitly assumed that the metrical groundform of Riemann's
space is definitely positive; in case of a different index of inertia a slight
modification is necessary in the ``Proof of~\textEq{I}''. In order that the second
derivatives be excluded from the volume integral~$J$ by means of partial
integration, it is necessary that the~$\lambda_{ik,rs}$'s depend only on the~$g_{ik}$'s and not
on their derivatives; we did not, however, require this fact at all in our
proof. Concerning the physical meaning entailed by the possibility, expressed
in~\Eq{(*)}, of adding to a multiple of~$R$ also a universal constant~$\lambda$,
we refer to §\,34. Concerning the theorem here proved, cf.\ Vermeil, \Title{Nachr.\
d.~Ges.\ d.~Wissensch.\ zu Göttingen}, 1917, pp.~334--344.
In the same way it may be proved that $g_{ik}$,~$Rg_{ik}$,~$R_{ik}$ are the only tensors
of the second order that contain derivatives of the~$g_{ik}$'s only to the second
order, and these, indeed, only linearly.
\PageSep{318}
\PageSep{319}
\Bibliography{(The number of each note is followed by the number of the page on which
reference is made to it)}
\BibSection[I]{Introduction and Chapter I}
\Note{1.}{(5)} The detailed development of these ideas follows very closely
the lines of Husserl in his ``Ideen zu einer reinen Phäno\-men\-ologie und phäno\-men\-ologi\-schen
Philosophie'' (Jahrbuch f.~Philos.\ u.~phänomenol.\ Forschung,
Bd.~1, Halle, 1913).
\Note{2.}{(15)} Helmholtz in his dissertation, ``Über die Tatsachen, welche
der Geometrie zugrunde liegen'' (Nachr.\ d.~K. Gesellschaft d.~Wissenschaften
zu Göttingen, math.-physik.\ Kl., 1868), was the first to attempt to found geometry
on the properties of the group of motions. This ``Helmholtz space-problem''
was defined more sharply and solved by S.~Lie (Berichte d.~K. Sachs.\
Ges.\ d.~Wissenschaften zu Leipzig, math.-phys. Kl., 1890) by means of the
theory of transformation groups, which was created by Lie (cf.~Lie-Engel,
Theorie der Transformationsgruppen, Bd.~3, Abt.~5). Hilbert then introduced
great restrictions among the assumptions made by applying the ideas of the
theory of aggregates (Hilbert, Grundlagen der Geometrie, 3~Aufl., Leipzig, 1909,
Anhang~IV).
\Note{3.}{(20)} The systematic treatment of affine geometry not limited
to the dimensional number~$3$ as well as of the whole subject of the geometrical
calculus is contained in the epoch-making work of Grassmann, Lineale
Ausdehnungslehre (Leipzig, 1844). In forming the conception of a manifold
of more than three dimensions, Grassmann as well as Riemann was influenced
by the philosophic ideas of Herbart.
\Note{4.}{(53)} The systematic form which we have here given to the
tensor calculus is derived essentially from Ricci and Levi-Civita: Méthodes de
calcul différentiel absolu et leurs applications, Math.\ Ann., Bd.~54 (1901).
\BibSection[II]{Chapter II}
\Note{1.}{(77)} For more detailed information reference may be made
to Die Nicht-Euklidische Geometrie, Bonola and Liebmann, published by
Teubner.
\Note{2.}{(80)} F.~Klein, Über die sogenannte Nicht-Euklidische Geometrie,
Math.\ Ann., Bd.~4 (1871), p.~573. Cf.\ also later papers in the Math.\
Ann., Bd.~6 (1873), p.~112, and Bd.~37 (1890), p.~544.
\Note{3.}{(82)} Sixth Memoir upon Quantics, Philosophical Transactions,
t.~149 (1859).
\Note{4.}{(90)} Mathematische Werke (2~Aufl., Leipzig, 1892), Nr.~XIII,
p.~272. Als besondere Schrift herausgegeben und kommentiert vom Verf.\
(2~Aufl., Springer, 1920).
\PageSep{320}
\Note{5.}{(93)} Saggio di interpretazione della geometria non euclidea,
Giorn.\ di Matem., t.~6 (1868), p.~204; Opere Matem.\ (Höpli, 1902), t.~1, p.~374.
\Note{6.}{(93)} Grundlagen der Geometrie (3~Aufl., Leipzig, 1909), Anhang~V\@.
\Note{7.}{(96)} Cf.\ the references in Chap.~I.\Sup{2} Christoffel, Über die Transformation
der homogenen Differentialausdrücke zweiten Grades, Journ.\ f.~d.\
reine und angew.\ Mathemathik, Bd.~70 (1869): Lipschitz, in the same journal,
Bd.~70 (1869), p.~71, and Bd.~72 (1870), p.~1.
\Note{8.}{(102)} Christoffel (l.c.\Sup{7}). Ricci and Levi-Civita, Méthodes de
calcul différentiel absolu et leurs applications, Math.\ Ann., Bd.~54 (1901).
\Note{9.}{(102)} The development of this geometry was strongly influenced
by the following works which were created in the light of Einstein's Theory of
Gravitation: Levi-Civita, Nozione di parallelismo in una varietà qualunque~\dots,
Rend.\ del Circ.\ Mat.\ di~Palermo, t.~42 (1917), and Hessenberg, Vektorielle
Begründung der Differentialgeometrie, Math.\ Ann., Bd.~78 (1917). It assumed
a perfectly definite form in the dissertation by Weyl, Reine Infinitesimalgeometrie,
Math.\ Zeitschrift, Bd.~2 (1918).
\Note{10.}{(112)} The conception of parallel displacement of a vector was
set up for Riemann's geometry in the dissertation quoted in Note~9; to derive
it, however, Levi-Civita assumed that Riemann's space is embedded in a Euclidean
space of higher dimensions. A direct explanation of the conception was
given by Weyl in the first edition of this book with the help of the geodetic co-ordinate
system; it was elevated to the rank of a fundamental axiomatic conception,
which is characteristic of the degree of the affine geometry, in the
paper ``Reine Infinitesimalgeometrie,'' mentioned in Note~9.
\Note{11.}{(133)} Hessenberg (l.c.\Sup{9}), p.~190.
\Note{12.}{(144)} Cf.\ the large work of Lie-Engel, Theorie der Transformationsgruppen,
Leipzig, 1888--93; concerning this so-called ``second fundamental
theorem'' and its converse, \textit{vide} Bd.~1, p.~156, Bd.~3, pp.~583,~659,
and also Fr.~Schur, Math.\ Ann., Bd.~33 (1888), p.~54.
\Note{13.}{(147)} A second view of the problem of space in the light of the
theory of groups forms the basis of the investigations of Helmholtz and Lie
quoted in Chapter~I.\Sup{2}
\BibSection[III]{Chapter III}
\Note{1.}{(149)} All further references to the special theory of relativity
will be found in Laue, Die Relativitätstheorie~I (3~Aufl., Braunschweig, 1919).
\Note{2.}{(161)} Helmholtz, Monatsber.\ d.~Berliner Akademie, Marz, 1876,
or Ges.\ Abhandlungen, Bd.~1 (1882), p.~791. Eichenwald, Annalen der Physik,
Bd.~11 (1903), p.~1.
\Note{3.}{(169)} This is true, only subject to certain limitations; \textit{vide}
A.~Korn, Mechanische Theorie des elektromagnetischen Feldes, Phys.\ Zeitschr.,
Bd.~18,~19 and~20 (1917--19).
\Note{4.}{(170)} A.~A. Michelson, Sill.\ Journ., Bd.~22 (1881), p.~120. A.~A.
Michelson and E.~W. Morley, \textit{idem}, Bd.~34 (1887), p.~333. E.~W. Morley and
D.~C. Miller, Philosophical Magazine, vol.~viii (1904), p.~753, and Bd.~9 (1905),
p.~680. H.~A. Lorentz, Arch.\ Néerl., Bd.~21 (1887), p.~103, or Ges.\ Abhandl.,
Bd.~1, p.~341. Since the enunciation of the theory of relativity by Einstein,
the experiment has been discussed repeatedly.
\Note{5.}{(172)} Cf.\ Trouton and Noble, Proc.\ Roy.\ Soc., vol.~lxxii (1903),
p.~132. Lord Rayleigh, Phil.\ Mag., vol.~iv (1902), p.~678. D.~B. Brace, \textit{idem}
\PageSep{321}
(1904), p.~317, vol.~x (1905), pp.~71,~591. B.~Strasser, Annal.\ d.~Physik, Bd.~24
(1907), p.~137. Des Coudres, Wiedemanns Annalen, Bd.~38 (1889), p.~71.
Trouton and Rankine, Proc.\ Roy.\ Soc., vol.~viii. (1908), p.~420.
\Note{6.}{(173)} Zur Elektrodynamik bewegter Körper, Annal.\ d.~Physik,
Bd.~17 (1905), p.~891.
\Note{7.}{(173)} Minkowski, Die Grundgleichungen für die elektromagnetischen
Vorgänge in bewegten Körpern, Nachr.\ d.~K. Ges.\ d.~Wissensch.\
zu Göttingen, 1908, p.~53, or Ges.\ Abhandl., Bd.~2, p.~352.
% [** TN: Title spelling taken from title page of Möbius]
\Note{8.}{(179)} Möbius, Der \Typo{baryzentrische Calcül}{barycentrische Calcul} (Leipzig, 1827; or
Werke, Bd.~1), Kap.~6 u.~7.
\Note{9.}{(186)} In taking account of the dispersion it is to be noticed that
$q'$~is the velocity of propagation for the frequency~$\nu'$ in water at rest, and not
for the frequency~$\nu$ (which exists inside and outside the water). Careful experimental
confirmations of the result have been given by Michelson and
Morley, Amer.\ Jour.\ of Science, \Vol{31}~(1886), p.~377, Zeeman, Versl.\ d.~K. Akad.\
v.~Wetensch., Amsterdam, \Vol{23}~(1914), p.~245; \Vol{24}~(1915), p.~18. There is a new
interference experiment by Zeeman similar to that performed by Fizeau:
Zeeman, Versl.\ Akad.\ v.~Wetensch., Amsterdam, \Vol{28}~(1919), p.~1451; Zeeman
and Snethlage, \textit{idem}, p.~1462. Concerning interference experiments
with rotating bodies, \textit{vide} Laue, Annal.\ d.~Physik, \Vol{62}~(1920), p.~448.
\Note{10.}{(192)} Wilson, Phil.\ Trans.~(A), vol.~204 (1904), p.~121.
\Note{11.}{(196)} Röntgen, Sitzungsber.\ d.~Berliner Akademie, 1885, p.~195;
Wied.\ Annalen, Bd.~35 (1888), p.~264, and Bd.~40 (1890), p.~93. Eichenwald,
Annalen d.~Physik, Bd.~11 (1903), p.~421.
\Note{12.}{(196)} Minkowski (l.c.\Sup{7}).
\Note{13.}{(199)} W.~Kaufmann, Nachr.\ d.~K. Gesellsch.\ d.~Wissensch.\ zu
Göttingen, 1902, p.~291; Ann.\ d.~Physik, Bd.~19 (1906), p.~487, and Bd.~20 (1906),
p.~639. A.~H. Bucherer, Ann.\ d.~Physik, Bd.~28 (1909), p.~513, and Bd.~29 (1919),
p.~1063. S.~Ratnowsky, Determination experimentale de la variation d'inertie
des corpuscules cathodiques en fonction de la vitesse, Dissertation, Geneva, 1911.
E.~Hupka, Ann.\ d.~Physik, Bd.~31 (1910), p.~169. G.~Neumann, Ann.\ d.~Physik,
Bd.~45 (1914), p.~529, mit Nachtrag von C.~Schaefer, \textit{ibid}., Bd.~49, p.~934.
Concerning the atomic theory, \textit{vide} K.~Glitscher, Spektroskopischer Vergleich
zwischen den Theorien des starren und des deformierbaren Elektrons, Ann.\ d.~Physik,
Bd.~52 (1917), p.~608.
\Note{14.}{(204)} Die Relativitätstheorie~I (3~Aufl., 1919), p.~229.
\Note{15.}{(205)} Einstein (l.c.\Sup{6}). Planck, Bemerkungen zum Prinzip der
Aktion und Reaktion in der allgemeinen Dynamik, Physik.\ Zeitschr., Bd.~9
(1908), p.~828; Zur Dynamik bewegter Systeme, Ann.\ d.~Physik, Bd.~26 (1908),
p.~1.
\Note{16.}{(205)} Herglotz, Ann.\ d.~Physik, Bd.~36 (1911), p.~453.
\Note{17.}{(206)} Ann.\ d.~Physik, Bd.~37, 39,~40 (1912--13).
\BibSection[IV]{Chapter IV}
\Note{1.}{(218)} Concerning this paragraph, and indeed the whole chapter
up to §\,34, \textit{vide} A.~Einstein, Die Grundlagen der allgemeinen Relativitätstheorie
(Leipzig, Joh.\ Ambr.\ Barth, 1916); Über die spezielle und die aligemeine Relativitätstheorie
(gemeinverständlich; Sammlung Vieweg, 10~Aufl., 1910). E.~Freundlich,
Die Grundlagen der Einsteinschen Gravitationstheorie (4~Aufl.,
Springer, 1920). M.~Schlick, Raum und Zeit in der gegenwärtigen Physik
(3~Aufl., Springer, 1920). A.~S. Eddington, Space, Time, and Gravitation
%[** TN: http://www.gutenberg.org/ebooks/29782]
\PageSep{322}
(Cambridge, 1920), an excellent, popular, and comprehensive exposition of the
general theory of relativity, including the development described in §§\,35,~36.
Eddington, Report on the Relativity Theory of Gravitation (London, Fleetway
Press, 1919). M.~Born, Die Relativitätstheorie Einsteins (Springer, 1920).
E.~Cassirer, Zur Einsteinschen Relativitätstheorie (Berlin, Cassirer, 1921).
E.~Kretschmann, Über den physikalischen Sinn der Relativitätspostulate,
Ann.\ Phys., Bd.~53 (1917), p.~575. G.~Mie, Die Einsteinsche Gravitationstheorie
und das Problem der Materie, Phys.\ Zeitschr., Bd.~18 (1917), pp.~551--56, 574--80
and 596--602. F.~Kottler, Über die physikalischen Grundlagen der allgemeinen
Relativitätstheorie, Ann.\ d.~Physik, Bd.~56 (1918), p.~401. \Typo{Einsten}{Einstein}, Prinzipielles
zur allgemeinen Relativitätstheorie, Ann.\ d.~Physik, Bd.~55 (1918), p.~241.
\Note{2.}{(218)} Even Newton felt this difficulty; it was stated most clearly
and emphatically by E.~Mach. Cf.~the detailed references in A.~Voss, Die
Prinzipien der rationellen Mechanik, in der Mathematischen Enzyklopädie,
Bd.~4, Art.~1, Absatz 13--17 (phoronomische Grundbegriffe).
\Note{3.}{(225)} Mathematische und naturwissenschaftliche Berichte aus
Ungarn~VIII (1890).
\Note{4.}{(227)} Concerning other attempts (by Abraham, Mie, Nordström)
to adapt the theory of gravitation to the results arising from the special theory
of relativity, full references are given in M.~Abraham, Neuere Gravitationstheorien,
Jahrbuch der Radioaktivität und Elektronik, Bd.~11 (1915), p.~470.
\Note{5.}{(233)} F.~Klein, Über die Differentialgesetze für die Erhaltung
von Impuls und Energie in der Einsteinschen Gravitationstheorie, Nachr.\ d.~Ges.\
d.~Wissensch.\ zu Göttingen, 1918. Cf.,~in the same periodical, the
general formulations given by E.~Noether, Invariante Variationsprobleme.
\Note{6.}{(238)} Following A.~Palatini, Deduzione invariantiva delle equazioni
gravitazionali dal principio di~Hamilton, Rend.\ del Circ.\ Matem.\ di~Palermo,
t.~43 (1919), pp.~203--12.
\Note{7.}{(239)} Einstein, Zur allgemeinen Relativitätstheorie, Sitzungsber.\ d.~Preuss.\
Akad.\ d.~Wissenschaften, 1915, \Vol{44}, p.~778, and an appendix on p.~799.
Also Einstein, Die Feldgleichungen der Gravitation, \textit{idem}, 1915, p.~844.
\Note{8.}{(239)} H.~A. Lorentz, Het beginsel van Hamilton in Einstein's
theorie der zwaartekracht, Versl.\ d.~Akad.\ v.~Wetensch.\ te Amsterdam, XXIII,
p.~1073: Over Einstein's theorie der zwaartekracht I,~II,~III, \textit{ibid}., XXIV, pp.~1389,
1759, XXV, p.~468. Trestling, \textit{ibid}., Nov., 1916; Fokker, \textit{ibid}., Jan.,
1917, p.~1067. Hilbert, Die Grundlagen der Physik, 1~Mitteilung, Nachr.\ d.~Gesellsch.\
d.~Wissensch.\ zu Göttingen, 1915, 2~Mitteilung, 1917. Einstein,
Hamiltonsches Prinzip und allgemeine Relativitätstheorie, Sitzungsber.\ d.~Preuss.\
Akad.\ d.~Wissensch., 1916, \Vol{42}, p.~1111. Klein, Zu Hilberts erster Note über die
Grundlagen der Physik, Nachr.\ d.~Ges.\ d.~Wissensch.\ zu Göttingen, 1918, and
the paper quoted in Note~5, also Weyl, Zur Gravitationstheorie, Ann.\ d.~Physik,
Bd.~54 (1917), p.~117.
\Note{9.}{(240)} Following Levi-Civita, Statica Einsteiniana, Rend.\ della R.~Accad.\
dei~Linceï, 1917, vol.~xxvi., ser.~5a, 1$^{\circ}$~sem., p.~458.
\Note{10.}{(244)} Cf.~also Levi-Civita, La teoria di Einstein e il principio di
Fermat, Nuovo Cimento, ser.~6, vol.~xvi. (1918), pp.~105--14.
\Note{11.}{(246)} F.~W. Dyson, A.~S. Eddington, C.~Davidson, A Determination
of the Deflection of Light by the Sun's Gravitational Field, from Observations
made at the Total Eclipse of May~29th, 1919; Phil.\ Trans.\ of the Royal
Society of London, Ser.~A, vol.~220 (1920), pp.~291--333. Cf.\ E.~Freundlich, Die
Naturwissenschaften, 1920, pp.~667--73.
\Note{12.}{(247)} Schwarzschild, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissenschaften,
\PageSep{323}
1914, p.~1201. Ch.~E. St.~John, Astrophys.\ Journal, \Vol{46}~(1917), p.~249
(vgl.\ auch die dort zitierten Arbeiten von Halm und Adams). Evershed and
Royds, Kodaik.\ Obs.\ Bull., \Vol{39}. L.~Grebe and A.~Bachem, Verhandl.\ d.~Deutsch.\
Physik.\ Ges., \Vol{21} (1919), p.~454; Zeitschrift für Physik, \Vol{1}~(1920), p.~51. E.~Freundlich,
Physik.\ Zeitschr., \Vol{20}~(1919), p.~561.
\Note{13.}{(247)} Einstein, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch., 1915,
\Vol{47}, p.~831. Schwarzschild, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch., 1916, \Vol{7},
p.~189.
\Note{14.}{(247)} The following hypothesis claimed most favour. H.~Seeliger,
Das Zodiakallicht und die empirischen Glieder in der Bewegung der
inneren Planeten, Münch.\ Akad., Ber.~36 (1906). Cf.\ E.~Freundlich, Astr.\
Nachr., Bd.~201 (June, 1915), p.~48.
\Note{15.}{(248)} Einstein, Sitzungsber.\ d.~\Typo{Preusz}{Preuss}.\ Akad.\ d.~Wissensch.,
1916, p.~688; and the appendix: Über Gravitationswellen, \textit{idem}, 1918, p.~154.
Also Hilbert (l.c.\Sup{8}), 2~Mitteilung.
\Note{16.}{(252)} Phys.\ Zeitschr., Bd.~19 (1918), pp.~33 and~156. Cf.~also
de~Sitter, Planetary motion and the motion of the moon according to Einstein's
theory, Amsterdam Proc., Bd.~19, 1916.
\Note{17.}{(252)} Cf.\ Schwarzschild (l.c.\Sup{12}); Hilbert (l.c.\Sup{8}), 2~Mitt.; J.~Droste,
Versl.\ K.~Akad.\ v.~Wetensch., Bd.~25 (1916), p.~163.
\Note{18.}{(258)} Concerning the problem of $n$~bodies, \textit{vide} J.~Droste, Versl.\
K.~Akad.\ v.~Wetensch., Bd.~25 (1916), p.~460.
\Note{19.}{(259)} Cf.\ A.~S. Eddington, Report, §§\,29,~30.
\Note{20.}{(260)} L.~Flamm, Beiträge zur Einsteinschen Gravitationstheorie,
Physik.\ Zeitschr., Bd.~17 (1916), p.~449.
\Note{21.}{(260)} H.~Reistner, Ann.\ Physik, Bd.~50 (1916), pp.~106--20. Weyl
\Typo{}{(}l.c.\Sup{8}). G.~Nordström, On the Energy of the Gravitation Field in Einstein's
Theory, Versl.\ d.~K. Akad.\ v.~Wetensch., Amsterdam, vol.~xx., Nr.~9,~10 (Jan.~26th,
1918). C.~Longo, Legge elettrostatica elementare nella teoria di Einstein,
Nuovo Cimento, ser.~6, vol.~xv. (1918). p.~191.
\Note{22.}{(266)} Sitzungsber.\ d.~\Typo{Preusz}{Preuss}.\ Akad.\ d.~Wissensch., 1916, \Vol{18}, p.~424.
Also H.~Bauer, Kugelsymmetrische Lösungssysteme der Einsteinschen
Feldgleichungen der Gravitation für eine ruhende, gravitierende Flüssigkeit mit
linearer Zustandsgleichung, Sitzungsber.\ d.~Akad.\ d.~Wissensch.\ in Wien,
math.-naturw.\ Kl., Abt.~IIa, Bd.~127 (1918).
\Note{23.}{(266)} Weyl (l.c.\Sup{8}), §§\,5,~6. And a remark in Ann.\ d.~Physik, Bd.~59
(1919).
\Note{24.}{(268)} Levi-Civita: $ds^{2}$~einsteiniani in campi newtoniani, Rend.\
Accad.\ dei Linceï, 1917--19.
\Note{25.}{(268)} A.~De-Zuani, Equilibrio relativo ed equazioni gravitazionali
di Einstein nel caso stazionario, Nuovo Cimento, ser.~v, vol.~xviii. (1819), p.~5.
A.~Palatini, Moti Einsteiniani stazionari, Atti del R.~Instit.\ Veneto di scienze,
lett.\ ed~arti, t.~78~(2) (1919), p.~589.
\Note{26.}{(270)} Einstein, Grundlagen [(l.c.\Sup{1})] S.~49. The proof here is
according to Klein (l.c.\Sup{5}).
\Note{27.}{(271)} For a discussion of the physical meaning of these equations,
\textit{vide} Schrödinger, Phys.\ Zeitschr., Bd.~19 (1918), p.~4; H.~Bauer, \textit{idem}, p.~163;
Einstein, \textit{idem}, p.~115, and finally, Einstein, Der Energiesatz in der allgemeinen
Relativitätstheorie, in den Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch.,
1918, p.~448, which cleared away the difficulties, and which we have followed
in the text. Cf.~also F.~Klein, Über die Integralform der Erhaltungssätze und
die Theorie der räumlich geschlossenen Welt, Nachr.\ d.~Ges.\ d.~Wissensch.\ zu
Göttingen, 1918.
\PageSep{324}
\Note{28.}{(273)} Cf.\ G.~Nordström, On the mass of a material system according
to the Theory of Einstein, Akad.\ v.~Wetensch., Amsterdam, vol\Add{.}~xx.,
No.~7 (Dec.~29th, 1917).
\Note{29.}{(275)} Hilbert (l.c.\Sup{8}), 2~Mitt.
\Note{30.}{(276)} Einstein, Sitzungsber.\ d.~Preuss.\ Akad.\ d.~Wissensch., 1917
\Vol{6}, p.~142.
\Note{31.}{(280)} Weyl, Physik. Zeitschr., Bd.~20 (1919), p.~31.
\Note{32.}{(282)} Cf.\ de~Sitter's Mitteilungen im Versl.\ d.~Akad.\ v.~Wetensch.\
te Amsterdam, 1917, as also his series of concise articles: On Einstein's theory
of gravitation and its astronomical consequences (Monthly Notices of the R.~Astronom.\
Society); also F.~Klein (l.c.\Sup{27}).
\Note{33.}{(282)} The theory contained in the two following articles were
developed by Weyl in the Note ``Gravitation und Elektrizität,'' Sitzungsber.\
d.~Preuss.\ Akad.\ d.~Wissensch., 1918, p.~465. Cf.~also Weyl, Eine neue Erweiterung
der Relativitätstheorie, Ann.\ d.~Physik, Bd.~59 (1919). A similar
tendency is displayed (although obscure to the present author in essential
points) in E.~Reichenbächer (Grundzüge zu einer Theorie der Elektrizität und
Gravitation, Ann.\ d.~Physik, Bd.~52 [1917], p.~135; also Ann.\ d.~Physik, Bd.~63
[1920], pp.~93--144). Concerning other attempts to derive Electricity and
Gravitation from a common root cf.~the articles of Abraham quoted in Note~4;
also G.~Nordström, Physik.\ Zeitschr., \Vol{15} (1914), p.~504; E.~Wiechert, Die
Gravitation als elektrodynamische Erscheinung, Ann.\ d.~Physik, Bd.~63 (1920),
p.~301.
\Note{34.}{(286)} This theorem was proved by Liouville: Note~IV in the
appendix to G.~Monge, Application de l'analyse à la géométrie (1850), p.~609.
\Note{35.}{(286)} This fact, which here appears as a self-evident result, had
been previously noted: E.~Cunningham, Proc.\ of the London Mathem.\ Society~(2),
vol.~viii. (1910), pp.~77--98; H.~Bateman, \textit{idem}, pp.~223--64.
\Note{36.}{(295)} Cf.\ also W.~Pauli, Zur Theorie der Gravitation und der
Elektrizität von H.~Weyl, Physik.\ Zeitschr., Bd.~20 (1919), pp.~457--67. Einstein
arrived at partly similar results by means of a further modification of his
gravitational equations in his essay: Spielen Gravitationsfelder im Aufbau der
materiellen Elementarteilchen eine wesentliche Rolle? Sitzungsber.\ d.~Preuss.\
Akad.\ d.~Wissensch., 1919, pp.~349--56.
\Note{37.}{(299)} Concerning such existence theorems at a point of singularity,
\textit{vide} Picard, Traité d'Analyse, t.~3, p.~21.
\Note{38.}{(302)} Ann.\ d.~Physik, Bd.~39 (1913).
\Note{39.}{(303)} As described in the book by Sommerfeld, Atombau and
Spektrallinien, Vieweg, 1919 and~1921.
\Note{40.}{(309)} This was proved by R.~Weitzenböck in a letter to the
present author; his investigation will appear soon in the Sitzungsber.\ d.~Akad.\
d.~Wissensch.\ in Wien.
\Note{41.}{(310)} W.~Pauli, Merkur-Perihelbewegung und Strahlenablenkung
in Weyl's Gravitationstheorie, Verhandl.\ d.~Deutschen physik.\ Ges., Bd.~21 (1919),
p.~742.
\Note{42.}{(310)} Pauli (l.c.\Sup{36}).
\PageSep{325}
\printindex
% [** TN: Commented index text]
\iffalse
INDEX
(The numbers refer to the pages)
Aberration 160, 186
Abscissa 9
Acceleration 115
Action@\emph{Action}
(cf.\ Hamilton's Function) 210
principle of 211
quantum of 284, 285
Active past and future 175
Addition of tensors 43
of tensor-densities 110
of vectors 17
Adjustment@{\emph{Adjustment} and \emph{persistence}} 308
Aether@{Æther}
(as a substance) 160
(in a generalised sense) 169, 311
Affine
geometry
(infinitesimal) 112
(linear Euclidean) 16
manifold 102
relationship of a metrical space 125
transformation 21
Allowable systems 177
Analysis situs@{\emph{Analysis situs}} 273, 279
Angles
measurement of 13, 29
right 13, 29
Angular
momentum 46
velocity 47
Associative law 17
Asymptotic straight line 77, 78
Atom, Bohr's 71, 303
Axioms
of affine geometry 17
of metrical geometry
(Euclidean) 27
(infinitesimal) 124
Axis of rotation 13
Between@{\emph{Between}} 12
Bilinear form 26
Biot and Savart's Law 73
Bohr's model of the atom 71, 303
Bolyai's geometry 79, 80
Calibration 121
(geodetic) 127
Canonical cylindrical co-ordinates 266
Cartesian co-ordinate systems 29
Cathode rays 198
Causality, principle of 207
Cayley's measure-determination 82
Centrifugal forces 222, 223
Charge
(\emph{as a substance}) 214
(\emph{generally}) 269, 294
Christoffel's $3$-indices symbols#Christoffel 132
Clocks 7, 307
Co-gredient transformations 41, 42
Commutative law 17
Components, co-variant, and contra-variant
displacement@{of a displacement} 35
tensor@{of a tensor} 37
generally@{(\emph{generally})} 103
linear@{(in a linear manifold)} 103
vector@{of a vector} 20
affine@{of the affine relationship} 142
Conduction 195
Conductivity 76
Configuration, linear point 20
Congruent 11, 81
transference 140
transformations 11, 28
Conservation, law of
electricity@{of electricity} 269, 271
energy@{of energy and momentum} 292
Continuity, equation of
electricity@{of electricity} 161
mass@{of mass} 188
Continuous relationship 103, 104
Continuum 84, 85
Contraction-hypothesis of Lorentz and Fitzgerald 171
process of 48
Contra-gredient transformation 34
Contra-variant tensors 35
(generally) 103
Convection currents 195
Co-ordinate systems 9
Cartesian 29
normal 173, 313
Co-ordinates, curvilinear
Gaussian@{(or Gaussian)} 86
generally@{(generally)} 9
hexaspherical@{(hexaspherical)} 286
linear@{(in a linear manifold)} 17, 28
Coriolis forces 222
Coulomb's Law 73
Co-variant tensors 55
(generally) 103
Curl 60
Current
conduction 160
convection 195
electric 131
Curvature
direction 126
\PageSep{326}
distance 124
Gaussian 95
generally@{(generally)} 118
light@{of light rays in a gravitational field} 245
scalar of 134
vector 118
Curve 85
Definite@{\emph{Definite, positive}} 27
Density
based@{(based on the notion of substance)} 163, 291
general@{(general conception)} 197
electricity@{(of electricity and matter)} 167, 214, 311
Dielectric 70
constant 72
Differentiation of tensors and tensor-densities 58
Dimensions 19
(positive and negative, of a quadratic form) 31
Direction-curvature 126
Displacement current 162
dielectric 70
electrical 71
infinitesimal, of a point 103
vector@{of a vector} 110
space@{of space} 38
towards red due to presence of great masses 246
Distance (generally) 121
(in Euclidean geometry) 20
Distortion tensor 60
Distributive law 17
Divergence@{Divergence (\emph{div})} 60
(more general) 163, 188
Doppler's Principle 185
Earlier@{\emph{Earlier} and \emph{later}} 7, 175
Einstein's Law of Gravitation 236
(in its modified form) 291
Electrical
charge
flux@{(as a flux of force)} 294
substance@{(as a substance)} 214
current 131
displacement 162
intensity of field 65, 161
momentum 208
pressure 208
Electricity, positive and negative 212
Electromagnetic field 64
and electrostatic units 161
origin@{(origin in the metrics of the world)} 282
potential 165
Electromotive force 76
Electron 213, 260
Electrostatic potential 73
Energy
(acts gravitationally) 232, 237
(possesses inertia) 204
(total energy of a system) 301
Energy-density
(in the electric field) 70, 167
(in the magnetic field) 73
Energy-momentum, tensor@{Energy-momentum, tensor (cf.\ Energy-momentum)} 168
Energy-momentum, tensor
(for the whole system, including gravitation) 269
(general) 199
(in the electromagnetic field) 168
(in the general theory of relativity) 269
(in physical events) 292
(kinetic and potential) 199
(of an incompressible fluid) 205
(of the electromagnetic field) 291
(of the gravitational field) 269
theorem of (in the special theory of relativity) 168
Energy-steam or energy-flux 163
Eotvos@{Eötvös' experiment} 225
Equality
of time-lengths 7
of vectors 118
Ether, |See{æther}.
Euclidean
geometry 11-33 %[** TN: Sections 1-4 listed in the original]
group of rotations 138
manifolds, Chapter I (from the point of view of infinitesimal geometry) 119
Euler's equations 51
Faraday's Law of Induction 161, 191
Fermat's Principle 244
Field action of electricity 216
electromagnetic@{(electromagnetic)} 194
energy 166
gravitation@{of gravitation} 231
forces (contrasted with inertial forces) 282
general@{(general conception)} 68
guiding@{(``guiding'' or gravitational)} 283
intensity of electrical 65
magnetic@{of magnetic} 75
metrical@{(metrical)} 100
momentum 168
Finitude of space 278
Fluid, incompressible 262
Force 38
(electric) 68
(field force andinertial force) 282
(ponderomotive, of electrical field) 68
(ponderomotive, of magnetic field) 73
(ponderomotive, of electromagnetic field) 208
(ponderomotive, of gravitational field) 222
Form
bilinear 26
linear 22
quadratic 27
\PageSep{327}
Four-current ($4$-current)#current 165
Four-force ($4$-force)#force 167
Fresnel's convection co-efficient 186
Future, active and passive 177
Galilei's Principle of Relativity and Newton's Law of Inertia 149
Gaussian curvature 95
General principle of relativity 227, 236
Geodetic calibration 127
co-ordinate system 112
line (general) 114 %[** TN: "lime" in the original.]
(in Riemann's space 128
null-line 127
systems of reference 127
Geometry
affine 16
Euclidean 11-33 %[** TN: Sections 1-4 listed in the original]
infinitesimal 142
metrical 27
n-dimensional@{$n$-dimensional} 19, 25
non-Euclidean (Bolyai-Lobatschefsky) 79, 80
surface@{on a surface} 87
Riemann's 84
spherical 266
Gradient 59
(generalised) 106
Gravitation
Einstein's Law of (modified form) 291
Einstein's Law of (general form) 236
Newton's Law of 229
Gravitational
constant 243
energy 268
field 240
mass 225
potential 243
radius of a great mass 255
waves 248-252 %[** TN: Section 30 listed in the original]
Groundform, metrical
linear@{(of a linear manifold)} 28
general@{(in general)} 140
Groups 9
infinitesimal 144
of rotations 138
of translations 15
Hamilton's
function 209
principle
special@{(in the special theory of relativity)} 216
Maxwell@{(according to Maxwell and Lorentz)} 236
Mie@{(according to Mie)} 209
general@{(in the general theory of relativity)} 292
Height of displacement 158
Hexaspherical co-ordinates 286
Homogeneity
of space 91
of the world 155
Homogeneous linear equations 24
Homologous points 11
% [** TN: Next two entries hyphenated in the original (text usage inconsistent)]
Hydrodynamics 205, 263
Hydrostatic pressure 205, 263
Impulse (momentum) 44
Independent vectors 19
Induction, magnetic 75
law of 161, 191
Inertia
(as property of energy) 202
moment of 48
principle of (Galilei's and Newton's) 152
Inertial force 301
index 30
law of quadratic forms 30
mass 225
moment 48
Infinitesimal
displacement 110
geometry 142
group 144
operation of a group 142
rotations 146
Integrable 108
Intensity of field 65, 161
quantities 109
Joule (heat-equivalent) 162
Klein's model 80
Later@{\emph{Later}} 5
Light
electromagnetic theory of 164
ray 183
(curved in gravitational field) 245
Line, straight
Euclidean@{(in Euclidean geometry)} 12
generally@{(generally)} 18
geodetic 114
Line-element
Euclidean@{(in Euclidean geometry)} 56
generally@{(generally)} 103
Linear equation
point-configuration 20
tensor 57, 104
tensor-density 105, 109
vector manifold 19
transformation 21, 22
Linearly independent 19
Lobatschefsky's geometry 79, 80
Lorentz
Einstein@{-Einstein Theorem of Relativity} 165
Fitzgerald@{-Fitzgerald contraction} 171
transformation 166
Magnetic
induction 75
intensity of field 75
permeability 75
Magnetisation 75
Magnetism 74
Magnitudes 99
Manifold
affinely connected 112
discrete 97
\PageSep{328}
metrical 102, 121
Mass
energy@{(as energy)} 204
flux@{(as a flux of force)} 305
inertial and gravitational 225
producing@{(producing a gravitational field)} 303, 306
Matrix 39
Matter 68, 203, 272
flux of 188
Maxwell's
application of stationary case to Riemann's space 130
density of action 286
stresses 75
theory
(derived from the world's metrics) 285
(general case) 161
(in the light of the general theory of relativity) 222
(stationary case) 64
Measure
electrostatic and electromagnetic 161
relativity of 282
unit of 40
Measure-index of a distance 121
Measurement 176
Mechanics
fundamental law of
derived@{(derived from field laws)} 290, 293
general@{(in general theory of relativity)} 222, 226
special@{(in special theory of relativity)} 197
Newton@{of Newton's} 44, 66
of the principle of relativity 24
Metrical groundform 28, 140
Metrics or metrical structure 156
(general) 121, 207, 282
Michelson-Morley experiment 170
Mie's Theory 206
Minor space 157
Molecular currents 74
Moment
electrical 208
mechanical 44, 200
of momentum 48
Momentum 44, 200
density 168
flux 168
Motion
(in mathematical sense) 105
(under no forces) 51, 229
Multiplication
of a tensor by a number 43
of a tensor-density
by a number 109
by a tensor 110
of tensors 44
of a vector by a number 17
Newton's Law of Gravitation 229
Non-degenerate bilinear and quadratic forms 17
Non-Euclidean
geometry 77
plane
(Beltrami's model) 93
(Klein's model) 80
(metrical groundform of) 94
Non-homogeneous linear equations 24
Normal calibration of Riemann's space 124
system of co-ordinates 173, 313
Now@{\emph{Now}} 143
Null-lines, geodetic 127
Number 8, 39
Ohm's Law 76
One-sided surfaces 274
Order of tensors 36
Orthogonal transformations 34
Parallel 14, 21
displacement
infinitesimal@{(infinitesimal, of a contra-variant vector)} 113
co-variant vector 115
projection 157
Parallelepiped 20
Parallelogram 88
Parallels, postulate of 78
Partial integration (principle of) 110
Passive past and future 175
Past, active and passive 175
Perihelion, motion of Mercury's 247
Permeability, magnetic 75
Perpendicularity 121
(in general) 29
Persistence@{\emph{Persistence}} 308
Phase 219
Plane 18
(Beltrami's model) 93
(in Euclidean space) 13
(Klein's model) 82
(metrical groundform) 94
(non-Euclidean) 80
Planetary motion 256
Polarisation 71
Ponderomotive force
of the electric, magnetic and electromagnetic field 67, 73, 194
of the gravitational field 222, 223
Positive definite 27
Potential
electromagnetic 165
electrostatic 164
energy-momentum tensor of 199, 200
of the gravitational field 230
retarded 164, 165, 250
vector- 74, 163
Poynting's vector 163
Pressure, on all sides
electrical 208
hydrostatic 205, 263
Problem of one body 254
Product@{Product, etc., |See Multiplication}.
Product
tensor@{of a tensor and a number} 43
scalar 27
vectorial 45
\PageSep{329}
Projection 157
Propagation
of electromagnetic disturbances 164
of gravitational disturbances 251
of light 164
Proper-time 178, 180, 197
Pythagoras' Theorem 91, 228
Quadratic forms 31
Quantities
intensity 109
magnitude 109
Quantum Theory 285, 303
Radial symmetry 252
Reality 213
Red, displacement towards the 246
Relationship
affine 112
continuous 103, 104
metrical 142
of a manifold as a whole (conditions of) 114
of the world 273
Relativity
of magnitude 283
of motion 152, 282
principle of
(Einstein's special) 169
(general) 227, 236
Galilei's 149
theorem of (Lorentz-Einstein) 165
Resolution of tensors into space and time of vectors 158, 180
Rest 150
Retarded potential 164, 165, 250
Riemann's
curvature 132
geometry 84
space 132
Right angle 29, 121
Rotation
curl@{(or curl)} 60
general@{(general)} 155
geometrical@{(in geometrical sense)} 13
kinematical@{(in kinematical sense)} 47
relativity of 155
Rotations, group of 138, 146
Scalar-Density 109
Scalar
field 58
product 27
Similar representation or transformation 140
Simultaneity 174, 183
Skew-symmetrical 39, 55
Space
form of@{(as form of phenomena)} 1, 96
projection@{(as projection of the world)} 158, 180
element@{-element} 56
Euclidean 1-4
like@{-like} vector 179
metrical 33, 37
n-dimensional@{$n$-dimensional} 24
Special principle of relativity 169
Sphere, charged 260
Spherical
geometry 83, 266
transformations 286
Static
density 197
gravitational field 29, 240
length 176
volume 183
Stationary
field 114, 240
orbits in the atom 303
vectors 114
Stokes' Theorem 108
Stresses
elastic 58, 60
Maxwell's 75
Substance 214, 273
Substance-action of electricity and gravitation 215
mass@{($=$~mass)} 300
Subtraction of vectors 17
Sum of
tensor-densities 109
tensors 43
vectors 17
Surface 85, 274
Symmetry 26
Systems of reference 177
geodetic 127
Tensor
general@{(general)} 50, 103
linear@{(in linear space)} 33
density 109
field 105
(general) 58
Time 246
-like vectors 179
Top, spinning 51
Torque of a force 46
Trace of a matrix 49, 146
Tractrix 93
Transference, congruent 140
Transformation or representation
affine 21
congruent 11, 28
linear-vector 21, 22
similar 140
Translation of a point
(in the geometrical sense) 10
(in the kinematical sense) 115
Turning-moment of a force 46
Twists 13
Two-sided surfaces 274
Unit vectors 104
Vector 16, 24
curvature 126
density@{-density} 109
manifold@{-manifold, linear} 19
potential 74, 163
product 45
transference 117
transformation, linear 21, 22
Velocity 105
gravitation@{of propagation of gravitation} 251
light@{of light} 164
\PageSep{330}
rotation@{of rotation} 47
Volume-element 210
Weight of tensors and tensor-densities 127
Wilson's experiment 192
World ($=$ space-time) 189
-canal 268
-law 212, 273, 276
-line 149
-point 149
-vectors 155
PRINTED IN GREAT BRITAIN AT THE UNIVERSITY PRESS, ABERDEEN
\fi
%** End of commented index text
%[** TN: Methuen catalogue text removed]
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