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Subject: Invariant Galilean Transformations On All Laws
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Summary: All laws/equations are Galilean invariant when expressed
in the generalized cartesian coordinates demanded by basic
analytic geometry, vector algebra, and measurement theory.
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Invariant Galilean Transformations On All Laws
(c) Eleaticus/Oren C. Webster
Thnktank@concentric.net
An obvious typo or two corrected.
The Brittanica section revised to less
'pussyfooting' and to more directly
anticipate the elementary measurement
theory and basic analytic geometry
that is applied to the transformation
concept.

Subject: 1. Purpose
The purpose of this document is to provide the student of Physics,
especially Relativity and Electromagnetism, the most basic princ
iples and logic with which to evaluate the historic justification
of Relativity Theory as a necessary alternative to the classical
physics of Newton and Galileo.
We will prove that all laws are invariant under the Galilean
transformation, rather than some being noninvariant, after
we show you what that means.
We shall also show that another primal requirement that SR
exist is nonsense: MichelsonMorley and KennedyThorndike do
indeed fit Galilean (c+v) physics.

Subject: 2. Table of Contents
1. Foreword and Intent
2. Table of Contents
3. The Principle of Relativity
4. The Encyclopedia Brittanica Incompetency.
5. Transformations on Generalized Coordinate Laws
6. The data scale degradation absurdity.
7. The Crackpots' Version of the Transforms.
8. What does sci.math have to say about x0'=x0vt?
9. But Doesn't x.c'=x.c?
10. But Isn't (x'x.c')=(xx.c) Actually Two Transformations?
11. But Doesn't (x'x.c+vt) Prove The Transformation Time
Dependent?
12. But Isn't (x'x.c')=(xx.c) a Tautology?
13. But Isn't (x'x.c')=(xx.c) Almost the Definition of
a Linear Transform?
14. But The Transform Won't Work On Time Dependent Equations?
15. But The Transform Won't Work On Wave Equations?
16. But Maxwell's Equations Aren't Galilean Invariant?
17. First and Second Derivative differential equations.

Subject: 3. The Principle of Relativity and Transformation
If a law is different over there than it is here,
it is not one law, but at least two, and leaves us
in doubt about any third location. This is the
Principle of Relativity: a natural law must be the
same relative to any location at which a given event
may be perceived or measured, and whether or not the
observer is moving.
The idea of location translates to a coordinate
system, largely because any object in motion could
be considered as having a coordinate system origin
moving with it. If you perceive me moving relative
to you  who have your own coordinate system  will
your measurements of my position and velocity fit
the same laws my own, different measurements fit?
If a law has the same form in both cases it is
called covariant. If it is identical in form, var
ables, and output values, it is called invariant.
What we're asking is that if the xcoordinate, x,
on one coordinate axis works in an equation, does
the coordinate, x', on some other, parallel axis
work? Speaking in terms of the axis on which x is
the coordinate, x' is the 'transformed' coordinate.
The situation is complicated because we're talking
about coordinates  locations  but in most mean
ingful laws/equations, it is lengths/distances (and
time intervals) the equations are about, and x coord
inates that represent good, ratio scale measures of
distances are only interval scale measures on the x'
axis. [See Table of Contents for discussion of scales.]
So, if we have an xcoordinate in one system, then
we can call the x' value that corresponds to the same
point/location the transform of x.
In particular, the Principle of Relativity is embodied
in the form of the Galilean transformation, which
relates the original x, y, z, t to x', y', z', t' by
the transform equations x'=xvt, y'=y, z'=z, t'=t in
the simplified case where attention is focused only
on transforming the xaxis, and not y and z. In the
case of Special Relativity, the x' transform is the
same except that x' is then divided by sqrt(1(v/c)^2),
and t'=(txv/cc)/sqrt(1(v/c)^2). In either case, v
is the relative velocity of the coordinate systems;
if there is already a v in the equations being trans
formed use u or some other variable name.

Subject: 4. The Encyclopedia Brittanica Incompetency.
One example of the traditional fallacious idea
that an equation is not invariant under the galilean
transformation comes from the Encyclopedia Brittanica:
"Before Einstein's special theory of relativity
was published in 1905, it was usually assumed
that the time coordinates measured in all inertial
frames were identical and equal to an 'absolute
time'. Thus,
t = t'. (97)
"The position coordinates x and x' were then
assumed to be related by
x' = x  vt. (98)
"The two formulas (97) and (98) are called a
Galilean transformation. The laws of nonrelativ
istic mechanics take the same form in all frames
related by Galilean transformations. This is the
restricted, or Galilean, principle of relativity.
"The position of a light wave front speeding from
the origin at time zero should satisfy
x^2  (ct)^2 = 0 (99)
in the frame (t,x) and
(x')^2  (ct')^2 = 0 (100)
in the frame (t',x'). Formula (100) does not
transform into formula (99) using the transform
ations (97) and (98), however."
.................................................
Besides the trivially correct statement of what the
Galilean 'transform' equations are, there is exactly
one thing they got right.
I. Eq100 is indeed the correct basis for discussing
the question of invariance, given that eq99 is
the correct 'stationary' (observer S) equation.
[Let observer M be the 'moving'system observer.]
In particular, eq100 is of exactly the same
form [the square of argument one minus the square
of argument two equals zero (argument three).]
II. It is nonsense to say eq99 should be derivable from
eq100; for one thing, the transforms are TO x' and
t' from x and t, not the other way around, and the
idea that either observer's equation should contain
within itself the terms to simplify or rearrange to
get to the other is ridiculous. As the transform
equations say, the relationship of t', x' to t, x
is based on the relative velocity between the two
systems, but neither the original (eq99) equation
nor the M observer equation is about a relationship
between coordinate systems or observers. One might
as well expect the two equations to contain banana
export/import data; there is no relevancy. The
'transform' equations are the relationships between
x' and x, t' and t and have nothing to do with what
one equation or the other ought to 'say'. The
equations' content is the rate at which light emitted
along the xaxes moves.
III. Most remarkable, the True Believer SR crackpots who
most despise the consequences of measurement theory
(demonstrable fact) contained in this document are
those who want to argue against our saying the Britt
anica got eq100 right;
They insist that the correct equation is derived
directly from x'=xvt and t'=t. Solve for x=x'+vt
and replace t with t', then substitute the result
in eq99: (x'+vt')^2  (ct')^2 = 0.
Besides the fact that this results in an equation
with arguments exactly equal to eq99, they will
insist the transform is not invariant.
IV. A major justification they have for their idea of
the correct M system equation on which to base the
the discussion of invariance, is that the variables
are M system variables, never mind the fact that
the arguments are S system values.
That argument of theirs is arrant nonsense. The
velocity v that S sees for the M system relative
to herself is the negative of what the M system
sees for the S system relative to himself.
In other words, x'+vt' is a mixed frame expression
and it is x'+(v)t' that would be strictly M frame
notation, and that equation is far off base. [Work
it out for yourself, but make sure you try out an
S frame negative v so as not to mislead yourself.]
V. In I. we said: "given that eq99 is the correct
'stationary' equation. Let's look at it closely:
x^2  (ct)^2 = 0 (99)
This whole matter is supposed to be about coordinate
transforms. Is that what t is, just a coordinate?
No. It isn't, in general. Suppose you and I are both modelling
the same light event and you are using EST and I'm using PST.
'Just a time coordinate' is just a clock reading amd your t clock
reading says the light has been moving three hours longer
than my clock reading says. Well, that's what the idea that
t is a coordinate means.
Eq99 works if and only if t is a time interval, and in
particular the elapsed time since the light was emitted.
Thus, that equation works only if we understand just
what t is, an elapsed time, with emissioon at t=0.
However, we don't have to 'understand' anything if we use
a more intelligent and insightful form of the equation:
(x)^2  [ c(tt.e) ]^2 = 0,
where t.e is anyone's clock reading at the time of light
emission, and t is any subsequent time on the same clock.
Similarly, x is not just a coordinate, but a distance
since emission.
(xx.e)^2  [ c(tt.e) ]^2 = 0 (99a)
VI. In the spirit of 'there is exactly one thing
they got right', the correct M system version
of eq99a is eq100a:
(x'x.e')^2  [ c(t't.e') ]^2 = 0 (100a)
Every observer in the universe can derive their
eq100a from eq99a and vice versa, not to mention to and
from every other observer's eq99a.
Now, THAT's invariance. [You do realize that every
eq100a reduces to eq99a, when you back substitute
from the transforms, right? t.e'=t.e, x.e'=x.evt.]

Subject: 5. Transformations on Generalized Coordinate Laws
The traditional Gallilean transform is correct:
t' = t
x' = x  vt.
But remember this: a transform of x doesn't effect
just some values of x, but all of them, whether they
are in the formula or not. This is important if you
want to do things right. The crackpot position is
strongly against this sci.math verified position, and
the apparently standard coordinate pseudotransformation
they suggest is perhaps the result. {See Table of
Contents.]
Let's use a simple equation: x^2 + y^2 = r^2, which is
the formula for a circle with radius r, centered at a
location where x=0.
But what if the circle center isn't at x=0? Well, we'd
want to use the form analytic geometry, vector algebra,
and elementary measurement theory tells us to use, a form
where we make explicit just where the circle center is,
even if it is at x=x0=0:
(xx0)^2 + (yy0)^2 = r^2.
The circle center coordinate, x0, is an xaxis coordinate,
just like all the xvalues of points on the circle.
So, in proper generalized cartesian coordinate forms
of laws/equations we want to transform every occurence
of x and x0  by whatever name we call it: x.c, x_e,
whatever.
So, what is the transformed version of (xx0)? Why,
(x'x0'); both x and x0 are xcoordinates, and every
xcoordinate has a new value on the new axis.
So, what is the value of (x'x0') in terms of the original
x data?
From the transform equations we see that x'=xvt, which
is also true for x0'=x0vt:
(x'x0')=[ (xvt)(x0vt) ]=(xx0).
In other words, when we use the generalized coordinate form
specified by analytic geometry, we find that the value of
(x'x0') does not depend on either time or velocity in any
way, shape, form, or fashion.
Similarly for (yy0).
We can treat time the same way if necessary: (tt0).
The above is a proof that any equation in x,y,z,t is
invariant under the galilean transforms. Just use the
generalized coordinate form, with (xx0)/etc, in the
transformation process, not the incompetently selected
privileged form, with just x/etc.
[The form is "privileged" because it assumes the circle
center, point of emission, whatever, is at the origin of
the axes instead at some less convenient point. After
transform the coordinate(s) of the circle center/origin
are also changed but the privileged form doesn't make
this explicit and screws up the calculations, which
should be based on (x'x0') but are calculated as (x'0).]
The value of (x'x0') is the same as (xx0). That makes
sense.
Draw a circle on a piece of paper, maybe to the right
side of the paper. On a transparent sheet, draw x and y
coordinate axes, plus x to the right, plus y at the top.
Place this axis sheet so the yaxis is at the left side
of the circle sheet.
Now answer two questions after noting the xcoordinate of
the circle center and then moving the axis sheet to the right:
(a) did the circle change in any way because you moved
the axis sheet (ie because you transformed the coordin
nate axis)?
(b) did the coordinate of the circle center change?
The circle didn't change [although SR will say it did];
that means that (x'x0') does indeed equal (xx0).
The coordinate of the circle center did change, and it
changed at the same rate (vt) as did every point on
the circle. That means that x0'<>x0, and the fact the
circle center didn't change wrt the circle, means that
the relationship of x0' with x0 is the same as that of
any x' on the circle with the corresponding x: x'=xvt;
x0'=x0vt.
This is to prepare you for the True Believer crackpots that
say 'constant' coordinates can't be transformed; some even
say they aren't coordinates. These crackpots include some
that brag about how they were childhood geniuses, btw.
QED: The galilean transformation for any law on
generalized Cartesian coordinates is invariant under
the Galilean transform.
The use of the privileged form explains HOW the transformed
equation can be messed up, the next Subject explains what
the screwed up effect of the transform is, and how use
of the generalized form corrects the screwup.

Subject: 6. The data scale degradation absurdity.
The SR transforms and the Galilean transforms both
convert good, ratio scale data to inferior interval
scale data. The effect is corrected, allowed for,
when the transforms are conducted on the generalized
coordinate forms specified by analytic geometry and
vector algebra.
Both sets of transforms are 'translations'  lateral
movements of an axis, increasing over time in these
cases  but with the SR transform also involving a
rescaling. It is the translation term, vt in the x
transform to x', and xv/cc in the t transform to t',
that degrades the ratio scale data to interval scale
data. In general, rescaling does not effect scale
quality in the sizeofunits sense we have here.
SR likes to consider its transforms just rotations,
however  in spite of the fact Einstein correctly said
they were 'translations' (movements)  and in the case
of 'good' rotations, ratio scale data quality is indeed
preserved, but SR violates the conditions of good ro
tations; they are not rigid rotations and they don't
appropriately rescale all the axes that must be rescaled
to preserve compatibility.
The proof is in the pudding, and the pudding is the
combination of simple tests of the transformations.
We can tell if the transformed data are ratio scale
or interval.
Ratio scale data are like absolute Kelvin. A measure
ment of zero means there is zero quantity of the
stuff being measured. Ratio scale data support add
ition, subtraction, multiplication, and division.
The test of a ratio scale is that if one measure
looks like twice as much as another, the stuff
being measured is actually twice as much. With
absolute Kelvin, 100 degrees really is twice the
heat as 50 degrees. 200 degrees really is twice
as much as 100.
Interval scale data are like relative Celsius, which
is why your science teacher wouldn't let you use it
in gas law problems. There is only one mathematical
operation interval scales support, and that has to
be between two measures on the same scale: subtraction.
100 degrees relative (household) Celsius is not twice
as much as 50; we have to convert the data to absolute
Kelvin to tell us what the real ratio of temperatures
is.
However, whether we use absolute Kelvin or relative
Celsius, the difference in the two temperature readings
is the same: 50 degrees.
Thus, if we know the real quantities of the 'stuff'
being measured, we can tell if two measures are on
a ratio scale by seeing if the ratio of the two
measures is the same as the ratio of the known quant
ities.
If a scale passes the ratio test, the interval scale test
is automatically a pass.
If the scale fails the ratio test, the interval scale
test becomes the next in line.
It isn't just the bare differences on an interval
scale that provides the test, however. Differences
in two interval scale measures are ratio scale, so
it is ratios of two differences that tell the tale.
Let's do some testing, and remember as we do that our
concern is for whether or not the data are messed up,
not with 'reasons', excuses, or avoidance.

Are we going to take a transformed length (difference)
and see whether that length fits ratio or interval scale
definitions?
Of course, not. Interval scale data are ratio after
one measure is subtracted from another. That is the
major reason the SR transforms can be used in science.
Let there be three rods, A, B, C, of length 10, 20, 40,
respectively. These lengths are on a known ratio scale,
our original xaxis, with one end of each rod at the
origin, where x=0, and the other end at the coordinate
that tells us the correct lengths.
Note that these xvalues are ratio scale only because
one end of each rod is at x=0. That may remind you of
the correct way to use a ruler or yard/meterstick:
put the zero end at one end of the thing you are
measuring. Put the 1.00 mark there instead of the zero,
and you have interval scale measures.
Let A,B,C, be 10, 20, 40.
Let a,b,c be x' at v=.5, t=10.
x'=xvt.
A B C a b c
 
10 20 40 5 15 35
 
B/A = 2 b/a = 3
C/A = 4 c/a = 7
C/B = 2 c/b = 2.333
Obviously, the transformed
values are no longer ratio
scale. The effect is less on
the greater values.
CA = 10 ba = 10
CA = 30 ca = 30
CB = 20 cb = 20
Obviously, the transformed
values are now interval scale.
This will hold true for any
value of time or velocity.
(CA)/(BA) = 3 (ca)/(ba) = 3
(CB)/(BA) = 2 (cb)/(ba) = 2
Obviously, the ratios of the
differences are ratio scale,
being identical to the ratios
of the corresponding original
 ratio scale  differences.
The main difference between these results and the SR
results is that the differences do not correspond so
neatly to the original, ratio scale, differences.
This is due only to the rescaling by 1/sqrt(1(v/c)^2).
The ratios of the differences on the transformed values
do correspond neatly and exactly to the ratio scale
results.
Using the generalized coordinate form, such as (xx0),
the transform produces an interval scale x' and an
interval scale x0'. That gives us a ratio scale (x'x0'),
just like  and equal to  (xx0).

Subject: 7. The Crackpots' Version of the Transforms.
It has become apparent  whether misleading or not 
that the crackpot responses to the obvious derive from
a common source, whether it be bandwagoning or their
SR instructors.
Below, in the sci.math subject, we see that all sci.math
respondents agree with the basic "controversial" position
of this faq: every coordinate is transformed, whether a
supposed "constant" or not.
Think about it, the generalized coordinate of a circle
center, x0, applies to infinities upon infinities of
circle locations (given y and z, too); it is a constant
only for a given circle, and even then only on a given
coordinate axis.
And even "variables" are often held 'constant' during
either integration or differentiation.
The utility of a "variable" is that you can discuss all
possible particular values without having to single out
just one. That utility does not make particular  singled
out  values on the variable's axis not values of the
variable just because they have become named values.
In any case, all that is preamble to the incompetent idea
they have proposed for a transform of coordinates. It is
based on the idea that the circle center, point of emission,
whatever, has coordinates that cannot be transformed.
Let there be an equation, say (x)^2  (ict)^2 = 0.
What is the transformed version of that equation?
Answer: (x')^2  (ict')^2 = 0. That's the one thing the
Brittanica got right. Note that the leading crackpot just
criticized this faq for presuming to correct the Britt
anica, but it then and before poses the incompetent pseudo
transform we analyze here in this section.
x to x' and t to t' are obviously coordinate transforms;
the x and t coordinates have been replaced by the coord
inates in the primed system.
A tranform of an equation from one coordinate system to
another is NOT a substitution of the/a definition of x
for itself; that is not a coordinate transformation.
The most that can said for such a substitution is that
it is a change of variable.
But the crackpots are calling this a coordinate trans
form of the original equation:
(x'+vt)^2  (ict')^2 = 0.
It is not a coordinate transform, of course, except
accidentally. (x'+vt) is not the primed system
coordinate, it is another form/expression of x. They
get that substitution by solving x'=xvt for x; x=x'+vt.
So, by incompetent misnomer, they accomplish what they
have been railing against all along.
It has been the generalized coordinate form in question all
this time:
(xx0)^2  (ict)^2 = 0.
Here they substitute for x instead of transforming to the
primed frame:
(x'+vtx0)^2  (ict')^2.

^

^

It is still x ^ but see what they have accomplished
by their mis/malfeasance:
[x'+vtx0]=[x'+(vtx0)]=[x'(x0vt)].
=[x'x0']
The crackpots have been bragging about how you don't
have to transform the circle center's coordinate to
transform the circle center's coordinate. Bragging
that what they were doing was not what they said
they were doing.
This does give us insight as to some of the crackpot
variations on their x0'<>x0vt theme, which in all the
variations will be discussed in later sections..
They are used to seeing the mixed coordinate form,
(x'+vtx0) without realizing what it respresented,
so  accompanied with a lack of understanding of
the term 'dependent'  they are used to seeing just
the one vt term, and not the one hidden in the defi
nition of x' and are used to imagining it makes the
whole expression time dependent and thus not invariant.
About which, let x=10, let, x0=20, v=10, and t
variously 10 and 23:
(xx0)=10. Using their (x'+vtx0):
For t=10, we have (x'+vtx0) = [ (1010*10) + (10*10)  (20) ]
= 90 + 100  20
= 10
= (xx0)
For t=23, we have (x'+vtx0) = [ (1010*23) + (10*23)  (20) ]
= 220 + 230  20
= 10
= (xx0)
The result depends in no way on the value of time;
we showed the obvious for a couple of instances of t
just so you can see that the crackpots not only do
not understand the obvious logic of the algebra
{ (x'x0')=[ (vt)(x0vt) ]=(xx0) }  which shows
that the transform has no possible time term effect 
but they don't understand even a simple arithmetic
demonstration of the facts.
Oh. Their (x'+vtx0) or (x'+vt'x0) reduces the same
way since t'=t:
(xvt+vtx0)=(xx0).
Their process, which says (x'+vt') is the transform
of x, says that (x'+vt') is the moving system location
of x, but it can't be because x is moving further in
the negative direction from the moving viewpoint.
That formula will only work out with v<0 which is indeed
the velocity the primed system sees the other moving at.
However, that formula cannot be derived from x'=xvt,
the formula for transformation of the coordinates from
the unprimed to the primed,

Subject: 8. What does sci.math have to say about x0'=x0vt?
The crackpots' positions/arguments were put to sci.math
in such a way that at least two or three who posted re
sponses thought it was your faqer who was on the idiot's
side of the questions.
Their responses:

I. x0' = x0. In other words: x0' <> x0vt, or "constant
values on the xaxis are not subject to the transform".
AA: ====================================================================
No. x0' = x0  vt.
Well, if you want, you could define "constant values on the xaxis", but
in the context of the question that is not relevant. The relevant fact is
that if the unprimed observer holds an object at point x0, then the
primed observer assigns to that object a coordinate x0' which is
numerically related to x0 by x0'= x0 vt.
AA: ====================================================================
EE: ====================================================================
What does this mean? The line x=x0 will give x'=xv*t=x0vt', so if x0'
is to give the coordinate in the (x',t',)system, it will be given by
x0'=x0v*t': ie., it is not given by a constant. Thus, being at rest
(constant xcoordinate) is a coordinatedependent concept.
EE: ====================================================================
GG: ====================================================================
Sounds very false. We can say that the representation of the point X0 is
the number x0 in the unprimed system, and x0' in the primed system.
Clearly x0 and x0' are different, if vt is not zero. However one may say
that (though it sounds/is stupid) the point X0 itself "is the same
throughout the transformation". However that expression sounds
meaningless, since a transform (ok, maybe we should call it a change of
basis) is only a function that takes the point's representation in one
system into the same point's representation in another system. It is
preferrable to use three notations: X0 for the point itself and x0 and
x0' for the points' representations in some coordinate systems.
GG: ====================================================================

Subject: 9. But Doesn't x.c'=x.c?
That idea is one of the most idiotic to come up, and it does
so frequently. And in a number of guises.
The idea being that x.c' <> x.cvt, with x.c being what
we have called x0 above; the notation makes no difference.
Some crackpots have managed to maintain that position even
after graphs have illustrated that such an idea means that
after a while a circle center represented by x.c' could be
outside the circle.
The leading crackpot just make that explicit, as far as
one can tell from his befuddled post in response to a line
about "active" transforms, which are actually moving body
situations, not coordinate transformations:

e>An active transform is not a coordinate transform, ...
Right, it is a transform of the center (in the opposite direction)
done to effect the change of coordinates without a coordinate
transform. ...
E: Transform of the center? Center of a circle?
He really is saying a circle center moves in
the opposite direction of the circle! Right?

If r=10 and x.c was at x.c=0, then the points on the circle
(10,0), (10,0), (0,10) and (0,10) could at some time become
(10,0), (30,0), (20,10), and (20,10), but with x.c'=x.c,
the circle center would be at (0,0) still! The circle is here
but its center is way, way over there! Indeed, although a change
of coordinate systems is not movement of any object described in
the coordinates, the x.c'=x.c crackpottery is tantamount to the
circle staying put but the center moving away. Or vice versa.

Subject: 10. But Isn't (x'x.c')=(xx.c) Actually Two Transformations?
One crackpot puts the (x'x.c')=(xvt  x.c+vt) relationship
like this:
(xvt+vt  x.c).
See, he says, that is transforming x (with xvt  x.c) and then
reversing the transform (xvt+vt  x.c).
That's just another crackpot form of the idiocy that
x.c' <> x.cvt. You'll have noticed the implication
is that there is no transform vt term relating to x.c.

Subject: 11. But Doesn't (x'x.c+vt) Prove The Transformation
Time Dependent?
That particular crackpottery is perhaps more corrupt than
moronic, since it includes deliberately hiding a vt term from
view, and pretending it isn't there. [However, we have seen
above that it is a familiar incompetency, and not likely an
original.]
"Look," the crackpots say, "there is a time term in the
transformed (x'  x.c+vt). The transform isn't invariant!
It's time dependent!"
Just put x' in its original axis form, also, which reveals
the other time term, the one they hide:
(x'x.c+vt) = (xvt  x.c+vt) = (xx.c).
So, at any and all times, the transform reduces to the
original expression, with no time term on which to be
dependent.
Then there is the fact that if you leave the equation
in any of the various notation forms  with or without
reducing them algebraicly  the arithmetic always comes
down to the same as (xx.c). That means nothing to crack
pots, but may mean something to you.

Subject: 12. But Isn't (x'x.c')=(xx.c) a Tautology?
My dictionary relates 'tautology' to needless repetition.
That's another form of the x.c' <> x.cvt idiocy.
The repetition involved is the vt transformation term.
Apply the vt term to the x term, and it is needless
repetition to apply it anywhere again? The 'again' is
to the x.c term. The x.c' = x.c crackpot idiocy.
The repetition of the vt terms is required by the presence
of two x values to be transformed.
Be sure to note the next section.

Subject: 13. But Isn't (x'x.c')=(xx.c) Almost the Definition of
a Linear Transform?
Now, how on earth can we relate a tautology to a basic
definition in math?
From the top, bottom, middle, and other books in the stack
we get this definition:

A linear transformation, A, on the space is a method of corr
esponding to each vector of the space another vector of the
space such that for any vectors U and V, and any scalars
a and b,
A(aU+bV) = aAU + bAV.

Let points on the sphere satisfy the vector X={x,y,z,1},
and the circle center satisfy C={x.c,y.c,z.c,1}. Let a=1,
and b=1.
Let A= ( 1 0 0 ut )
( 0 1 0 vt )
( 0 0 1 wt )
( 0 0 0 1 )
A(aX+bC) = aAX + bAC.
aX+bC = (xx.c, yy.c, zz.c, 0 ).
The left hand side:
A( x  x.c , y  y.c, z  z.c, 0 )
= ( xx.c , yy.c, zz.c, 0 ).
The right hand side:
aAX= ( xut, yvt, zwt, 1 ).
bAC= (x.c+ut, y.c+vt, z.c+wt, 1 ).
and
aAX+bAC = ( xx.c, yy.c, zz.c, 0 ).
Need it be said?
Sure: QED. On the galilean transform the
definition of a linear transform,
A(aU+bV)=aAU + bAV,
is completely satisfied.
The generalized form transforms exactly and
nonredundantly  with ONE TRANSFORM, not a
transform and reverse transform  and non
tautologically, just as the very definition
of a linear transform says it should.
And does so with absolute invariance, with this
galilean transformation.

Subject: 14. But The Transform Won't Work On Time Dependent Equations?
The main crackpot that has asserted such a thing was referring
to equations such as in Subject 4, above. The Light Sphere
equation; for which we have shown repeatedly elsewhere that the
numerical calculations are identical for any primed values as
for the unprimed values.
The presence  before transformation  of a velocity term
seems to confuse the crackpots. It turns out there is ex
treme historical reason for this, as you will see in the
subject on Maxwell's equations.

Subject: 15. But The Transform Won't Work On Wave Equations?
See Subject 17, below, for a discussion of Second Derivative
forms and the galilean transforms.

Subject: 16. But Maxwell's Equations Aren't Galilean Invariant?
Oh? Just what is the magical term in them that prevents
(x'x.c')=(xvt  x.c+vt)=(xx.c) from holding true?
It turns out not to be magic, but reality, that interferes
with the application of the galilean transforms to the gen
eralized coordinate form(s) of Maxwell: there are no coordi
nates to transform!
When True Believer crackpots are shown the simple
demonstration that the galilean transform on
generalized cartesian coordinates is invariant,
their first defense is usually an incredibly stupid
"x0'=x0, because the coordinate of a circle center,
or point of emission, etc, is a constant and can't
be transformed."
The last defense is "but Maxwell's equations are not
invariant under that coordinate transform." When
asked just what magic occurs in Maxwell that would
prevent the simple algebra
(x'x0')=[ (xvt)(x0vt) ]=(xx0)
from working, and when asked them for a demonstration,
they will never do so, however many hundreds of
times their defense is asserted.
The reason may help you understand part of Einstein's
1905 paper in which he gave us his absurd Special
Relativity derivation:
THERE ARE NO COORDINATES IN THE EQUATIONS TO BE TRANSFORMED.
Einstein gave the electric force vector as E=(X,Y,Z)
and the magnetic force vector as B=(L,M,N), where the
force components in the direction of the x axis are
X and L, Y and M are in the y direction, Z and N in
the z direction.
Those values are not, however, coordinates, but values
very much like acceleration values.
BTW, the current fad is that E and B are 'fields', having
been 'force fields' for a while, after being 'forces'.
So, when Einstein says he is applying his coordinate
transforms to the Maxwell form he presented, he is
either delusive or lying.
(a) there are no coordinates in the transform equations
he gives us for the Maxwell transforms, where
B=beta=1/sqrt(1(v/c)^2):
X'=X. L'=L.
Y'=B(Y(v/c)N). M'=B(M+(v/c)Z).
Z'=B(Z+(v/c)M). N'=B(N(v/c)Y).
X is in the same direction as x, but is not a coordinate.
Ditto for L. They are not locations, coordinates on the
xaxis, but force magnitudes in that direction.
Similarly for Y and M and y, Z and N and z.
(b) the v of the "coordinate transforms" is in Maxwell
before any transform is imposed; Einstein's transform
v is the velocity of a coordinate axis, not the velocity
of a particle, which is what was in the equation before
he touched it.
(c) if they were honest Einsteinian transforms, they'd be
incompetent. The direction of the particle's movement is
x, which means it is X and L that are supposed to be
transformed, not Y and M, and Z and N. And when SR does
transform more than one axis, each axis has its own
velocity term; using the v along the xaxis as the v
for a yaxis and zaxis transform is thus trebly absurd:
the axes perpendicular to the motion are not changed
according to SR, the v used is not their v, and the v
is not a transform velocity anyway.
(d) as everyone knows, the effect of E and B are on the
particle's velocity, which is a speed in a particular
direction. Both the speed and direction are changed
by E and B, but v  the speed  is a constant in SR.
As absurd as are the previously demonstrated Einsteinian
blunders, this one transcends error and is an incredible
example of True Believer delusion propagating over decades.
The components of E and B do differ from point to point,
and in the variations that are not coordinate free,
they are subject to the usual invariant galilean trans
formation when put in the generalized coordinate form.

The SR crackpots don't know what coordinates are. The
various things they call coordinates include coordin
nates, but also include a variety of other quantities.

1. One may express coordinates in a oneaxisatatime
manner [like x^2+y^2=r^2] but it is the use of vector
notation that shows us what is going on. In vector
notation the triplet x,y,z [or x1,x2,x3, whatever]
represents the three spatial coordinates, but there
are socalled basis vectors that underlie them. Those
may be called i,j,k. Thus, what we normally treat as
x,y,z is a set of three numbers TIMES a basis vector
each.
2. These e*i, f*j, g*k products can have a lot of meanings.
If e, f, j are distances from the origin of i,j,k then
e*i, f*j, g*k are coordinates: distances in the directions
of i,j,k respectively, from their origin. That makes the
triplet a coordinate vector that we describe as being an
x,y,z triplet; perhaps X=(x,y,z).
The e*i, f*j, g*k products could be directions; take any
of the other vectors described above or below and divide the
e,f,g numbers by the length of the vector [sqrt(e^2+f^2+g^2)].
That gives us a vector of length=1.0, the e,f,g values of
which show us the direction of the original vector. That
makes the triplet a direction vector that we describe as
being an x,y,z triplet; perhaps D=(x,y,z).
The e*i, f*j, g*k products could be velocities; take any
of the unit direction vectors described above and multiply
by a given speed, perhaps v. That gives a vector of length
v in the direction specified. That makes the triplet a
velocity vector that we describe as being an x,y,z triplet;
perhaps V=(x,y,z). Each of the three values, e,f,g, is the
velocity in the direction of i,j,k respectively.
The e*i, f*j, g*k products could be accelerations; take any
of the unit direction vectors described above and multiply
by a given acceleration, perhaps a. That gives a vector of
length a in the direction specified. That makes the triplet
an acceleration vector that we describe as being an x,y,z
triplet; perhaps A=(x,y,z). Each of the three values, e,f,g,
is the acceleration in the direction of i,j,k respectively.
The e*i, f*j, g*k products could be forces (much like accel
erations); take any of the unit direction vectors described
above and multiply by a given force, perhaps E or B. That
gives a vector of length E or B in the direction specified.
That makes the triplet a force vector that we describe as
being an x,y,z triplet; perhaps E=(x,y,z) or B=(x,y,z). Each
of the three values, e,f,g, is the force in the direction of
i,j,k respectively.
Einstein's  and Maxwell's  E and B are
not coordinate vectors.
============================================================
There is another variety of intellectual befuddlement that
misinforms the idea that Maxwell isn't invariant under the
galilean transform: confusions about velocities.
Velocities With Respect to Coordinate Systems.

Aaron Bergman supplied the background in a post to a sci.physics.*
newsgroup:
===============================================================
Imagine two wires next to each other with a current I in each.
Now, according to simple E&M, each current generates a magnetic
field and this causes either a repulsion or attraction between
the wires due to the interaction of the magnetic field and the
current. Let's just use the case where the currents are parallel.
Now, suppose you are running at the speed of the current between
the wires. If you simply use a galilean transform, each wire,
having an equal number of protons and electrons is neutral. So,
in this frame, there is no force between the wires. But this is a
contradiction.
================================================================
First of all, the invariance of the galilean transform (x'x.c')
=(xx.c), insures that it is an error to imagine there is any
difference between the data and law in one frame and in another;
the usual, convenient rest frame is the best frame and only frame
required for universal analysis. [Well, (x'<>x, x,c'<>x.c, but
(x'x.c')=(xx.c).]
Second, given that you decide unnecessarily to adapt a law to
a moving frame, don't confuse coordinate systems with meaningful
physical objects, like the velocity relative to a coordinate
system instead of relative to a physical body or field.
In other words, what does current velocity with respect to a
coordinate system have to do with physics?
Nothing. Certainly not anything in the example Bergman gave.
What is relevant is not current velocity with respect to a
coordinate system, but current velocity with respect to wires
and/or a medium. The velocity of an imaginary coordinate sys
tem has absolutely nothing to do with meaningful physical vel
ocity. You can  if you are insightful enough and don't violate
item (e)  identify a coordinate system and a relevant physical
object, but where some v term in the pretransformed law is
in use, don't confuse it with the velocity of the coordinate
transform.
Velocities With Respect to ... What?

Albert Einstein opened his 1905 paper on Special Relativity
with this ancient incompetency:
===============================================================
The equations of the day had a velocity term that was taken
as meaning that moving a magnet near a conductor would create
a current in the conductor, but moving a conductor near a
wire would not. This was belied by fact, of course.
The important velocity quantity is the velocity of the
magnet and conductor with respect to each other, not to
some absolute coordinate frame (as far as we know) and
not to an arbitrary coordinate system.
One possible cause was the idea: "but the equation says the magnet
must be moving wrt the coordinate system" or "... the absolute
rest frame".
There not being anything in the equation(s) to say either of
those, it is amazing that folk will still insist the velocity
term has nothing to do with velocity of the two bodies wrt
each other.


Subject: 17. First and Second Derivative differential equations.
One of the intellectually corrupt ways of
denying the very simple demonstration of
galilean invariance of all laws expressed
in the generalized coordinate form demanded
by analytic geometry, vector analysis, and
measurement theory
[ (x'x.c')=[ (xvt)(x.cvt) ]=(xx.c) ]
is the assertion that those equations 'over there'
(usually Maxwell or wave) are somehow immune to
the elementary laws of algebra used to demon
strate the invariance. [Unfortunately, the
assertions are never accompanied by reference
to the magical math that makes elementary al
gebra invalid. Wonder why that is?]
Part of the time it is based on the old lore
based on the incompetent transformation of
the privileged form of an equation instead
of the correct form. [Evidence of this is
any reference to an effect due to the velocity
of the transform; it falls out algebraicly
 as you see above  and cancels out arith
metically  as you can see above.]
But usually it is just whistling in the dark,
waving the cross (zwastika, I'd say) at
the mean old vampire.
The most general equation that could be conjured
up is a differential with either First or Second
Derivatives.
Let's examine the plausibility of such magical
magical, noninvariance assertions.
(a) to get a Second Derivative you must have
a First Derivative.
(b) to get a First Derivative you must have
a function to differentiate.
(c) to get a Second Derivative you must have
a function in the second degree.
So, let us examine the question as to whether
any such common Maxwell/wave equation will
differ for
(a) the common, privileged form, represented
as ax^2, with a being an unknown constant
function.
(b) the generalized cartesian form, represented
as a(xx.c)^2 = ax^2 2ax(x.c) + ax.c^2,
with a being an unknown constant function.
(c) the transformed generalized cartesian form,
represented as a(xvt x.c+vt)^2, same as for
(b), = ax^2 2ax(x.c) + ax.c^2, of course,
with a being an unknown constant function.
I. for (a), remembering that x.c is a constant,
and that this version is only correct because
x.c=0, otherwise (b) is the correct form:
d/dx ax^2 = 2ax
(d/dx)^2 ax^2 = 2a
II. for (b), remembering that x.c is a constant.
d/dx (ax^2 2ax(x.c) + ax.c^2) = 2ax  2ax.c
(d/dx)^2 (ax^2 2ax(x.c) + ax.c^2) = 2a
III. for (c); same as for (b).
So, what we have seen so far is
(1) differential equations in the second degree
 the wave equations  must clearly be the same for
all forms: the privileged form in x, the generalized
cartesian form in x and the centroid, x.c, or the
transformed generalized cartesian form.
That is, anyone who imagines that correct usage
gives different results for galilean transformed
frames is at first showing his ignorance, and in
the end showing his intellectual corruption.
(2) As far as the First Derivatives are concerned, the
only cases in which there really is a difference between
the two forms is where x.c <> 0, and in that case, the
use of the privileged form is obviously incompetent.
So, how do you correctly use the differential equations?
If you are using rest frame data with the centroid
at x=0, etc, you can't go wrong without trying to
go wrong.
If you are using rest frame data with the centroid
not at x=0, you must use (xx.c) anyplace x appears
in the equation.
If you are using moving frame data, you must use the
moving frame centroid as well as the light front
(or whatever) moving frame data itself, perhaps first
calculating (x'x.c'), which equals (xx.c) which is
obviously correct, and which is obviously the plain old
correct x of the privileged form.
Unless, of course, there really is some magical term
or expression that invalidates the obvious and elemen
tary algebra of the invariance demonstration.
Or maybe you just whistle when you don't want basic
algebra to hold true.
Eleaticus
!?!?!?!?!?!?!?!?!?
! Eleaticus Oren C. Webster ThnkTank@concentric.net ?
! "Anything and everything that requires or encourages systematic ?
! examination of premises, logic, and conclusions" ?
!?!?!?!?!?!?!?!?!?