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Invariant Galilean Transformations On All Laws
Section - 4. The Encyclopedia Brittanica Incompetency.

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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.   Eq-100 is indeed the correct basis for discussing
     the question of invariance, given that eq-99 is
     the correct 'stationary' (observer S) equation.
     [Let observer M be the 'moving'system observer.]

     In particular, eq-100 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 eq-99 should be derivable from
     eq-100; 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 (eq-99) 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 x-axes 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 eq-100 right;

     They insist that the correct equation is derived 
     directly from x'=x-vt and t'=t. Solve for x=x'+vt
     and replace t with t', then substitute the result
     in eq-99: (x'+vt')^2 - (ct')^2 = 0.

     Besides the fact that this results in an equation
     with arguments exactly equal to eq-99, 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 eq-99 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. 

     Eq-99 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(t-t.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.

     (x-x.e)^2 - [ c(t-t.e) ]^2 = 0        (99a)

VI.  In the spirit of 'there is exactly one thing
     they got right', the correct M system version
     of eq-99a is eq-100a:

     (x'-x.e')^2 - [ c(t'-t.e') ]^2 = 0   (100a)

     Every observer in the universe can derive their
     eq-100a from eq-99a and vice versa, not to mention to and
     from every other observer's eq-99a.

     Now, THAT's invariance. [You do realize that every
     eq-100a reduces to eq-99a, when you back substitute
     from the transforms, right? t.e'=t.e, x.e'=x.e-vt.]

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