Re: Teaching logic is urgent (the only reasonable transformations)

[Pentcho Valev <pvalev@BAS.BG>]

Pentcho Valev wrote:

> --- Bob LaMontagne <rlamont@POSTOFFICE.PROVIDENCE.EDU> wrote:
>
>  At this very elementary level,
> > your statements (A), (B) and (C) appear logically compatible when
> the
> > Lorentz transformations are used.
> >
> > You proposed this as a question to be presented to students. Could
> you
> > please present what you would consider an acceptable student
> solution -
> > something more detailed than simple assertions that (A), (B) and
> (C) are
> > incompatible.
>
> This is the strangest problem in science I know of. Let us start with
> the linearity condition
>
> x' = ax + by + cz + dt
>
> which is reduced to
>
> x' = ax + dt                /1/
>
> Then Einstein introduces
>
> x' = 0  <->  x = vt            /2/
>
> which, combined with /1/, gives
>
> d = -av                       /3/
>
> The condition symmetrical to /2/ is
>
> x = 0  <->  x' = -vt'         /4/
>
> which, combined with /1/ and /3/ gives
>
> t' = at                       /5/
>
> The last result is obviously incompatible with the second Lorentz
> equation, i.e. Lorentz transformations are incompatible with the
> basic conditions /1/, /2/ and /4/.

This forces us to advance THE ONLY REASONABLE TRANSFORMATIONS compatible
with /1/, /2/ and /4/. These are

x' = a(x - vt)                   /6/

t' = at                             /7/

Obviously the transformations /6/ and /7/ are incompatible with the
postulate of the constancy of the speed of light (x=ct  <->  x'=ct'). As
for the coefficient a, it could be determined as follows. If time
dilation is experimentally proved and processes in the moving system are
slower than in the system at rest, we can substitute, in /7/, a = 1/gamma
and then

x' = (1/gamma)(x - vt)                  /8/

t' = (1/gamma)t                            /9/

If time dilation is an unreliable experimental finding, we substitute a =
1 and return to Galilean transformations.

Pentcho