Einstein's third axiom (was: ...affirming the consequent)

[Pentcho Valev <pvalev@BAS.BG>]

Ken Caviness wrote:

> Pentcho Valev wrote:
>
> > However Einstein needs something different:
> >
> > B -> A, A, therefore B                                      /4/
> >
> > He proceeds in accordance with /4/ - builds Lorentz equations on B and
> > so creates the sequence (A therefore B therefore Lorentz equations) -
> > the illusion is that
> > Lorentz equations ultimately stem from A. In fact, A CAN be a corollary
> > of B or Lorentz equations, in accordance with /3/, but Lorentz equations
> > can BY NO MEANS be
> > deduced from A.
>
> I vividly remember an assignment in my freshman year Engineering Physics class
> where we were asked to _derive_ the Lorentz transformation equations from
> Einstein's postulates that the laws of physics and the speed of light are the
> same for all inertial observers.  I have since used this in my classes or as a
> homework assignment.  The only additional assumption needed is that the
> transformations be linear in all the variables (x,y,z,t,x',y',z',t').  That
> was handled by the statement that we would _first_ seek a set of
> transformation equations which was linear, but if necessary we could back off
> from that requirement.
>
> Briefly, if you let
>
> x = A x' + B y' + C z' + D t', y = E x' + ..., etc., the unknown constants can
> be identified by symmetry arguments and the requirement that the speed of
> light be measured by both primed and unprimed observers as c.  The Lorentz
> transformation falls out in your lap, i.e., the Lorentz equations can indeed
> "be deduced from A" + the additional requirement that the dependence be
> linear.

In his "Relativity: The special and general theory" Einstein does not postulate
linearity but almost explicitly introduces a third axiom:

1. Principle of special relativity

2. Postulate of the constancy of the speed of light: If and only if the speed of
light in the first inertial frame is c >> v, then in another inertial frame having
a speed v with respect to the first it is c as well.

3. Postulate of the variability of the speed of light: If and only if the speed of
light in the first inertial frame is as low as v (x = vt), then it is zero in the
other inertial frame (x' = 0).

Physically, the third axiom sounds silly but mathematically it is equipollent to
the second - it is indispensable for the determination of initially unknown
parameters. Clearly, axioms 2 and 3 are, independently, corollaries of Lorentz
equations. For the moment, it seems to me that Lorentz equations CAN be derived
from the three axioms, but I am still not sure - there may be some hidden fourth
axiom. In any event, Lorentz equations cannot be derived from axioms 1 and 2
alone.

Pentcho Valev