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

Pentcho Valev wrote:

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.

Wild. But couldn't 3 be rephrased as:

3. If and only if in a reference frame S the speed v of a reference frame S'
(x = vt) is equal to the speed of light, then the speed of light is zero in S'
(x' = 0). But if v = c, then gamma = infinity, we'd have the product of
infinity and zero in Lorentz transformation. Basically all lengths are
contracted to zero, all positions and times measured as zero in the frame
moving at the speed of light. Since this only deals with a limiting case, I
really can't see this as a viable postulate on the same level as the two
canonical postulates of special relativity.

Clearly, axioms 2 and 3 are, independently, corollaries of Lorentz
equations.

Yes, 2 can be derived from the Lorentz transformation, and it's great fun to
show that the wave equation for E-M waves is invariant under a Lorentz
transformation -- the laws of physics are the same for all observers, etc.
But the Lorentz transformation is not sufficiently intuitive for me to be
happy with treating it as an axiom.

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.

Has that actually been shown? Granted that I've only seen it done with the
additional linearity condition (or first-order linearity, as John S. Denker
has pointed out), has some _non-linear_ tranformation been found that obeys
axioms 1 & 2? That would show that the Lorentz transformation is not uniquely
determined by the two axioms of special relativity. Or maybe an argument can
be made that first-order linearity is "clearly necessary" (for what reasons?)
and so can be assumed to be fundamental?

It may just be a case for Occam's Razor. It'd take linearity as an assumption
long before #3 above!

Ken