I am going to present the axioms REALLY used in the deduction of Lorentz equations in
Einstein's "Relativity: The special and general theory". The first axiom is:
Axiom 1: If and only if x' - ct' = P, then x - ct = P/lambda. Also, for light moving along
the negative axis, if and only if x' + ct' = Q, then x + ct = Q/mu. P and Q can be any
numbers including zero. In contrast, lambda and mu cannot be zeros.
Obviously, the postulate of the constancy of the speed of light is a corollary of Axiom 1
but itself is not an axiom for the deduction of Lorentz equations as I previously believed.
By defining a = (lambda + mu)/2 and b = (lambda - mu)/2, one obtains the following
equations equivalent to Axiom 1:
Axiom 1: x' = ax - bct ; ct' = act - bx
The second axiom is the one I formulated before:
Axiom 2: 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).
Ken Caviness' version is equivalent:
Axiom 2: 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).
The third axiom should have been the principle of special relativity but here Einstein
is particularly obscure. There is a mysterious story about a snapshot such that t = 0 (or t'
= 0) whereas all other variables remain unrestricted. In fact, when t = 0, x = x' = t' = 0,
in acordance with the initial assumptions. The only solution to the problem is to take the
result Einstein obviously strives for and declare it as the third axiom:
Axiom 3: a^2 = 1/((1 - v^2)/c^2)
If I am wrong and the above formula CAN be validly deduced from the principle of special
relativity, Axiom 3 will be:
??? Axiom 3: The principle of special relativity
These are the real axioms Einstein uses in this book - Lorentz equations CAN be deduced from
them. But in other sources another set of axioms could be relevant.