Special Relativity (alternative Formulations) - "Single Postulate" Approaches

"Single Postulate" Approaches

Equivalent to the original ? Yes.

According to some references, the theory of special relativity can be derived from a single postulate: the principle of relativity. This claim can be misleading because actually these formulations rely on various unsaid assumptions such as isotropy and homogeneity of space. The question here is not about the exact number of postulates. The phrase "single postulate" is just used in comparison with the original "two postulate" formulation. The real question here is whether universal lightspeed can be deduced rather than assumed.

The Lorentz transformations, up to a nonnegative free parameter, can be derived without first postulating the universal lightspeed. Experiment rules out the validity of the Galilean transformations and this means the parameter in the Lorentz transformations is nonzero hence there is a finite maximum speed before anything has been said about light. Combining this with Maxwell's equations shows that light travels at this maximum speed. The numerical value of the parameter in these transformations is determined by experiment, just as the numerical values of the parameter pair c and the permittivity of free space are left to be determined by experiment even when using Einstein's original postulates. When the numerical values in both Einstein's and these other approaches have been found then these different approaches result in the same theory. So the end result of the interlocking trio of theory+Maxwell+experiment is the same either way. This is the sense in which universal lightspeed can be deduced rather than postulated.

For some historical information, see: History of special relativity#Spacetime physics and the section "Lorentz transformation without second postulate" for the approaches of Ignatowski and Frank/Rothe. However, according to Pauli (1921), Resnick (1967), and Miller (1981), those models were insufficient. But the constancy of the speed of light is contained in Maxwell's equations. That section includes the phrase "Ignatowski was forced to recourse to electrodynamics to include the speed of light.". So, the trio of "principle of relativity+Maxwell+numerical values from experiment" gives special relativity and this should be compared with "principle of relativity+second postulate+Maxwell+numerical values from experiment". Since Einstein's 1905 paper is all about electrodynamics he is assuming Maxwell's equations, and the theory isn't practically applicable without numerical values. When compared like with like, from the point of view of asking what is knowable, the second postulate can be deduced. If you restrict your attention to just the standalone theory of relativity then yes you need the postulate. But given all the available knowledge we don't need to postulate it. In other words different domains of knowledge are overlapping and thus taken together have more information than necessary.

This can be summarized as follows:

1. Experimental results rule out the validity of the Galiliean transformations.
2. That just leaves the Lorentz transformations with a finite maximal speed V.
3. Given a maximal speed V, the only consistent way of combining PofR with Maxwell's equations is to identify Maxwell's parameter : with the aforementioned maximal speed V.
4. We are now at the same starting point as if we had postulated the constancy of light, so we proceed to develop all the usual results of special relativity.

There are references which discuss in more detail the principle of relativity