Getting the Lorentz transformations without requiring an invariant speed

TL;DR

This paper demonstrates that Lorentz transformations can be derived solely from the absence of privileged reference frames and their group structure, without assuming an invariant speed, clarifying foundational aspects of special relativity.

Contribution

It provides an elementary proof that the invariant speed in Lorentz transformations is a consequence, not an assumption, of the relativity principle and group properties.

Findings

Lorentz transformations follow from group structure and relativity principle

Invariant speed is a derived consequence, not an initial assumption

Elementary proof suitable for introductory courses

Abstract

The structure of the Lorentz transformations follows purely from the absence of privileged inertial reference frames and the group structure (closure under composition) of the transformations---two assumptions that are simple and physically necessary. The existence of an invariant speed is \textit{not} a necessary assumption, and in fact is a consequence of the principle of relativity (though the finite value of this speed must, of course, be obtained from experiment). Von Ignatowsky derived this result in 1911, but it is still not widely known and is absent from most textbooks. Here we present a completely elementary proof of the result, suitable for use in an introductory course in special relativity.