The necessity of the second postulate in special relativity

TL;DR

The paper argues that the second postulate of special relativity, concerning the constancy of the speed of light, is essential and cannot be derived from other principles without additional assumptions.

Contribution

It demonstrates that the distinction between zero and non-zero k in Lorentz-like transformations is a fundamental postulate, not just empirical data, and critiques derivations claiming to eliminate the second postulate.

Findings

The value of k is a fundamental postulate, not empirical.

Derivations claiming to derive the second postulate make hidden assumptions.

Dropping isotropy leads to unresolved fundamental questions.

Abstract

Many authors noted that the principle of relativity together with space-time homogeneity and isotropy restrict the form of the coordinate transformations from one inertial frame to another to being Lorentz-like. The equations contain a free parameter, $k$ (equal to $c^{-2}$ in special relativity), which value is claimed to be merely an empirical matter, so that special relativity does not need the postulate of constancy of the speed of light. I analyze this claim and argue that the distinction between the cases $k = 0$ and $k \neq 0$ is on the level of a postulate and that until we assume one or the other, we have an incomplete structure that leaves many fundamental questions undecided, including basic prerequisites of experimentation. I examine an analogous case in which isotropy is the postulate dropped and use it to illustrate the problem. Finally I analyze two attempts by Sfarti, and Behera and Mukhopadhyay to derive the constancy of the speed of light from the principle of relativity. I show that these attempts make hidden assumptions that are equivalent to the second postulate.