The Function of the Second Postulate in Special Relativity

TL;DR

The paper argues that measuring the parameter in Lorentz transformations without the second postulate is impossible, and that assuming light's properties is necessary to fully establish special relativity.

Contribution

It demonstrates that the second postulate is logically weaker than previously thought but still essential for deriving special relativity.

Findings

Measuring the parameter $k$ requires a signal traveling identically in both directions.

Methods to determine $k$ without the second postulate fail without additional assumptions.

Assuming light's symmetric propagation suffices to recover special relativity.

Abstract

Many authors noted that the principle of relativity, together with space-time symmetries, suffices to derive Lorentz-like coordinate transformations between inertial frames. These contain a free parameter, $k$, (equal to $c^{-2}$ in special relativity) which is usually claimed to be empirically determinable, so that special relativity does not need the postulate of constancy of the speed of light. I analyze this claim and find that all methods destined to measure $k$ fail without further assumptions, similar to the second postulate. Specifically, measuring $k$ requires a signal that travels identically in opposite directions (this is unrelated to the conventionality of synchronization, as the one-postulate program implicitly selects the standard synchronization convention). Positing such a property about light is logically weaker than Einstein's second postulate but suffices to recover special relativity in full.