The wave equation in the birth of spacetime symmetries

TL;DR

This paper explores the historical development of spacetime symmetries by deriving Lorentz and Voigt transformations from wave equation invariance principles, highlighting their mathematical relationship and implications for special relativity.

Contribution

It derives Lorentz and Voigt transformations from wave equation invariance and clarifies their relation, offering a new perspective on special relativity's foundational symmetries.

Findings

Lorentz transformations derived from D'Alembert operator invariance

Voigt transformations derived from conformal invariance of wave equation

Voigt transformations are conformally invariant, related to Lorentz transformations

Abstract

In 1887 Voigt published a paper dedicated to the Doppler effect in which he demanded form invariance to the wave equation in inertial frames and obtained a set of spacetime transformations now known as the Voigt transformations. In 1905 Poincar\'e showed that the wave equation was also invariant under the Lorentz transformations. Voigt and Lorentz transformations are then closely related, but this relation is not widely known in the standard literature. In this paper we derive the Lorentz transformations from the invariance of the D'Alembert operator $\big(\Box^{2}=\Box'^{2}\big)$ and the Voigt transformations from the conformal invariance of the D'Alembert operator $\big(\Box^{2}=(1/\gamma^2)\Box'^{2},$ where $\gamma=1/\sqrt{1-v^2/c^2}\big).$ The homogeneous scalar wave equation is then invariant under the Lorentz transformations and conformally invariant under the Voigt transformations. We suggest a presentation of special relativity in which the Voigt transformations are commented after discussing the Galilean transformations but before presenting the Lorentz transformations.