Didactic derivation of the special theory of relativity from the Klein-Gordon equation

TL;DR

This paper provides an educational derivation of special relativity by deriving Lorentz transformations as symmetries of the Klein-Gordon equation, revealing their natural emergence without prior assumptions.

Contribution

It introduces a didactic approach to derive Lorentz transformations from the Klein-Gordon equation, highlighting their role as symmetry transformations and explaining velocity composition.

Findings

Lorentz transformations are derived as symmetries of the Klein-Gordon equation.

The relative velocity bound |v|<c is proven as a theorem.

The noncommutativity of relativistic velocity addition is explained via polar decomposition.

Abstract

We present a didactic derivation of the special theory of relativity in which Lorentz transformations are `discovered' as symmetry transformations of the Klein-Gordon equation. The interpretation of Lorentz boosts as transformations to moving inertial reference frames is not assumed at the start, but it naturally appears at a later stage. The relative velocity $\textbf{v}$ of two inertial reference frames is defined in terms of the elements of the pertinent Lorentz matrix, and the bound $|\textbf{v}| <c\;$ is obtained as a simple theorem that follows from the structure of the Lorentz group. The polar decomposition of Lorentz matrices is used to explain noncommutativity and nonassociativity of the relativistic composition (`addition') of velocities.