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

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.