From Symmetry and Continuity to the Lorentz Transformation

Abstract

We derive the Lorentz transformation from spacetime symmetries and continuity, without assuming the invariance of the speed of light. Spacetime homogeneity yields an additive coordinate map; continuity along four linearly independent inertial worldlines, together with additivity, establishes linearity throughout spacetime. Spatial isotropy and the assumed continuous local one-parameter group structure of collinear boosts yield velocity reciprocity and determine a generalized Lorentz-type family parameterized by $\kappa$. The velocity law and the empirical identification of its invariant speed with the coordinate speed of light in vacuum select the Lorentzian branch, fixing $\kappa =-1/c^2$ and recovering the Lorentz transformation.