A Symmetry-First Elementary Derivation of the Lorentz Transformation

Abstract

We present an elementary, symmetry-first derivation of the Lorentz transformation without assuming the invariance of the speed of light at the outset. The argument proceeds in three stages. First, spacetime homogeneity yields additivity, and uniformity of free motion then supplies the along-line continuity needed to pass from additivity to linearity. Second, spatial isotropy determines the parity of the coefficient functions, and the Abelian structure of the continuous local one-parameter group of collinear boosts yields velocity reciprocity, rather than presupposing it, together with a universal constant $\kappa$. These constraints produce a local one-parameter family of generalized Lorentz-type transformations. Third, analysis of the collinear velocity law, combined with the experimentally supported frame independence of the speed of light, selects the physical branch. Identifying its invariant speed with the vacuum speed of light fixes $\kappa=-1/c^2$ and gives the Lorentz transformation. This staged derivation makes the linearity and reciprocity arguments explicit while separating the mathematical construction of the kinematical family from its empirical selection.