A Unified Derivation of the Lorentz Transformation and the Relativistic Energy and Momentum

Abstract

I present a unified derivation of the Lorentz transformation and the relativistic formulations of energy and momentum. The derivation begins by defining momentum as a general vector function of velocity, consistent with the principles of space-time homogeneity and spatial isotropy. By applying the postulate of relativity, this momentum formulation remains valid in all inertial reference frames. Relativistic energy is then identified through a comparison with the work-energy theorem. The existence of an invariant speed follows from symmetry arguments, and its sign is determined by the requirement of energy positivity. This approach leads to a unified derivation of many fundamental formulas of special relativity.