Energy-mass equivalence from Maxwell equations

Abstract

The celebrated mass-energy equivalence relation, [Formula: see text], was derived by Einstein using thought experiments involving the kinematics of the emission of light and the relativity principle. Textbook derivations often follow paths similar to Einstein's or analyze the kinematics of particle collisions in different inertial frames. In such derivations, the direct dynamical link with fundamental fields describing matter (e.g., Maxwell theory or other) is overshadowed. Here, we show that the formula can be derived directly from the dynamical equations of electromagnetic energy confined in a box (which can be thought of as a toy model of a composite particle). The mass-energy equivalence relation follows from the inertia or (taking the equivalence principle for granted) weight of confined radiation. The present derivation offers an interesting pedagogical perspective on the formula, provides a simple toy model for the origin of mass, and can serve as a bridge to general relativity.