Energy-mass equivalence from Maxwell equations

TL;DR

This paper derives Einstein's mass-energy equivalence directly from Maxwell's equations using a simple dynamical model of confined electromagnetic fields, offering new pedagogical insights into the origin of mass.

Contribution

It provides a novel derivation of $E=mc^2$ directly from Maxwell's equations without relying on symmetry or kinematic arguments, emphasizing the dynamical link with electromagnetic fields.

Findings

Derivation of $E=mc^2$ from Maxwell's equations and energy-momentum tensor.

Shows mass-energy equivalence as a consequence of confined electromagnetic fields.

Provides a pedagogical model connecting electromagnetism and the concept of mass.

Abstract

Since the appearance of Einstein's paper {\em"On the Electrodynamics of Moving Bodies"} and the birth of special relativity, it is understood that the theory was basically coded within Maxwell's equations. The celebrated mass-energy equivalence relation, $E=mc^2$, is derived by Einstein using thought experiments involving the kinematics of the emission of light (electromagnetic energy) and the relativity principle. Text book derivations often follow paths similar to Einstein's, or the analysis of the kinematics of particle collisions interpreted from the perspective of different inertial frames. All the same, in such derivations the direct dynamical link with hypothetical fundamental fields describing matter (e.g. Maxwell theory or other) is overshadowed by the use of powerful symmetry arguments, kinematics, and the relativity principle. Here we show that the formula can be derived directly form the dynamical equations of a massless matter model confined in a box (which can be thought of as a toy model of a composite particle). The only assumptions in the derivation are that the field equations hold and the energy-momentum tensor admits a universal interpretation in arbitrary coordinate systems. The mass-energy equivalence relation follows from the inertia or (taking the equivalence principle for granted) weight of confined field radiation. The present derivation offers an interesting pedagogical perspective on the formula providing a simple toy model on the origin of mass and a natural bridge to the foundations of general relativity.