Energy-momentum tensor from diffeomorphism invariance in classical electrodynamics

TL;DR

The paper addresses energy–momentum localization in classical electrodynamics, where canonical and improvement-based tensors yield gauge-dependent or ambiguous local densities. It adopts a diffeomorphism-invariance perspective with spacetime-dependent translations in flat spacetime to derive a unique, symmetric, gauge-invariant energy–momentum tensor that satisfies an off-shell Noether identity for the free field. For interacting systems, it shows that the field and particle each acquire well-defined EMTs while the interaction term itself contributes no independent local density; the exchange is encoded in the coupled equations of motion, resolving prior ambiguities. The total EMT, $T_T^{\mu\nu} = T_D^{\mu\nu} + T_p^{\mu\nu}$, is conserved off-shell and coincides with the Hilbert definition, providing a geometric, gravity-inspired foundation for energy–momentum localization in flat spacetime with interacting subsystems.

Abstract

We reexamine the energy-momentum tensor in classical electrodynamics from the perspective of spacetime-dependent translations, i.e., diffeomorphism invariance in flat spacetime. When energy-momentum is identified through local translations rather than constant ones, a unique, symmetric, and gauge-invariant energy-momentum tensor emerges that satisfies a genuine off shell Noether identity without invoking the equations of motion. For the free electromagnetic field, this tensor coincides with the familiar Belinfante-Rosenfeld and Bessel-Hagen expressions, but arises here directly from spacetime-dependent translation symmetry rather than from improvement procedures or compensating gauge transformations. In interacting classical electrodynamics, comprising a point charge coupled to the electromagnetic field, diffeomorphism invariance yields well-defined energy-momentum tensors for the field and the particle, while the interaction term itself generates no independent local energy-momentum tensor. Its role is instead entirely encoded in the coupled equations of motion governing energy-momentum exchange, thereby resolving ambiguities in energy-momentum localization present in canonical and improvement-based approaches.