Energy-momentum tensor from diffeomorphism invariance in classical electrodynamics

TL;DR

This paper derives a unique, symmetric, and gauge-invariant energy-momentum tensor in classical electrodynamics using diffeomorphism invariance, clarifying energy-momentum localization and exchange in both free and interacting fields.

Contribution

It introduces a novel derivation of the energy-momentum tensor from spacetime-dependent translations, avoiding traditional improvement procedures and clarifying energy-momentum exchange in interactions.

Findings

The energy-momentum tensor is symmetric, gauge-invariant, and derived from diffeomorphism invariance.

In free fields, it matches known expressions like Belinfante-Rosenfeld.

In interacting systems, it clarifies the role of the interaction term in energy-momentum exchange.

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.