Energy and electromagnetism of a differential form

TL;DR

This paper characterizes the electromagnetic energy tensor on Lorentzian manifolds as the unique natural tensor satisfying specific divergence conditions, and extends the theory to higher forms and p-dimensional charged particles.

Contribution

It provides a unique characterization of the electromagnetic energy tensor and generalizes electromagnetism to higher differential forms and extended charged particles.

Findings

Characterization of the electromagnetic energy tensor as the only natural tensor with specific properties.

Extension of electromagnetic theory to differential forms of arbitrary order.

Development of a generalized electromagnetism for p-dimensional charged particles.

Abstract

Let X be a smooth manifold of dimension 1+n endowed with a lorentzian metric g, and let T be the electromagnetic energy tensor associated to a 2-form F. In this paper we characterize this tensor T as the only 2-covariant natural tensor associated to a lorentzian metric and a 2-form that is independent of the unit of scale and satisfies certain condition on its divergence. This characterization is motivated on physical grounds, and can be used to justify the Einstein-Maxwell field equations. More generally, we characterize in a similar manner the energy tensor associated to a differential form of arbitrary order k. Finally, we develop a generalized theory of electromagnetism where charged particles are not punctual, but of an arbitrary fixed dimension p. In this theory, the electromagnetic field F is a differential form of order 2+p and its electromagnetic energy tensor is precisely the energy tensor associated to F.