Low-dimensional electric charges. Covariant description

TL;DR

This paper presents a covariant, metric-free formalism for describing low-dimensional electric charges and currents using differential forms, applicable in arbitrary spacetimes and involving Dirac delta-forms for singular densities.

Contribution

It introduces a covariant, metric-free framework for low-dimensional electric charge densities using differential forms and Dirac delta-forms, extending previous descriptions.

Findings

Covariant description of surface, string, and point charges.

Use of Dirac delta-forms for singular densities.

Derivation of covariant conservation laws for low-dimensional charges.

Abstract

A compact and elegant description of the electromagnetic fields in media and in vacuum is attained in the differential forms formalism. This description is explicitly invariant under diffeomorphisms of the spacetime so it is suitable for arbitrary curvilinear coordinates. Moreover, it is independent of the geometry of the underline spacetime. The bulk electric charge and current densities are represented by twisted non-singular differential 3-forms. The charge and current densities with a support on the low dimensional submanifolds (surfaces, strings and points) naturally require singular differential forms. In this paper, we present a covariant metric-free description of the surface, string and point densities. It is shown that a covariant description requires Dirac's delta-forms instead of delta-functions. Covariant metric-free conservation laws for the low-dimensional densities are derived.