Derivation of the potential, field, and locally-conserved charge-current density of an arbitrarily moving point-charge

TL;DR

This paper explicitly derives the complete charge-current density and electromagnetic field of an arbitrarily accelerated relativistic point-charge, revealing additional delta-contributions necessary for local charge conservation and interpreting the fields as nonlinear generalized functions.

Contribution

It introduces new delta-contributions to the charge-current density and field strength, ensuring local conservation and providing a unique four-potential consistent with classical electrodynamics.

Findings

Charge-current density includes derivatives of the charge's trajectory.

Field strength contains additional delta-contributions for local conservation.

The four-potential is uniquely determined by local charge conservation.

Abstract

The complete charge-current density and field strength of an arbitrarily accelerated relativistic point-charge are explicitly calculated. The current density includes, apart from the well-established three-dimensional delta-function which is sufficient for its global conservation, additional delta-contributions depending on the second and third proper-time derivatives of the position, which are necessary for its local conservation as required by the internal consistency of classical electrodynamics which implies that local charge-conservation is an {identity}. Similarly, the field strength includes an additional delta-contribution which is necessary for obtaining this result. The Lienard-Wiechert field and charge-current density must therefore be interpreted as nonlinear generalized functions, i.e., not just as distributions, even though only linear operations are necessary to verify charge-current conservation. The four-potential from which this field and the conserved charge-current density derive is found to be unique in the sense that it is the only one reducing to an invariant scalar function in the instantaneous rest frame of the point-charge that leads to a point-like locally-conserved charge-current density.