Derivation of the self-interaction force on an arbitrarily moving point-charge and of its related energy-momentum radiation rate: The Lorentz-Dirac equation of motion in a Colombeau algebra

TL;DR

This paper rigorously derives the self-interaction force and energy-momentum radiation rate for an arbitrarily moving point-charge using Colombeau algebra, clarifying longstanding issues in classical electrodynamics and deriving the Lorentz-Dirac equation.

Contribution

It introduces a rigorous mathematical framework in Colombeau algebra to derive the Lorentz-Dirac equation and resolve issues related to the Schott term in classical electrodynamics.

Findings

Self-energy of a point charge is a delta-squared integral, not Coulomb energy.

Energy-momentum radiation rate equals negative self-interaction force.

Provides a clear derivation of the Lorentz-Dirac equation of motion.

Abstract

The classical theory of radiating point-charges is revisited: the retarded potentials, fields, and currents are defined as nonlinear generalized functions. All calculations are made in a Colombeau algebra, and the spinor representations provided by the biquaternion formulation of classical electrodynamics are used to make all four-dimensional integrations exactly and in closed-form. The total rate of energy-momentum radiated by an arbitrarily moving relativistic point-charge under the effect of its own field is shown to be rigorously equal to minus the self-interaction force due to that field. This solves, without changing anything in Maxwell's theory, numerous long-standing problems going back to more than a century. As an immediate application an unambiguous derivation of the Lorentz-Dirac equation of motion is given, and the origin of the problem with the Schott term is explained: it was due to the fact that the correct self-energy of a point charge is not the Coulomb self-energy, but an integral over a delta-squared function which yields a finite contribution to the Schott term that is either absent or incorrect in the customary formulations.