The self-interaction force on an arbitrarily moving point-charge and its energy-momentum radiation rate: A mathematically rigorous derivation of the Lorentz-Dirac equation of motion

TL;DR

This paper rigorously derives the Lorentz-Dirac equation for a moving point-charge by defining fields as nonlinear generalized functions in Colombeau algebra, resolving longstanding issues in classical electrodynamics without altering Maxwell's theory.

Contribution

It provides a mathematically rigorous derivation of the Lorentz-Dirac equation using Colombeau algebra, clarifying the origin of the Schott term and resolving classical inconsistencies.

Findings

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

Correct self-energy involves delta-squared functions, not Coulomb energy.

Resolved longstanding problems in classical radiation theory.

Abstract

The classical theory of radiating point-charges is revisited: the retarded potentials, fields, and currents are defined as nonlinear generalized functions and all calculations are made in a Colombeau algebra. 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.