Resolution of an ambiguity in the derivation of the Lorentz-Dirac equation

Abstract

An ambiguity in the derivation of the equation of motion of a charged point particle is exhibited. Using momentum conservation in the form $\int T^{\mu \nu }d{S}_{\mu }=0\mathrm{}$, one deduces the Lorentz-Dirac equation by integrating over one surface, and a different equation, with variable rest mass, by interating over another. In the second case, the variability of the rest mass exactly compensates the Larmor radiation, and the Schott term is absent. In both cases, infinite mass renormalizations, of identical form, are required. These renormalizations are both covariant, but since they are the only mathematically insecure parts of the derivations they are obviously responsible for the ambiguity. A distribution theory of the problem is set up to resolve the question. This theory, at its present stage of development, has its own types of arbitrariness, but making very natural choices, a perfectly definite and everywhere finite formulation results, out of which the Lorentz-Dirac equation emerges uniquely. And in this form, as an expression of ${\partial }_{\mu }{T}^{\mu \nu }=0\mathrm{}$, the theory clarifies the energy-momentum exchanges in the problem.