We resolve the longstanding paradox that in classical electrodynamics the energy and linear momentum of the Abraham-Lorentz (classical) electron do not transform as 4-vector components under Lorentz transformations. In our treatment these quantities transform properly and remain finite in the point particle limit.

Introduction

In Classical Electrodynamics the energy and linear momentum of the extended classical electron do not transform as components of a 4-vector (Leighton, 1959); also infinities arise in the point charge limit.

Abraham (1905) and Lorentz (1909) proposed the classical model of the electron. Lorentz suggested a model in which the electron consisted of a thin, uniformly charged shell. Poincaré (1909) added a stress in order to stabilize the electron. However, he used an expression for the cohesive stress-energy tensor which led to difficulties (Fermi, 1922; Pais, 1948).

Dirac (1938a, 1938b), Matthison (1931, 1940, 1942) and others have proposed point particle models for the classical electron. The lack of covariance associated with the Abraham-Lorentz model and other extended models of the electron does not arise in such models or in classical models based on non local generalisations of Maxwell electromagnetism; see for example, Bopp (1942), McManus (1948) and Feynmann (1948). Erber (1961) reviews much of the literature on models of the classical electron.