Unphysical and physical(?) solutions of the Lorentz-Dirac equation

Abstract

A simple proof of a weak version of Eliezer's theorem on unphysical solutions of the Lorentz-Dirac equation is given. This version concerns a free particle scattered by a spatially localized electric field in one space dimension. (The solutions are also solutions in three space dimensions.) It establishes that for certain physically reasonable localized fields, all solutions which are free (i.e., unaccelerated) before they enter the field have unbounded proper acceleration and velocity asymptotic to that of light in the future. For any given initial velocity, the fields yielding this unphysical behavior can be arbitrarily weak. The result is then extended to a class of static fields which need not be spatially localized, including Coulomb fields. For this case the same conclusion is obtained omitting the assumption that the particle be free in the past. The rest of the paper discusses solutions to the localized field problem which are assumed free in the future rather than the past. Some strange features of these solutions are noted. The possibility of experimentally detecting deviations from the Coulomb law for a particle obeying the Lorentz-Dirac equation is discussed.