Variant forms of Eliezer's Theorem

Abstract

Over 60 years ago, Eliezer proved the surprising result that an electron moving radially according to the Lorentz-Dirac equation in the Coulomb field of a proton will not be attracted to a collision with the proton as expected. Instead, it is repelled from the proton with proper acceleration increasing asymptotically with proper time. Proponents of the Lorentz-Dirac equation sometimes try to explain this away by speculation that the electron must approach so closely to the proton that the field strength would be beyond the domain of validity of the classical Lorentz-Dirac equation and therefore require a quantum-mechanical analysis. This note proves some variants of Eliezer's result which apply to *bounded*, compactly supported, spherically symmetric fields, and thus call into question such speculation. However, these variants do require the additional hypothesis (not required by Eliezer) that the electron is unaccelerated before entering the field--i.e., that "preacceleration" is impossible. Though these results may not appear in the literature, the methods of proof are well known. The motivation for writing down careful proofs was continued skepticism by proponents of the Lorentz-Dirac equation.