Introduction and Summary of Results

Abstract

Abstract The primary purpose of this work is to determine an equation of motion for the classical Lorentz model of the electron that is consistent with causal solutions to the Maxwell-Lorentz equations, the relativistic generalization of Newton's second law of motion, and Einstein's mass-energy relation. (The latter two laws of physics were not discovered until after the original works of Lorentz, Abraham, and Poincaré. The hope of Lorentz and Abraham for deriving the equation of motion of an electron from the self force determined by the Maxwell-Lorentz equations alone was not fully realized.) The work begins by reviewing the contributions of Lorentz, Abraham, Poincaré, and Schott to this century-old problem of finding the equation of motion of an extended electron. Their original derivations, which were based on the Maxwell-Lorentz equations and assumed a zero bare mass, are modified and generalized to obtain a nonzero bare mass and consistent force and power equations of motion. By looking at the Lorentz model of the electron as a charged insulator, general expressions are derived for the binding forces that Poincaré postulated to hold the charge distribution together. A careful examination of the classic Lorentz-Abraham derivation reveals that the self electromagnetic force must be modified during a short time interval after the external force is first applied and after all other nonanalytic points in time of the external force. The resulting modification to the equation of motion, although slight, eliminates the noncausal pre-acceleration (and pre-deceleration) that has plagued the solution to the Lorentz-Abraham equation of motion. As part of the analysis, general momentum and energy relations are derived and interpreted physically for the solutions to the equation of motion, including “hyperbolic” and “runaway” solutions. Also, a stress-momentum-energy tensor that includes the binding, bare-mass, and electromagnetic momentum-energy densities is derived for the charged insulator model of the electron, and an assessment is made of the redefinitions of electromagnetic momentum-energy that have been proposed in the past to obtain a consistent equation of motion.