Absence of a consistent classical equation of motion for a mass-renormalized point charge

TL;DR

This paper investigates the classical equations of motion for charged particles, revealing issues with causality, conservation, and the high acceleration catastrophe when mass renormalization is applied as the charge radius approaches zero.

Contribution

It explicitly accounts for analyticity, relativistic rigidity, and transition forces, showing the limitations of classical equations of motion for point charges with mass renormalization.

Findings

Transition forces restore causality and conservation for extended charges.

Mass renormalization leads to violations of conservation at high accelerations.

No consistent classical equation of motion exists for a point charge with finite mass and high external forces.

Abstract

The restrictions of analyticity, relativistic (Born) rigidity, and negligible O(a) terms involved in the evaluation of the self electromagnetic force on an extended charged sphere of radius "a" are explicitly revealed and taken into account in order to obtain a classical equation of motion of the extended charge that is both causal and conserves momentum-energy. Because the power-series expansion used in the evaluation of the self force becomes invalid during transition time intervals immediately following the application and termination of an otherwise analytic externally applied force, transition forces must be included during these transition time intervals to remove the noncausal pre-acceleration and pre-deceleration from the solutions to the equation of motion without the transition forces. For the extended charged sphere, the transition forces can be chosen to maintain conservation of momentum-energy in the causal solutions to the equation of motion within the restrictions of relativistic rigidity and negligible O(a) terms under which the equation of motion is derived. However, it is shown that renormalization of the electrostatic mass to a finite value as the radius of the charge approaches zero introduces a violation of momentum-energy conservation into the causal solutions to the equation of motion of the point charge if the magnitude of the external force becomes too large. That is, the causal classical equation of motion of a point charge with renormalized mass experiences a high acceleration catastrophe.