On the self-force in Bopp-Podolsky electrodynamics

TL;DR

This paper explores Bopp-Podolsky electrodynamics as a potential solution to the infinite self-force problem in classical electrodynamics, deriving a finite self-force and a new equation of motion for charged particles.

Contribution

It introduces a method to define a finite self-force in Bopp-Podolsky theory, leading to an integro-differential equation for particle motion with continuous 4-velocities.

Findings

The Bopp-Podolsky field can be bounded and well-defined on the particle's worldline.

A procedure is developed to assign finite values to the self-field and self-force.

Solutions have continuous 4-velocities, avoiding unphysical runaway solutions.

Abstract

In the classical vacuum Maxwell-Lorentz theory the self-force of a charged point particle is infinite. This makes classical mass renormalization necessary and, in the special relativistic domain, leads to the Abraham-Lorentz-Dirac equation of motion possessing unphysical run-away and pre-acceleration solutions. In this paper we investigate whether the higher-order modification of classical vacuum electrodynamics suggested by Bopp, Lande, Thomas and Podolsky in the 1940s, can provide a solution to this problem. Since the theory is linear, Green-function techniques enable one to write the field of a charged point particle on Minkowski spacetime as an integral over the particle's history. By introducing the notion of timelike worldlines that are "bounded away from the backward light-cone" we are able to prescribe criteria for the convergence of such integrals. We also exhibit a timelike worldline yielding singular fields on a lightlike hyperplane in spacetime. In this case the field is mildly singular at the event where the particle crosses the hyperplane. Even in the case when the Bopp-Podolsky field is bounded, it exhibits a directional discontinuity as one approaches the point particle. We describe a procedure for assigning a value to the field on the particle worldline which enables one to define a finite Lorentz self-force. This is explicitly derived leading to an integro-differential equation for the motion of the particle in an external electromagnetic field. We conclude that any worldline solutions to this equation belonging to the categories discussed in the paper have continuous 4-velocities.