Exact solution of the EM radiation-reaction problem for classical finite-size and Lorentzian charged particles

TL;DR

This paper provides an exact covariant solution to the classical electromagnetic radiation-reaction problem for finite-size charged particles, avoiding divergences and establishing a well-posed dynamical system.

Contribution

It introduces a variational formulation and a covariant self-field potential for finite-size charges, leading to a new delay-type RR equation with proven existence and uniqueness.

Findings

Derived a divergence-free self-potential for extended charges.

Obtained an exact delay-type RR equation consistent with relativity.

Proved local existence and uniqueness of the initial-value problem.

Abstract

An exact solution is given to the classical electromagnetic (EM) radiation-reaction (RR) problem, originally posed by Lorentz. This refers to the dynamics of classical non-rotating and quasi-rigid finite size particles subject to an external prescribed EM field. A variational formulation of the problem is presented. It is shown that a covariant representation for the EM potential of the self-field generated by the extended charge can be uniquely determined, consistent with the principles of classical electrodynamics and relativity. By construction, the retarded self 4-potential does not possess any divergence, contrary to the case of point charges. As a fundamental consequence, based on Hamilton variational principle, an exact representation is obtained for the relativistic equation describing the dynamics of a finite-size charged particle (RR equation), which is shown to be realized by a second-order delay-type ODE. Such equation is proved to apply also to the treatment of Lorentzian particles, i.e., point-masses with finite-size charge distributions, and to recover the usual LAD equation in a suitable asymptotic approximation. Remarkably, the RR equation admits both standard Lagrangian and conservative forms, expressed respectively in terms of a non-local effective Lagrangian and a stress-energy tensor. Finally, consistent with the Newton principle of determinacy, it is proved that the corresponding initial-value problem admits a local existence and uniqueness theorem, namely it defines a classical dynamical system.