Radiation reaction, over-reaction, and under-reaction

TL;DR

This paper clarifies the classical radiation reaction problem by deriving equations of motion, analyzing instabilities in point particles, and showing finite-size particles avoid these issues, connecting to broader theoretical physics concepts.

Contribution

Provides a rigorous, textbook-level treatment of radiation reaction, demonstrating how finite particle size resolves classical instabilities and clarifies paradoxes.

Findings

Pointlike particles lead to instability and self-acceleration.

Finite-size particles avoid instabilities and allow systematic expansion.

The reduced-order Abraham-Lorentz equation is stable and physically consistent.

Abstract

The subject of radiation reaction in classical electromagnetism remains controversial over 120 years after the pioneering work of Lorentz. We give a simple but rigorous treatment of the subject at the textbook level that explains the apparent paradoxes that are much discussed in the literature on the subject. We first derive the equation of motion of a charged particle from conservation of energy and momentum, which includes the self-force term. We then show that this theory is unstable if charged particles are pointlike: the energy is unbounded from below, and charged particles self-accelerate (`over-react') due to their negative `bare' mass. This theory clearly does not describe our world, but we show that these instabilities are absent if the particle has a finite size larger than its classical radius. For such finite-size charged particles, the effects of radiation reaction can be computed in a systematic expansion in the size of the particle. The leading term in this expansion is the reduced-order Abraham-Lorentz equation of motion, which has no stability problems. We also discuss the apparent paradox that a particle with constant acceleration radiates, but does not suffer radiation reaction (`under-reaction'). Along the way, we introduce the ideas of renormalization and effective theories, which are important in many areas of modern theoretical physics. We hope that this will be a useful addition to the literature that will remove some of the air of mystery and paradox surrounding the subject.