Point Charges in Classical Electrodynamics

TL;DR

This work surveys the classical treatment of point charges in electrodynamics, focusing on radiation reaction and the Lorentz–Dirac equation. It contrasts Dirac’s approach, which invokes both retarded and advanced fields, with Teitelboim’s retarded-field splitting into near and far components to define bound and emitted momentum, and discusses mass renormalization as a historical obstacle. By deriving the Lorentz–Dirac equation from momentum balance and examining self-interaction through retarded averaging and analytic continuation, the paper illuminates the trade-offs between causality, infinite self-energy, and the need for a consistent field definition on the world-line. The study highlights strategies to avoid mass renormalization and emphasizes the ongoing conceptual challenges in reconciling point charges with a finite, physically meaningful mass, radiation reaction, and self-field dynamics.

Abstract

LaTeX transcription (2025) of a 1989 honours thesis (University of Adelaide) on point charges in classical electrodynamics and the Lorentz-Dirac radiation-reaction equation. The thesis reviews the retarded field of an arbitrarily moving charge, energy-momentum conservation, and derives the Lorentz-Dirac equation via momentum balance. It discusses self-interaction and mass renormalization, and presents world-line self-field definitions including retarded averaging and an analytic continuation approach. Appendices include Mathematica listings used to obtain near-world-line expansions of the field and related quantities.

Figures (8)

• Figure 1: Light-cone and World-line of a point charge: $\tau_a$ and $\tau_r$ are the advanced and retarded proper-times corresponding to $x^\mu$; $u^\mu$ is the charge's four-velocity; $\rho$ is the spatial distance between the retarded point of the charge and the point $x^\mu$, in the instantaneous retarded rest-frame (i.r.r.f).
• Figure 2: Spacelike surface intersecting world-line. $(x^\mu - z^\mu(\tau_2))(x_\mu - z_\mu(\tau_2)) = 0$.
• Figure 3: Intersection of light-cone from $z(\tau_2)$ with spacelike surface $S$
• Figure 4: Intersection of future light-cone from $z(\tau_1)$ and $z(\tau_1 + d\tau)$ with two spacelike surfaces cutting the world-line at $z(\tau)$.
• Figure 5: Evaluation of the rate of change of bound four-momentum, $P^\mu_s$.