Reduction of the two-body dynamics to a one-body description in classical electrodynamics

TL;DR

This paper explores reducing two-body classical electrodynamics to a one-body problem, revealing the necessity of external parameters and considering non-flat spacetime to achieve a consistent mapping.

Contribution

It demonstrates that a straightforward one-body reduction in classical electrodynamics requires external parameters and suggests relaxing flat spacetime assumptions for a better mapping.

Findings

External parameters are needed in the effective electromagnetic field.

Relinquishing flat spacetime assumption enables better two-body to one-body mapping.

Recoil and relativistic effects are incorporated up to second post-Coulombian order.

Abstract

We discuss the mapping of the conservative part of two-body electrodynamics onto that of a test charged particle moving in some external electromagnetic field, taking into account recoil effects and relativistic corrections up to second post-Coulombian order. Unlike the results recently obtained in general relativity, we find that in classical electrodynamics it is not possible to implement the matching without introducing external parameters in the effective electromagnetic field. Relaxing the assumption that the effective test particle moves in a flat spacetime provides a feasible way out.