# The Einstein-Infeld-Hoffmann legacy in mathematical relativity. Part I: The classical motion of charged point particles

## TL;DR

This paper examines the classical motion of charged point particles in general relativity, highlighting the limitations of Einstein-Infeld-Hoffmann's original approach and proposing conditions for well-posed initial value problems using alternative electromagnetic laws.

## Contribution

It identifies necessary conditions for a well-defined initial value problem for charged particles in general relativity, emphasizing the role of electromagnetic vacuum laws and proposing a modified framework.

## Key findings

- Maxwell(--Lorentz) law disqualifies finite energy-momentum for point charges
- Using BLTP law with non-zero rest mass yields well-posed problems in special relativity
- Gravitational coupling must also be modified for a well-defined joint initial value problem

## Abstract

Einstein, Infeld, and Hoffmann (EIH) claimed that the field equations of general relativity theory alone imply the equations of motion of neutral matter particles, viewed as point singularities in space-like slices of spacetime; they also claimed that they had generalized their results to charged point singularities. While their analysis falls apart upon closer scrutiny, the key idea merits our attention. This rapport identifies necessary conditions for a well-defined general-relativistic joint initial value problem of $N$ classical point charges and their electromagnetic and gravitational fields. Among them, in particular, is the requirement that the electromagnetic vacuum law guarantees a finite field energy-momentum of a point charge. This disqualifies the Maxwell(--Lorentz) law used by EIH. On the positive side, if the electromagnetic vacuum law of Bopp, Land\'e--Thomas, and Podolsky (BLTP) is used, and the singularities equipped with a non-zero bare rest mass, then a joint initial value problem can be formulated in the spirit of the EIH proposal, and shown to be locally well-posed --- \emph{in the special-relativistic zero-$G$ limit}. With gravitational coupling (i.e. $G>0$), though, changing Maxwell's into the BLTP law and assigning a bare rest mass to the singularities is by itself not sufficient to obtain even a merely well-defined joint initial value problem: the gravitational coupling also needs to be changed, conceivably in the manner of Jordan and Brans--Dicke.