On the initial-value problem of the Maxwell-Lorentz equations

TL;DR

This paper proves that the initial-value problem for the Maxwell-Lorentz equations in general-relativistic spacetime is well-posed by showing the evolution equations are strongly hyperbolic, with specific analysis on Minkowski spacetime.

Contribution

It demonstrates the strong hyperbolicity of the Maxwell-Lorentz system and establishes well-posedness of the initial-value problem in a general-relativistic setting.

Findings

Evolution equations are strongly hyperbolic.

Initial-value problem is well-posed.

Analysis includes spherically symmetric solutions on Minkowski spacetime.

Abstract

We consider the Maxwell-Lorentz equations, i.e., the equation of motion of a charged dust coupled to Maxwell's equations, on an arbitrary general-relativistic spacetime. We decompose this system of equations into evolution equations and constraints, and we demonstrate that the evolution equations are strongly hyperbolic. This result guarantees that the initial-value problem of the Maxwell-Lorentz equations is well-posed. We illustrate this general result with a discussion of spherically symmetric solutions on Minkowski spacetime.