Joint evolution of a Lorentz-covariant massless scalar field and its point-charge source in one space dimension

TL;DR

This paper proves the asymptotic stability of a static solution for a massless scalar field coupled with a point charge in 1+1 dimensions, demonstrating the back-reaction effect and deriving the force from energy-momentum conservation.

Contribution

It rigorously establishes the global well-posedness and asymptotic stability of the coupled particle-field system, providing a closed-form expression for the self-action force.

Findings

The static solution is asymptotically stable under compactly-supported perturbations.

The self-action force is restorative, proportional to negative velocity.

The joint evolution problem is globally well-posed.

Abstract

In this paper we prove that the static solution of the Cauchy problem for a massless real scalar field that is sourced by a point charge in $1+1$ dimensions is asymptotically stable under perturbation by compactly-supported radiation. This behavior is due to the process of back-reaction. Taking the approach of Kiessling, we rigorously derive the expression for the force on the particle from the principle of total energy-momentum conservation. We provide a simple, closed form for the particle's self-action, and show that it is restorative in this model , i.e. proportional to negative velocity, and causes the charge to return to rest after the radiation passes through. We establish these results by studying the joint evolution problem for the particle-scalar field system, and proving its global well-posedness and the claimed asymptotic behavior.