On the Joint Evolution Problem for a Scalar Field and its Singularity

TL;DR

This paper investigates the joint evolution of a scalar field and its singularity in a simplified 1D model, analyzing stability and instability of static solutions under incoming radiation, with implications for understanding point charge dynamics.

Contribution

It introduces a simplified 1D model to analyze the joint evolution of a scalar field and its singularity, providing new insights into stability and instability phenomena.

Findings

Stable static solution for positive bare mass with incoming radiation causing return to rest.

Nonlinear instability for negative bare mass with arbitrarily small incoming radiation leading to speed of light.

Discussion of potential modifications to improve model realism.

Abstract

In the classical electrodynamics of point charges in vacuum, the electromagnetic field, and therefore the Lorentz force, is ill-defined at the locations of the charges. Kiessling resolved this problem by using the momentum balance between the field and the particles, extracting an equation for the force that is well-defined where the charges are located, so long as the field momentum density is locally integrable in a neighborhood of the charges. In this paper, we examine the effects of such a force by analyzing a simplified model in one space dimension. We study the joint evolution of a massless scalar field together with its singularity, which we identify with the trajectory of a particle. The static solution arises in the presence of no incoming radiation, in which case the particle remains at rest forever. We will prove the stability of the static solution for particles with positive bare mass by showing that a pulse of incoming radiation that is compactly supported away from the point charge will result in the particle eventually coming back to rest. We will also prove the nonlinear instability of the static solution for particles with negative bare mass by showing that an incoming radiation with arbitrarily small amplitude will cause the particle to reach the speed of light in finite time. We conclude by discussing modifications to this simple model that could make it more realistic.