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On the joint evolution problem for a scalar field and its singularity

Aditya Agashe, Ethan Lee and Shadi Tahvildar-Zadeh

Vol. 17 (2024), No. 1, 163–182

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.

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.

radiation-reaction problem, propagation of singularities, scalar fields, point-charge sources
Mathematical Subject Classification
Primary: 35A21, 70S10, 78A35
Received: 31 October 2022
Revised: 19 January 2023
Accepted: 21 January 2023
Published: 15 March 2024

Communicated by Martin Bohner
Aditya Agashe
Brown University
Providence, RI
United States
Ethan Lee
Rutgers University
Piscataway, NJ
United States
Shadi Tahvildar-Zadeh
Rutgers University
Piscataway, NJ
United States