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On the joint
evolution problem for a scalar field and its singularity
Aditya Agashe, Ethan Lee and Shadi Tahvildar-Zadeh |
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Vol. 17 (2024), No. 1, 163–182
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Abstract |
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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. |
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Keywords
radiation-reaction problem, propagation of singularities,
scalar fields, point-charge sources
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Mathematical Subject Classification
Primary: 35A21, 70S10, 78A35
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Milestones
Received: 31 October 2022
Revised: 19 January 2023
Accepted: 21 January 2023
Published: 15 March 2024
Communicated by Martin Bohner
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Authors |
| Aditya Agashe | |
| Brown University Providence, RI United States |
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| Ethan Lee | |
| Rutgers University Piscataway, NJ United States |
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| Shadi Tahvildar-Zadeh | |
| Rutgers University Piscataway, NJ United States |
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| © 2024 MSP (Mathematical Sciences Publishers). |
