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.
Keywords
radiation-reaction problem, propagation of singularities,
scalar fields, point-charge sources
