Self-force and the Schwarzschild star

TL;DR

This paper calculates the electromagnetic and scalar self-force on a charge in a Schwarzschild star's interior, introducing new regularization methods and analyzing divergence behaviors near the star's surface.

Contribution

It applies difference regularization inside a non-vacuum star and provides closed-form solutions and detailed analysis of the self-force near the star's center and boundary.

Findings

Agreement between direct and difference regularization methods.

Series expansions for self-force near the center and far field.

Identification of logarithmic divergence near the star's surface.

Abstract

We consider the self-force acting on a pointlike (electromagnetic or conformal-scalar) charge held fixed on a spacetime with a spherically-symmetric mass distribution of constant density (the Schwarzschild star). The Schwarzschild interior is shown to be conformal to a three-sphere geometry; we use this conformal symmetry to obtain closed-form expressions for mode solutions. We calculate the self-force with two complementary regularization methods, direct and difference regularization, showing agreement. For the first time, we show that difference regularization can be applied in the non-vacuum interior region, due to the vanishing of certain regularized mode sums. The new results for the self-force come in three forms: series expansions for the self-force near the centre of the star and in the far field; a new approximation that describes the divergence in the self-force near the star's boundary; and numerical data presented in a selection of plots. We conclude with a discussion of the logarithmic divergence in the self-force in the approach to the star's surface, and the effect of boundaries.