The self-force on a static scalar test-charge outside a Schwarzschild black hole

TL;DR

This paper calculates the self-force on a static scalar charge outside a Schwarzschild black hole, finding it to be zero after mass renormalization, contrasting with the electrostatic case.

Contribution

It provides a closed-form expression for the scalar field and demonstrates that the self-force on a static scalar charge is zero, advancing techniques for force calculations in black hole spacetimes.

Findings

Self-force on scalar charge is zero after renormalization

Closed-form scalar field expression derived from Hadamard's solution

Force calculation confirmed via Carter's mass-variation theorem

Abstract

The finite part of the self-force on a static scalar test-charge outside a Schwarzschild black hole is zero. By direct construction of Hadamard's elementary solution, we obtain a closed-form expression for the minimally coupled scalar field produced by a test-charge held fixed in Schwarzschild spacetime. Using the closed-form expression, we compute the necessary external force required to hold the charge stationary. Although the energy associated with the scalar field contributes to the renormalized mass of the particle (and thereby its weight), we find there is no additional self-force acting on the charge. This result is unlike the analogous electrostatic result, where, after a similar mass renormalization, there remains a finite repulsive self-force acting on a static electric test-charge outside a Schwarzschild black hole. We confirm our force calculation using Carter's mass-variation theorem for black holes. The primary motivation for this calculation is to develop techniques and formalism for computing all forces - dissipative and non-dissipative - acting on charges and masses moving in a black-hole spacetime. In the Appendix we recap the derivation of the closed-form electrostatic potential. We also show how the closed-form expressions for the fields are related to the infinite series solutions.