Self-force on a charge outside a five-dimensional black hole

Abstract

We compute the electromagnetic self-force acting on a charged particle held in place at a fixed position $r$ outside a five-dimensional black hole described by the Schwarzschild-Tangherlini metric. Using a spherical-harmonic decomposition of the electrostatic potential and a regularization prescription based on the Hadamard Green's function, we express the self-force as a convergent mode sum. The self-force is first evaluated numerically, and next presented as an analytical expansion in powers of $R/r$, with $R$ denoting the event-horizon radius. The power series is then summed to yield a closed-form expression. Unlike its four-dimensional version, the self-force features a dependence on a regularization parameter $s$ that can be interpreted as the particle's radius. The self-force is repulsive at large distances, and its behavior is related to a model according to which the force results from a gravitational interaction between the black hole and the distribution of electrostatic field energy attached to the particle. The model, however, is shown to become inadequate as $r$ becomes comparable to $R$, where the self-force changes sign and becomes attractive. We also calculate the self-force acting on a particle with a scalar charge, which we find to be everywhere attractive. This is to be contrasted with its four-dimensional counterpart, which vanishes at any $r$.