Charged massive particle at rest in the field of a Reissner-Nordstr\"om black hole

TL;DR

This paper investigates the gravitational and electromagnetic interactions between a charged massive particle and a Reissner-Nordström black hole using perturbation theory, revealing conditions for equilibrium and explicit solutions for the combined system.

Contribution

It provides exact first-order solutions for the perturbed Einstein-Maxwell system and identifies equilibrium conditions based on charge-to-mass ratios, extending understanding of charged black hole-particle systems.

Findings

Exact solutions for perturbed fields are derived in closed form.

Equilibrium conditions depend on charge-to-mass ratios of black hole and particle.

Special case of extremal charges reproduces Majumdar-Papapetrou solution.

Abstract

The interaction of a Reissner-Nordstr\"om black hole and a charged massive particle is studied in the framework of perturbation theory. The particle backreaction is taken into account, studying the effect of general static perturbations of the hole following the approach of Zerilli. The solutions of the combined Einstein-Maxwell equations for both perturbed gravitational and electromagnetic fields at first order of the perturbation are exactly reconstructed by summing all multipoles, and are given explicit closed form expressions. The existence of a singularity-free solution of the Einstein-Maxwell system requires that the charge to mass ratios of the black hole and of the particle satisfy an equilibrium condition which is in general dependent on the separation between the two bodies. If the black hole is undercritically charged (i.e. its charge to mass ratio is less than one), the particle must be overcritically charged, in the sense that the particle must have a charge to mass ratio greater than one. If the charge to mass ratios of the black hole and of the particle are both equal to one (so that they are both critically charged, or "extreme"), the equilibrium can exist for any separation distance, and the solution we find coincides with the linearization in the present context of the well known Majumdar-Papapetrou solution for two extreme Reissner-Nordstr\"om black holes. In addition to these singularity-free solutions, we also analyze the corresponding solution for the problem of a massive particle at rest near a Schwarzschild black hole, exhibiting a strut singularity on the axis between the two bodies. The relations between our perturbative solutions and the corresponding exact two-body solutions belonging to the Weyl class are also discussed.