Charged massive particle at rest in the field of a Reissner-Nordstr\"om black hole. II. Analysis of the field lines and the electric Meissner effect

TL;DR

This paper analyzes the electric field lines of a charged particle near a Reissner-Nordström black hole, revealing an electric Meissner effect at the perturbative level and connecting test field and exact solutions.

Contribution

It completes the analysis of the electric field of a charged particle near a Reissner-Nordström black hole by numerically constructing field lines and demonstrating the electric Meissner effect at the perturbative level.

Findings

Electric field lines are expelled from the black hole horizon as it becomes extremal.

The effective field of the particle is explicitly computed and visualized.

The perturbative solution aligns with the exact Belinski-Alekseev solution in the appropriate gauge.

Abstract

The properties of the electric field of a two-body system consisting of a Reissner-Nordstr\"om black hole and a charged massive particle at rest have recently been analyzed in the framework of first order perturbation theory following the standard approach of Regge, Wheeler and Zerilli. In the present paper we complete this analysis by numerically constructing and discussing the lines of force of the "effective" electric field of the sole particle with the subtraction of the dominant contribution of the black hole. We also give the total field due to the black hole and the particle. As the black hole becomes extreme an effect analogous to the Meissner effect arises for the electric field, with the "effective field" lines of the point charge being expelled by the outer horizon of the black hole. This effect existing at the level of test field approximation, i.e. by neglecting the backreaction on the background metric and electromagnetic field due to the particle's mass and charge, is here found also at the complete perturbative level. We point out analogies with similar considerations for magnetic fields by Bi{\v c}\'ak and Dvo{\v r}\'ak. We also explicitly show that the linearization of the recently obtained Belinski-Alekseev exact solution coincides with our solution in the Regge-Wheeler gauge. Our solution thus represents a "bridge" between the test field solution, which neglects all the feedback terms, and the exact two-body solution, which takes into account all the non-linearity of the interaction.