Black holes with current loops revisited

TL;DR

This paper revises the electromagnetic field solutions around Kerr black holes with current loops, providing corrected formulas and exploring configurations like multiple rings and discs to better understand particle orbits near black holes.

Contribution

It offers a corrected solution to Maxwell's equations for a charged current loop around a Kerr black hole, including dipole approximations and solutions in the extreme Kerr limit.

Findings

Corrected electromagnetic field solutions around Kerr black holes with current loops.

Finite potential outside the black hole when electromotive force vanishes.

Constructed solutions with multiple rings or current discs.

Abstract

The electromagnetic field around a Kerr black hole inside a current loop is sometimes used as the basis of a toy model for discussing the properties of particle orbits near astrophysical black holes. The motivation for the present paper is to correct the published solution to Maxwell's equations with a charged current loop. Dipole approximations and closed-form expressions in the extreme Kerr limit are also presented. Using the corrected solution, it turns out that imposing a vanishing electromotive force produces a loop with a potential which is finite everywhere outside the black hole. Ring solutions can be combined into solutions with multiple rings or current discs.