Five Dimensional Rotating Black Hole in a Uniform Magnetic Field. The Gyromagnetic Ratio

TL;DR

This paper extends the understanding of electromagnetic fields around rotating black holes to five dimensions, showing that such black holes have a gyromagnetic ratio of 3, similar to four-dimensional cases.

Contribution

It develops a method to construct electromagnetic potentials for five-dimensional rotating black holes in magnetic fields and determines their gyromagnetic ratio.

Findings

Five-dimensional Myers-Perry black holes produce an inductive potential difference.

Weakly charged five-dimensional black holes have two magnetic dipole moments.

The gyromagnetic ratio for these black holes is g=3.

Abstract

In four dimensional general relativity, the fact that a Killing vector in a vacuum spacetime serves as a vector potential for a test Maxwell field provides one with an elegant way of describing the behaviour of electromagnetic fields near a rotating Kerr black hole immersed in a uniform magnetic field. We use a similar approach to examine the case of a five dimensional rotating black hole placed in a uniform magnetic field of configuration with bi-azimuthal symmetry, that is aligned with the angular momenta of the Myers-Perry spacetime. Assuming that the black hole may also possess a small electric charge we construct the 5-vector potential of the electromagnetic field in the Myers-Perry metric using its three commuting Killing vector fields. We show that, like its four dimensional counterparts, the five dimensional Myers-Perry black hole rotating in a uniform magnetic field produces an inductive potential difference between the event horizon and an infinitely distant surface. This potential difference is determined by a superposition of two independent Coulomb fields consistent with the two angular momenta of the black hole and two nonvanishing components of the magnetic field. We also show that a weakly charged rotating black hole in five dimensions possesses two independent magnetic dipole moments specified in terms of its electric charge, mass, and angular momentum parameters. We prove that a five dimensional weakly charged Myers-Perry black hole must have the value of the gyromagnetic ratio g=3.