Stationary electromagnetic fields around black holes. III. General solutions and the fields of current loops near the Reissner-Nordström black hole

Abstract

The stationary multipole solutions of the Einstein-Maxwell equations representing coupled electromagnetic and gravitational perturbations of the extreme Reissner-Nordstr\"om black hole are simple rational functions of the radial coordinate. These solutions are used to construct the axially symmetric field of a current loop in the equatorial plane and of a small current loop on the polar axis. The magnetic lines of force are calculated numerically and exhibited graphically. The field representing a general Reissner-Nordstr\"om black hole in an asymptotically uniform magnetic field is also given, and its relation to the exact solutions of Ernst describing Kerr-Newman black holes in Melvin's magnetic universe is discussed.