Membrane viewpoint on black holes: Dynamical electromagnetic fields near the horizon

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Abstract
This paper is part of a series of papers with the aim of developing a complete self-consistent formalism for the treatment of electromagnetic and gravitational fields in the neighborhood of a black-hole horizon. In this membrane formalism, the horizon is treated as a closed two-dimensional membrane lying in a curved three-dimensional space, and endowed with familiar physical properties such as entropy and temperature, surface pressure and viscosity, and electrical conductivity, charge, and current. This paper develops the concept of the ‘‘stretched horizon,’’ which will be vital for both the electromagnetic and gravitational aspects of the formalism, and it presents several model problems illustrating the interaction of dynamical electromagnetic fields with stationary black-hole horizons: The field of a test charge in various states of motion outside the Schwarzschild horizon is analyzed in the near-horizon limit, where the spatial curvature may be ignored and the metric may be approximated by that of Rindler. This analysis elucidates the influence of the horizon on the shapes and motions of electric and magnetic field lines when external agents move the field lines in arbitrary manners. It also illustrates how the field lines interact with the horizon’s charge and current to produce an exchange of energy and momentum between the external agent and the horizon. A numerical calculation of the dynamical relaxation of a magnetic field threading a Schwarzschild black hole is also presented, illustrating the ‘‘cleaning’’ of a complicated field structure by a black-hole horizon, and elucidating the constraints on the location of the stretched horizon.

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