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This paper derives the ``membrane formalism'' for black holes. The membrane formalism rewrites the standard mathematical theory of black holes in a language and notation which (we hope) will facilitate research in black-hole astrophysics: The horizon of a black hole is replaced by a surrogate ``stretched horizon,'' which is viewed as a 2-dimensional membrane that resides in 3-dimensional space and evolves in response to driving forces from the external universe. This membrane, following ideas of Damour and Znajek, is regarded as made from a 2-dimensional viscous fluid that is electrically charged and electrically conducting and has finite entropy and temperature, but cannot conduct heat. The interaction of the stretched horizon with the external universe is described in terms of familiar laws for the horizon's fluid, e.g., the Navier-Stokes equation, Ohm's law, a tidal-force equation, and the first and second laws of thermodynamics. Because these laws have familiar forms, they are likely to help astrophysicists understand intuitively and compute quantitatively the behaviors of black holes in complex external environments. Previous papers have developed and elucidated electromagnetic aspects of the membrane formalism for time-independent rotating holes. This paper derives the full formalism for dynamical, evolving holes, with one exception: In its present form the formalism is not equipped to handle horizon caustics, where new generators attach themselves to the horizon.
Price et al. (Sat,) studied this question.