Membrane paradigm and entropy of black holes in the Euclidean action
  approach

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Abstract

The membrane paradigm approach to black holes fixes in the vicinity of the event horizon a fictitious surface, the stretched horizon, so that the spacetime outside remains unchanged and the spacetime inside is vacuum. Using this powerful method, several black hole properties have been found and settled, such as the horizon's viscosity, electrical conductivity, resistivity, as well as other properties. On the other hand the Euclidean action approach to black hole spacetimes has been very fruitful in understanding black hole entropy. Combining both the Euclidean action and membrane paradigm approaches a direct derivation of the black hole entropy is given. In the derivation it is considered that the only fields present are the gravitational and matter fields, with no electric field.

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Most cited references 6

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Path-integral derivation of black-hole radiance

J. B. Hartle, S. Hawking (1976)

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Quasilocal Energy and Conserved Charges Derived from the Gravitational Action

J. David Brown, James W. York (1992)

The quasilocal energy of gravitational and matter fields in a spatially bounded region is obtained by employing a Hamilton-Jacobi analysis of the action functional. First, a surface stress-energy-momentum tensor is defined by the functional derivative of the action with respect to the three-metric on \({}^3B\), the history of the system's boundary. Energy density, momentum density, and spatial stress are defined by projecting the surface stress tensor normally and tangentially to a family of spacelike two-surfaces that foliate \({}^3B\). The integral of the energy density over such a two-surface \(B\) is the quasilocal energy associated with a spacelike three-surface \(\Sigma\) whose intersection with \({}^3B\) is the boundary \(B\). The resulting expression for quasilocal energy is given in terms of the total mean curvature of the spatial boundary \(B\) as a surface embedded in \(\Sigma\). The quasilocal energy is also the value of the Hamiltonian that generates unit magnitude proper time translations on \({}^3B\) in the direction orthogonal to \(B\). Conserved charges such as angular momentum are defined using the surface stress tensor and Killing vector fields on \({}^3B\). For spacetimes that are asymptotically flat in spacelike directions, the quasilocal energy and angular momentum defined here agree with the results of Arnowitt-Deser-Misner in the limit that the boundary tends to spatial infinity. For spherically symmetric spacetimes, it is shown that the quasilocal energy has the correct Newtonian limit, and includes a negative contribution due to gravitational binding.

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The Stretched Horizon and Black Hole Complementarity

J. Uglum, L. Thorlacius, L. Susskind (1993)

Three postulates asserting the validity of conventional quantum theory, semi-classical general relativity and the statistical basis for thermodynamics are introduced as a foundation for the study of black hole evolution. We explain how these postulates may be implemented in a ``stretched horizon'' or membrane description of the black hole, appropriate to a distant observer. The technical analysis is illustrated in the simplified context of 1+1 dimensional dilaton gravity. Our postulates imply that the dissipative properties of the stretched horizon arise from a course graining of microphysical degrees of freedom that the horizon must possess. A principle of black hole complementarity is advocated. The overall viewpoint is similar to that pioneered by 't~Hooft but the detailed implementation is different.