Microcanonical functional integral for the gravitational field

J. David Brown and James W. York, Jr.
Phys. Rev. D 47, 1420 – Published 15 February 1993
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Abstract

The gravitational field in a spatially finite region is described as a microcanonical system. The density of states ν is expressed formally as a functional integral over Lorentzian metrics and is a functional of the geometrical boundary data that are fixed in the corresponding action. These boundary data are the thermodynamical extensive variables, including the energy and angular momentum of the system. When the boundary data are chosen such that the system is described semiclassically by any real stationary axisymmetric black hole, then in this same approximation lnν is shown to equal ¼ the area of the black-hole event horizon. The canonical and grand canonical partition functions are obtained by integral transforms of ν that lead to "imaginary-time" functional integrals. A general form of the first law of thermodynamics for stationary black holes is derived. For the simpler case of nonrelativistic mechanics, the density of states is expressed as a real-time functional integral and then used to deduce Feynman's imaginary-time functional integral for the canonical partition function.

  • Received 22 September 1992

DOI:https://doi.org/10.1103/PhysRevD.47.1420

©1993 American Physical Society

Authors & Affiliations

J. David Brown* and James W. York, Jr.

  • Institute of Field Physics and Theoretical Astrophysics and Relativity Group, Department of Physics and Astronomy, The University of North Carolina, Chapel Hill, North Carolina 27599-3255

  • *Present address: Departments of Physics and Mathematics, North Carolina State University, Raleigh, NC 27695-8202.

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Issue

Vol. 47, Iss. 4 — 15 February 1993

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