Microcanonical functional integral for the gravitational field
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 1/4 the area of the blackhole event horizon. The canonical and grand canonical partition functions are obtained by integral transforms of ν that lead to ``imaginarytime'' 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 realtime functional integral and then used to deduce Feynman's imaginarytime functional integral for the canonical partition function.
 Publication:

Physical Review D
 Pub Date:
 February 1993
 DOI:
 10.1103/PhysRevD.47.1420
 arXiv:
 arXiv:grqc/9209014
 Bibcode:
 1993PhRvD..47.1420B
 Keywords:

 04.20.Cv;
 04.60.+n;
 05.30.Ch;
 Fundamental problems and general formalism;
 Quantum ensemble theory;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory
 EPrint:
 29 pages, plain TeX