Hamiltonian thermodynamics of the Reissner-Nordström-anti-de Sitter black hole

Details Hamiltonian thermodynamics of the Reissner-Nordström-anti-de Sitter black hole Hamiltonian thermodynamics of the Reissner-Nordström-anti-de Sitter black hole

1996-02-02

arxiv.org/abs/gr-qc/9602003v2

Phys.Rev.D54:2647-2663,1996

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

34 pages, REVTeX v3.0. (Minor corrections and presentational revisions; added references.)

Scientific paper

10.1103/PhysRevD.54.2647

We consider the Hamiltonian dynamics and thermodynamics of spherically symmetric Einstein-Maxwell spacetimes with a negative cosmological constant. We impose boundary conditions that enforce every classical solution to be an exterior region of a Reissner-Nordstr\"om-anti-de Sitter black hole with a nondegenerate Killing horizon, with the spacelike hypersurfaces extending from the horizon bifurcation two-sphere to the asymptotically anti-de Sitter infinity. The constraints are simplified by a canonical transformation, which generalizes that given by Kucha\v{r} in the spherically symmetric vacuum Einstein theory, and the theory is reduced to its true dynamical degrees of freedom. After quantization, the grand partition function of a thermodynamical grand canonical ensemble is obtained by analytically continuing the Lorentzian time evolution operator to imaginary time and taking the trace. A~similar analysis under slightly modified boundary conditions leads to the partition function of a thermodynamical canonical ensemble. The thermodynamics in each ensemble is analyzed, and the conditions that the (grand) partition function be dominated by a classical Euclidean black hole solution are found. When these conditions are satisfied, we recover in particular the Bekenstein-Hawking entropy. The limit of a vanishing cosmological constant is briefly discussed. (This paper is dedicated to Karel Kucha\v{r} on the occasion of his sixtieth birthday.)

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