Charged black hole in a grand canonical ensemble

Harry W. Braden, J. David Brown, Bernard F. Whiting, and James W. York, Jr.
Phys. Rev. D 42, 3376 – Published 15 November 1990
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Abstract

A spherical charged black hole in thermal equilibrium is considered from the perspective of a grand canonical ensemble in which the electrostatic potential, temperature, and surface area are specified at a finite boundary. A correspondence is established between the boundary-value data of a well-posed problem in a finite region of Euclidean spacetime and the freely chosen thermodynamic data specifying the ensemble. The Hamiltonian and Gauss's-law constraints are solved and eliminated from the Einstein-Maxwell action, producing a "reduced action" that depends upon two remaining degrees of freedom (two free parameters), as well as on the thermodynamic data. The black-hole temperature, entropy, and corresponding electrostatic potential then follow from relations holding at the stationary points of the reduced action with respect to variation of the free parameters. Investigation of an appropriate eigenvalue problem shows that the criteria for local dynamical and thermodynamical stability are the same. The ensemble can be either stable or unstable, depending upon a certain relation involving mean charge, gravitational radius, and boundary radius. The role of the reduced action in determining the grand partition function, the thermodynamics of charged black holes, and the density of states is discussed.

  • Received 2 July 1990

DOI:https://doi.org/10.1103/PhysRevD.42.3376

©1990 American Physical Society

Authors & Affiliations

Harry W. Braden*, J. David Brown, Bernard F. Whiting, and James W. York, Jr.

  • Institute of Field Physics, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, North Carolina 27599-3255

  • *Present address: Department of Mathematics, James Clerk Maxwell Building, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, Scotland.
  • Present address: Department of Physics, Williamson Hall, University of Florida, Gainesville, FL 32611.

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Issue

Vol. 42, Iss. 10 — 15 November 1990

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