Charged black hole in a grand canonical ensemble

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.