HAMILTONIAN THERMODYNAMICS OF THE REISSNER-NORDSTROM-ANTI-DE SITTER BLACK-HOLE

We consider the Hamiltonian dynamics and thermodynamics of spherically symmetric Einstein-Maxwell spacetimes with a negative cosmological co nstant. We impose boundary conditions that enforce every classical sol ution to be an exterior region of a Reissner-Nordstrom-anti-de Sitter black hole with a nondegenerate Killing horizon, with the spacelike hy persurfaces extending from the horizon bifurcation two-sphere to the a symptotically anti-de Sitter infinity. The constraints are simplified by a canonical transformation, which generalizes that given by Kuchar 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 ens emble is obtained by analytically continuing the Lorentzian time evolu tion operator to imaginary time and taking the trace. A similar analys is under slightly modified boundary conditions leads to the partition function of a thermodynamical canonical ensemble. The thermodynamics i n each ensemble is analyzed, and the conditions that the (grand) parti tion function be dominated by a classical Euclidean black hole solutio n are found. When these conditions are satisfied, we recover,in partic ular, the Bekenstein Hawking entropy. The limit of a vanishing cosmolo gical constant is briefly discussed.