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

TL;DR

The paper develops a Hamiltonian framework for spherically symmetric Einstein–Maxwell theory with a negative cosmological constant, focusing on RNAdS black holes and their thermodynamics under grand canonical and canonical ensembles.A Kuchař-type canonical transformation reduces the theory to true dynamical degrees of freedom, enabling explicit quantization and the construction of grand/canonical partition functions via analytic continuation to imaginary time.Thermodynamic analysis shows that in regimes where a classical Euclidean RNAdS solution dominates, the energy and charge equal the black hole parameters and the entropy matches the Bekenstein–Hawking value, with phase transitions between black hole and hot AdS sectors.The negative cosmological constant plays a stabilizing role analogous to a finite bounding box, whereas the asymptotically flat limit fails to yield well-defined ensembles due to divergences.The work connects a concrete Hamiltonian treatment with Euclidean quantum gravity insights, reproducing standard black hole thermodynamics from a first-principles reduction and quantization approach.

Abstract

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-Nordström-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 Kuchař 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 Kuchař on the occasion of his sixtieth birthday.)