Controlling unboundedness in the gravitational path integral.

Abstract

New results are presented on the Euclidean path-integral formulation for the partition function and density of states pertinent to spherically symmetric black-hole systems in thermodynamic equilibrium. We extend the path-integral construction of Halliwell and Louko which has already been used by one of us (Louko and Whiting), and investigate further a lack of uniqueness in our previous formulation of the microcanonical density of states and in the canonical partition function. In that work, the method chosen for removing the ambiguity resulted in two specific path-integral contours having finite extent. Physically motivated criteria exercised a dominant influence on that choice, as did the need to overcome the unboundedness from below of the gravitational action. The new results presented here satisfy the same physical criteria, but differ in ways which are physically significant. The unboundedness is not now eliminated directly but, for positive temperatures only, it is dealt with by what may be viewed as the introduction of an effective measure, which nevertheless may be of exponential order. Having chosen to investigate alternative contours which, in fact, have infinite extent, we find that imposing the Wheeler-DeWitt equation automatically selects out particular finite end points for the contours, at which the singularity in the action is canceled. A further important outcome of this work is the emergence of a variational principle for the black hole entropy, which has already proved useful at the level of a zero-loop approximation to the coupling of a shell of quantum matter in equilibrium around a Schwarzschild black hole (Horwitz and Whiting). In the course of enquiring into the nature of the variables in which the path integral is constructed and evaluated, we were able to see how to give a unifying description of several previous results in the literature. A concise review of these separate approaches forms an integral part of our new synthesis, relating their various underlying ideas on Hamilton-Jacobi theory and Hamiltonian reduction in the context of path integration. The new insight we gain finally helps motivate the choice of the integration variables, identification of which has played an important role in our whole analysis.