CONTROLLING UNBOUNDEDNESS IN THE GRAVITATIONAL PATH-INTEGRAL

New results are presented on the Euclidean path-integral formulation f or the partition function and density of states pertinent to spherical ly symmetric black-hole systems in thermodynamic equilibrium. We exten d the path-integral construction of Halliwell and Louko which has alre ady been used by one of us (Louko and Whiting), and investigate furthe r a lack of uniqueness in our previous formulation of the microcanonic al density of states and in the canonical partition function. In that work, the method chosen for removing the ambiguity resulted in two spe cific path-integral contours having finite extent. Physically motivate d criteria exercised a dominant influence on that choice, as did the n eed to overcome the unboundedness from below of the gravitational acti on. The new results presented here satisfy the same physical criteria, but differ in ways which are physically significant. The unboundednes s is not now eliminated directly but, for positive temperatures only, it is dealt with by what may be viewed as the introduction of an effec tive measure, which nevertheless may be of exponential order. Having c hosen to investigate alternative contours which, in fact, have infinit e extent, re find that imposing the Wheeler-DeWitt equation automatica lly selects out particular finite end points for the contours, at whic h the singularity in the action is canceled. A further important outco me of this work is the emergence of a variational principle for the bl ack hole entropy, which has already proved useful at the level of a ze ro-loop approximation to the coupling of a shell of quantum matter in equilibrium around a Schwarzschild black hole (Horwitz and Whiting). I n the course of enquiring into the nature of the variables in which th e path integral is constructed and evaluated, we were able to see how to give a unifying description of several previous results in the lite rature. A concise review of these separate approaches forms an integra l part of our new synthesis, relating their various underlying ideas o n Hamilton-Jacobi theory and Hamiltonian reduction in the context of p ath integration. The new insight we gain finally helps motivate the ch oice of the integration variables, identification of which has played an important role in our whole analysis.