ENSEMBLE DEPENDENCE OF THE STABILITY OF THERMAL BLACK-HOLES

Within the Euclidean path integral approach to statistical mechanics we examine the question of the ensemble dependence of the stability of thermal black holes. In both of the ensembles considered it is found that there is only one system configuration which can satisfy the given thermodynamic boundary conditions (i.e. the appropriate Massieu function has only one extremum). We find that throughout the parameter space of one ensemble black holes are never stable (the extremum is a saddle point) whereas in the other ensemble they are nearly always stable (the extremum is a minimum). Using a proposal of Whiting and York for the integration measure of the path integral, we include the effects of quantum fluctuations in the partition function for the stable ensemble. This gives a gravitational field entropy which has terms not included in the Bekenstein-Hawking formula.