Action and free energy for black hole topologies

The gravitational fields of black holes have a thermal interpretation. The static cases correspond to equilibrium, so that the imaginary time or Euclidean method is appropriate. The boundary data required for the Euclidean action coincide with those that specify a canonical ensemble. Evaluation of this action for a finite region in a class of geometries that includes the classical black hole solutions, with the constraints or initial value equations taken into account, produces a “reduced” action and an associated free energy. The reduced action can have locally stable and unstable stationary points or none. In certain cases there are globally stable configurations. Their properties make it possible to understand black holes in the canonical ensemble, which was not possible in the original formulation of black hole thermodynamics. Thermodynamic reasoning in situations where gravity is important must be altered to fit the new point of view.