EUCLIDEAN ACTION AND THE THERMODYNAMICS OF MANIFOLDS WITHOUT BOUNDARY

I consider general, spherically symmetric spacetimes with cosmological and black-hole horizons. I find that a state of thermal equilibrium may exist in classical manifolds with two horizons as long as a matter distribution is present. The Euclidean action is calculated for general (nonclassical) manifolds without boundary. It is found to equal the grand canonical weighting factor for a spacetime with zero mean thermal energy. I also consider how the Euclidean action is modified by the imposition of fixed-temperature shells in the manifold. I find that the mean thermal energy of the cosmological horizon is negative. © 1990 The American Physical Society.