Entropy and energy of a class of spacetimes with horizon : a general derivation

Abstract

Euclidean continuation of several Lorentzian spacetimes with horizons requires treating the Eu- clidean time coordinate to be periodic with some period β. Such spacetimes (Schwarzschild, de- Sitter,Rindler .....) allow a temperature T = β−1 to be associated with the horizon. I construct a canonical ensemble of a subclass of such spacetimes with a fixed value for β and evaluate the par- tition function Z(β). For spherically symmetric spacetimes with a horizon at r = a, the partition function has the generic form Z ∝ exp[S −βE], where S = (1/4)4πa2 and |E| = (a/2). Both S and E are determined entirely by the properties of the metric near the horizon. This analysis reproduces the conventional result for the blackhole spacetimes and provides a simple and consistent interpre- tation of entropy and energy for deSitter spacetime. For the Rindler spacetime the entropy per unit transverse area turns out to be (1/4) while the energy is zero. The implications are discussed.