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

Euclidean continuation of several Lorentzian spacetimes with horizons requires treating the Euclidean time coordinate to be periodic with some period $\beta$. Such spacetimes (Schwarzschild, deSitter,Rindler …) allow a temperature $T={\beta }^{-1}$ to be associated with the horizon. I construct a canonical ensemble of a subclass of such spacetimes with a fixed value for $\beta$ and evaluate the partition function $Z(\beta )$. For spherically symmetric spacetimes with a horizon at r=a, the partition function has the generic form $Z\propto \exp [S-\beta E]$, where $S=(1\mathrm{/}4)4\pi a^2$ and $\mathrm{\mid }E\mathrm{\mid }=(a\mathrm{/}2)$. 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 interpretation 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.