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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 $β$. Such spacetimes (Schwarzschild, deSitter,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 partition function $Z(β)$. For spherically symmetric spacetimes with a horizon at r=a, the partition function has the generic form $Z\propto \exp[S-βE]$, where $S= (1/4) 4πa^2$ 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 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.