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

摘要

Euclidean continuation of several Lorentzian spacetimes with horizonsrequires 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 canonicalensemble of a subclass of such spacetimes with a fixed value for $\beta$ andevaluate the partition function $Z(\beta)$. For spherically symmetricspacetimes with a horizon at r=a, the partition function has the generic form$Z\propto \exp[S-\beta E]$, where $S= (1/4) 4\pi a^2$ and $|E|=(a/2)$. Both Sand E are determined entirely by the properties of the metric near the horizon.This analysis reproduces the conventional result for the blackhole spacetimesand provides a simple and consistent interpretation of entropy and energy fordeSitter spacetime. For the Rindler spacetime the entropy per unit transversearea turns out to be (1/4) while the energy is zero. The implications arediscussed.