Entropy of Static Spacetimes and Microscopic Density of States

A general ansatz for gravitational entropy can be provided using the criterion that, any patch of area which acts as a horizon for a suitably defined accelerated observer, must have an entropy proportional to its area. After providing a brief justification for this ansatz, several consequences are derived: (i) In any static spacetime with a horizon and associated temperature $β^{-1}$, this entropy satisfies the relation $S=(1/2)βE$ where $E$ is the energy source for gravitational acceleration, obtained as an integral of $(T_{ab}-(1/2)Tg_{ab})u^au^b$. (ii) With this ansatz of $S$, the minimization of Einstein-Hilbert action is equivalent to minimizing the free energy $F$ with $βF=βU-S$ where $U$ is the integral of $T_{ab}u^au^b$. We discuss the conditions under which these results imply $S\propto E^2$ and/or $S\propto U^2$ thereby generalizing the results known for black holes. This approach links with several other known results, especially the holographic views of spacetime.

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