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, this entropy satisfies the relation S = (1/2)βE where E is the energy source for gravitational acceleration, obtained as an integral of (Tab − (1/2)Tgab)u u. (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 Tabu u. We discuss the conditions under which these results imply S ∝ E and/or S ∝ U thereby generalizing the results known for black holes. This approach links with several other known results, especially the holographic views of spacetime.