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 beta(-1), this entropy satisfies the relation S = (1/2) betaE where E is the energy source for gravitational acceleration, obtained as an integral of (T-ab - (1/2)Tg(ab))u(a)u(b). (ii) With this ansatz of S, the minimization of Einstein-Hilbert action is equivalent to minimizing the free energy F with U - S where U is the integral of T(ab)u(a)u(b). We discuss the conditions under which these results imply S proportional to E-2 and/or S proportional to U-2 thereby generalizing the results known for black holes. This approach links with several other known results, especially the holographic views of spacetime.