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)β E 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 β F = β U − S where U is the integral of T ab u a u b . We discuss the conditions under which these results imply S ∝ E 2 and/or S ∝ U 2 thereby generalizing the results known for black holes. This approach links with several other known results, especially the holographic views of spacetime.