Gravitational Entropy o f static spacetimes and microscopic density o f states

Abstract

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 (Tab − (1/2)T gab)ua ub. (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 Tabua ub. We discuss the conditions under which these results imply S ∝ E2 and/or S ∝ U2 thereby generalizing the results known for black holes. This approach links with several other known results, especially the holographic views of spacetime.