Entropy of static spacetimes and microscopic density of 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 = (½)βE where E is the energy source for gravitational acceleration, obtained as an integral of (T_ab - (½)T_gab)μ^aμ^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_abu^au^b. We discuss the conditions under which these results imply S α E² and/or S α U thereby generalizing the results known for black holes. This approach links with several other known results, especially the holographic views of spacetime.