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<SUP>-1</SUP>, this entropy satisfies the relation S = (½)βE where E is the energy source for gravitational acceleration, obtained as an integral of (T<SUB>ab</SUB> - (½)T<SUB>g<SUB>ab</SUB></SUB>)μ<SUP>a</SUP>μ<SUP>b</SUP>. (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<SUB>ab</SUB>u<SUP>a</SUP>u<SUP>b</SUP>. We discuss the conditions under which these results imply S α E<SUP>2</SUP> 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