Entropy of Static Spacetimes and Microscopic Density of States

TL;DR

This paper proposes a general ansatz for gravitational entropy based on horizon area and derives its implications for static spacetimes, connecting entropy, energy, and free energy, and generalizing black hole thermodynamics.

Contribution

It introduces a new ansatz for gravitational entropy applicable to static spacetimes and explores its consequences, linking entropy to energy and free energy in a novel way.

Findings

Entropy S is proportional to half the product of temperature and energy.

Minimizing Einstein-Hilbert action is equivalent to minimizing free energy with this entropy.

Results suggest S can scale as the square of energy or internal energy under certain conditions.

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