Classical and Quantum Thermodynamics of horizons in spherically symmetric spacetimes

TL;DR

The paper develops a local thermodynamic framework for horizons in general spherically symmetric spacetimes, extending beyond black holes to de Sitter and multi-horizon geometries. It shows that near any horizon one can rewrite Einstein's equations as a first-law-like relation $TdS-dE=PdV$ with $S=\frac{1}{4}A_H$, $E=\frac{a}{2}$, and temperature set by the horizon's surface gravity, while pressure comes from the matter content and $dV$ corresponds to the infinitesimal horizon displacement. A parallel quantum treatment via a partition function independent of Einstein's equations reproduces the same thermodynamic quantities, giving $Z(\beta) \propto \exp[S-\beta E]$ and confirming the local interpretation in both (3+1) and (1+2) dimensions. The work highlights a consistent de Sitter horizon energy $E=-(1/2)H^{-1}$, discusses horizon dimensionality and the challenges of defining global temperatures for multi-horizon spacetimes, and suggests a deep link between the Einstein-Hilbert action and spacetime thermodynamics. Overall, it broadens horizon thermodynamics to non-asymptotically-flat spacetimes and multiple-horizon settings, with implications for quantum gravity and the nature of gravitational entropy.

Abstract

A general formalism for understanding the thermodynamics of horizons in spherically symmetric spacetimes is developed. The formalism reproduces known results in the case of black hole spacetimes. But its power lies in being able to handle more general situations like: (i) spacetimes which are not asymptotically flat (like the de Sitter spacetime) and (ii) spacetimes with multiple horizons having different temperatures (like the Schwarzschild-de Sitter spacetime) and provide a consistent interpretation for temperature, entropy and energy. I show that it is possible to write Einstein's equations for a spherically symmetric spacetime in the form $TdS-dE=PdV$ near {\it any} horizon of radius $a$ with $S=(1/4)(4πa^2), |E| = (a/2)$ and the temperature $T$ determined from the surface gravity at the horizon. The pressure $P$ is provided by the source of the Einstein's equations and $dV$ is the change in the volume when the horizon is displaced infinitesimally. The same results can be obtained by evaluating the quantum mechanical partition function {\it without using Einstein's equations or WKB approximation for the action}. Both the classical and quantum analysis provide a simple and consistent interpretation of entropy and energy for de Sitter spacetime as well as for $(1+2)$ dimensional gravity. For the Rindler spacetime the entropy per unit transverse area turns out to be $(1/4)$ while the energy is zero. The approach also shows that the de Sitter horizon -- like the Schwarzschild horizon -- is effectively one dimensional as far as the flow of information is concerned, while the Schwarzschild-de Sitter, Reissner-Nordstrom horizons are not. The implications for spacetimes with multiple horizons are discussed.