Classical and Quantum Thermodynamics of horizons in spherically symmetric spacetimes

TL;DR

This paper develops a general formalism for understanding horizon thermodynamics in spherically symmetric spacetimes, applicable to diverse scenarios including non-asymptotically flat and multi-horizon cases, with both classical and quantum insights.

Contribution

It introduces a unified approach to horizon thermodynamics that handles complex spacetimes with multiple or non-standard horizons, extending known results to new contexts.

Findings

Reproduces known black hole thermodynamics results.

Provides a consistent interpretation of entropy and energy in de Sitter spacetime.

Shows horizons can be described by a thermodynamic relation $TdS - dE = PdV$.

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 $S=(1/4)(4\pi a^2), |E| = (a/2)$ 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.