Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes

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 and can handle more general situations such as: (i) spacetimes which are not asymptotically flat (such as the de Sitter spacetime) and (ii) spacetimes with multiple horizons having different temperatures (such as 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 T dS − dE = P dV near any horizon of radius a with S equal; 1/4(4πa2), |E| = (a/2) and the temperature T determined from the surface gravity at the horizon. The pressure P is provided by the source of 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 without using Einstein's equations or the WKB approximation for the action. Both the classical and quantum analyses 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.