Temperature and entropy of Schwarzschild-de Sitter space-time

TL;DR

This paper extends semi-classical methods to Schwarzschild-de Sitter space-time with multiple horizons, deriving particle spectra, temperature relations, and proposing a new entropy definition consistent with the D-bound conjecture.

Contribution

It introduces an extension of the complex path method to multiple-horizon space-times and proposes a novel entropy definition based on effective surface gravity.

Findings

The radiation temperature is proportional to the inverse harmonic sum of horizon surface gravities.

The equilibrium temperature is the harmonic mean of the cosmological and event horizon temperatures.

The new entropy definition satisfies the D-bound conjecture.

Abstract

In the light of recent interest in quantum gravity in de Sitter space, we investigate semi-classical aspects of 4-dimensional Schwarzschild-de Sitter space-time using the method of complex paths. The standard semi-classical techniques (such as Bogoliubov coefficients and Euclidean field theory) have been useful to study quantum effects in space-times with single horizons; however, none of these approaches seem to work for Schwarzschild-de Sitter or, in general, for space-times with multiple horizons. We extend the method of complex paths to space-times with multiple horizons and obtain the spectrum of particles produced in these space-times. We show that the temperature of radiation in these space-times is proportional to the effective surface gravity -- inverse harmonic sum of surface gravity of each horizon. For the Schwarzschild-de Sitter, we apply the method of complex paths to three different coordinate systems -- spherically symmetric, Painleve and Lemaitre. We show that the equilibrium temperature in Schwarzschild-de Sitter is the harmonic mean of cosmological and event horizon temperatures. We obtain Bogoliubov coefficients for space-times with multiple horizons by analyzing the mode functions of the quantum fields near the horizons. We propose a new definition of entropy for space-times with multiple horizons analogous to the entropic definition for space-times with a single horizon. We define entropy for these space-times to be inversely proportional to the square of the effective surface gravity. We show that this definition of entropy for Schwarzschild-de Sitter satisfies the D-bound conjecture.