Temperature and entropy of Schwarzschild-de Sitter space-time

Details Temperature and entropy of Schwarzschild-de Sitter space-time Temperature and entropy of Schwarzschild-de Sitter space-time

2003-01-22

arxiv.org/abs/gr-qc/0301090v2

Phys.Rev. D67 (2003) 084026

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Final version; To appear in Phys. Rev. D; 12 pages, 1 figure, RevTex-4; Typos corrected; References added

Scientific paper

10.1103/PhysRevD.67.084026

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.

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