Temperature and entropy of Schwarzschild–de Sitter space-time

S. Shankaranarayanan
Phys. Rev. D 67, 084026 – Published 24 April 2003
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Abstract

In the light of recent interest in quantum gravity in de Sitter space, we investigate semiclassical aspects of four-dimensional Schwarzschild–de Sitter space-time using the method of complex paths. The standard semiclassical 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 space-time 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—the inverse harmonic sum of surface gravity of each horizon. For the Schwarzschild–de Sitter space-time, we apply the method of complex paths to three different coordinate systems—spherically symmetric, Painlevé, and Lemaître. We show that the equilibrium temperature in Schwarzschild–de Sitter space-time 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 space-time satisfies the D-bound conjecture.

  • Received 23 January 2003

DOI:https://doi.org/10.1103/PhysRevD.67.084026

©2003 American Physical Society

Authors & Affiliations

S. Shankaranarayanan*

  • DCTD, University of Azores, 9500 Ponta Delgada, Portugal

  • *Email address: shanki@notes.uac.pt

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Issue

Vol. 67, Iss. 8 — 15 April 2003

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