Temperature and entropy of Schwarzschild–de Sitter space-time

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.

Hawking radiation as quantum tunneling from a noncommutative Schwarzschild black hole

“…On the other hand, Shankaranarayanan et al have applied the tunneling approach to obtain the Hawking temperature in different coordinates within a Complex paths scenario [20]. This technique has been successfully applied to obtain a global temperature for multi-horizon spacetimes [21]. In this paper, we are going to develop Parikh-Wilczek method to noncommutative coordinate coherent states.…”

Near-extremal black holes

“…On the positive sides, this method has been successfully applied to different types of horizons including anti-de Sitter(AdS) [24], de Sitter(dS) [20,21,22,23], BTZ [25,26],higher dimensional black holes and some exotic spacetimes [27,29,30,31] apart from the more conventional types of black holes, in each case producing the corresponding Hawking temperature correctly. Also, as the derivation involves only the local geometry, the tunnelling method can be applied to any local horizon, in particular to cosmological and weakly isolated horizons [28,29].…”

Quantum Tunneling Radiation from Loop Quantum Black Holes and the Information Loss Paradox

“…In this way, the tunneling approach turns out to be very useful when one wishes to incorporate back-reaction effects in order to describe black hole evaporation. In addition, even though calculations in the tunneling formalism are straightforward and relatively simple, they are robust in the sense that they can be applied to a wide variety of spacetimes [12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”