Concept of temperature in multi-horizon spacetimes: analysis of Schwarzschild–De Sitter metric

Abstract

In case of spacetimes with single horizon, there exist several wellestablished procedures for relating the surface gravity of the horizon to a thermodynamic temperature. Such procedures, however, cannot be extended in a straightforward manner when a spacetime has multiple horizons. In particular, it is not clear whether there exists a notion of global temperature characterizing the multi-horizon spacetimes. We examine the conditions under which a global temperature can exist for a spacetime with two horizons using the example of Schwarzschild-De Sitter (SDS) spacetime. We systematically extend different procedures (like the expectation value of stress tensor, response of particle detectors, periodicity in the Euclidean time etc.) for identifying a temperature in the case of spacetimes with single horizon to the SDS spacetime. This analysis is facilitated by using a global coordinate chart which covers the entire SDS manifold. We find that all the procedures lead to a consistent picture characterized by the following features: (a) In general, SDS spacetime behaves like a non-equilibrium system characterized by two temperatures. (b) It is not possible to associate a global temperature with SDS spacetime except when the ratio of the two surface gravities is rational. (c) Even when the ratio of the two surface gravities is rational, the thermal nature depends on the coordinate chart used. There exists a global coordinate chart in which there is global equilibrium temperature while there exist other charts in which SDS behaves as though it has two different temperatures. The coordinate dependence of the thermal nature is reminiscent of the flat spacetime in Minkowski and Rindler coordinate charts. The implications are discussed.

Thermodynamics of phase transition in higher-dimensional Reissner–Nordström–de Sitter black hole

“…This result is the same as that of the RN-AdS black hole [38,57,58]. In fact, one can also introduce a "thermally opaque" membrane [45,50,59] between the black hole horizon and the cosmological horizon to construct two different thermal equilibrium states. The two states satisfy Eqs.…”

Generic Features of Thermodynamics of Horizons in Regular Spherical Space-Times of the Kerr-Schild Class

“…This gave the birth to thermodynamics of black holes [4][5][6]. In 1976 Gibbons and Hawking found that also cosmological horizon emits thermal radiation [7], and this gave rise to thermodynamics of horizons [8][9][10][11][12][13][14][15][16]. The basic point is that an observer for whom a horizon prevents seeing the whole space-time, does not have an access to the complete quantum state of the system, and the loss of information about quantum state is responsible for the thermal character of radiation [5,6].…”

“…A general approach to defining thermodynamical variables in this case for spherical space-times with the Kerr-Schild metrics (1) was developed by Padmanabhan [10][11][12] who considered a canonical ensemble of space-time metrics (1) at the constant temperature of a horizon determined by the periodicity of the Euclidean time in the Euclidean continuation of the Einstein action [10]. In this approach the first law of thermodynamics for black holes described by metrics (1) is obtained in agreement with the second law as a consequence of the Einstein equations [10][11][12]. A thorough overview of the concept of honrizon is give in [56].…”

Thermodynamics of Regular Cosmological Black Holes with the de Sitter Interior

“…A year later Hawking found a quantum evaporation of a black hole [3,4] which gave the birth to thermodynamics of black holes [5][6][7] (for a review see [8]). In 1976 Gibbons and Hawking found that also cosmological horizon can radiate [9], and this gave rise to thermodynamics of horizons [10][11][12][13][14][15][16][17][18].…”

“…A general approach for defining thermodynamical variables was proposed by Padmanabhan [12][13][14]. He considered the spherically symmetric Einstein equations as the thermodynamical identity for the class of solutions described by the line element (1) which leads to the results confirmed by consideration of a canonical ensemble of space-time metrics from the class (1) at the constant temperature of the horizon determined by the periodicity of the Euclidean time in the Euclidean continuation of the Einstein action; the partition function for this ensemble calculated as the path integral sum [12], can be written as…”