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

摘要

In case of spacetimes with single horizon, there exist severalwell-established procedures for relating the surface gravity of the horizon toa thermodynamic temperature. Such procedures, however, cannot be extended in astraightforward manner when a spacetime has multiple horizons. In particular,it is not clear whether there exists a notion of global temperaturecharacterizing the multi-horizon spacetimes. We examine the conditions underwhich a global temperature can exist for a spacetime with two horizons usingthe example of Schwarzschild-De Sitter (SDS) spacetime. We systematicallyextend different procedures (like the expectation value of stress tensor,response of particle detectors, periodicity in the Euclidean time etc.) foridentifying a temperature in the case of spacetimes with single horizon to theSDS spacetime. This analysis is facilitated by using a global coordinate chartwhich covers the entire SDS manifold. We find that all the procedures lead to aconsistent picture characterized by the following features: (a) In general, SDSspacetime behaves like a non-equilibrium system characterized by twotemperatures. (b) It is not possible to associate a global temperature with SDSspacetime except when the ratio of the two surface gravities is rational (c)Even when the ratio of the two surface gravities is rational, the thermalnature depends on the coordinate chart used. There exists a global coordinatechart in which there is global equilibrium temperature while there exist othercharts in which SDS behaves as though it has two different temperatures. Thecoordinate dependence of the thermal nature is reminiscent of the flatspacetime in Minkowski and Rindler coordinate charts. The implications arediscussed.