Concept of temperature in multi-horizon spacetimes: Analysis of Schwarzschild-De Sitter metric
T. Roy Choudhury, T. Padmanabhan
TL;DR
The paper addresses whether a single global temperature can characterize Schwarzschild–De Sitter spacetimes with two horizons. By extending standard single-horizon techniques—stress-tensor vacua, detector responses, and Euclidean-time periodicity—to SDS in a 1+1D setting and by constructing a global coordinate chart covering the full manifold, it shows that SDS generally behaves as a two-temperature, non-equilibrium system. A global temperature can exist only when the ratio of the two horizon surface gravities, $\kappa_+/\kappa_-$, is rational, and even then its existence depends on the chosen coordinate chart, with a global chart yielding $\beta^{-1}=\kappa_-/(2\pi n_-)=\kappa_+/ (2\pi n_+)$ for integers $n_-,n_+$. Otherwise, different local charts exhibit two distinct temperatures, and the Euclidean approach yields a conical singularity, indicating no universal thermal state. The work highlights the crucial role of coordinate choice in the thermal interpretation of multi-horizon spacetimes, drawing an analogy to Minkowski/Rindler thermodynamics and informing semiclassical gravity analyses and potential horizon-quantization considerations.
Abstract
In case of spacetimes with single horizon, there exist several well-established 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.