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

TL;DR

This paper investigates whether a meaningful global temperature exists in multi-horizon spacetimes, specifically Schwarzschild-De Sitter, and finds it generally does not unless certain conditions are met, highlighting coordinate dependence.

Contribution

It extends single-horizon temperature procedures to multi-horizon spacetimes and analyzes the conditions for global temperature existence in Schwarzschild-De Sitter spacetime.

Findings

SDS spacetime generally behaves as a non-equilibrium system with two temperatures.

A global temperature exists only when the ratio of surface gravities is rational.

The thermal nature depends on the choice of coordinate chart, indicating coordinate dependence.

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.