Abstract
We study quantum fields on spacetimes having a bifurcate Killing horizon by allowing the possibility that left- and right- (in-going and out-going) modes have different temperatures. We consider in particular the Rindler for both massless and massive fields, the static de Sitter and Schwarzschild black hole backgrounds for massive fields. We find that in all three cases, when any of the temperatures is different from the canonical one (Unruh, Hawking and Gibbons--Hawking, correspondingly) the correlation functions have extra singularities at the horizon.
Highlights
An example of such a state is the Unruh vacuum in a black hole background, in which ingoing modes are in the zero temperature state, while the outgoing ones are at the Hawking temperature
There is a well-known relation between the existence of a bifurcate Killing horizon on certain spacetime manifolds and the temperature of some specially privileged equilibrium thermal states
While there is vast literature devoted to the study of the above states, little attention has been paid to other possible thermal or pseudothermal states, let alone to what happens when interactions are switched on: does the fluctuationdissipation theorem still work as in flat space? Are the privileged states attractor solutions of some sort of kinetic equations with exact modes instead of plane waves? Does thermalization of a given initial state happen in strongly curved space-times without substantial backreaction on the gravitational background? The backreaction during thermalization seems to be negligible practically for any reasonable (Hadamard) initial state in flat space-time [2,3]
Summary
There is a well-known relation between the existence of a bifurcate Killing horizon on certain spacetime manifolds and the temperature of some specially privileged equilibrium thermal states. As regards to the Schwarzschild geometry and the black hole radiation, there are three distinguished states which are usually considered: the Boulware [14], Unruh [15], and Hartle-Hawking states. Even without performing loop calculations (where the heating process is seen), we will argue that the answer to the first question in the last paragraph seems to be negative Both in the static de Sitter space (see [6]) and in the black hole background all correlation functions with temperatures different from the Hawking one have anomalous singularities at the horizon. In all cases when the temperatures are different from the Unruh, Gibbons-Hawking, and Hawking ones in the corresponding situations there are anomalous singularities of the correlation functions at the horizon
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