Research PaperNo Access

The Hartle–Hawking–Israel state on spacetimes with stationary bifurcate Killing horizons

Christian Gérard

Département de Mathématiques, Université Paris-Sud XI, 91405 Orsay Cedex, France

E-mail Address: christian.gerard@math.u-psud.fr

https://doi.org/10.1142/S0129055X21500288Cited by:3 (Source: Crossref)

Abstract

We consider a free massive quantized Klein–Gordon field in a spacetime $(M, g)$ containing a stationary bifurcate Killing horizon, i.e. a bifurcate Killing horizon whose Killing vector field is globally time-like in the right wedge $ℳ^{+}$ associated to the horizon.

We prove the existence of the Hartle–Hawking–Israel (HHI) vacuum state, which is a pure state on the whole spacetime whose restriction to $ℳ^{+}$ is a thermal state $ω_{T_{H}}$ for the time-like Killing field at Hawking temperature $T_{H} = κ {(2 π)}^{- 1}$ , where $κ$ is the surface gravity of the horizon.

We show that the HHI state is a Hadamard state and is the unique Hadamard state which is equal to the double $T_{H}^{- 1}$ -KMS state in the double wedge $ℳ^{-} \cup ℳ^{+}$ . We construct the HHI state by Wick rotation in Killing time coordinates, using the notion of the Calderón projector for elliptic boundary value problems.

Keywords:

AMSC: 81T20, 35S05, 35S15

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