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

Abstract

We consider a free massive quantized Klein–Gordon field in a spacetime [Formula: see text] containing a stationary bifurcate Killing horizon, i.e. a bifurcate Killing horizon whose Killing vector field is globally time-like in the right wedge [Formula: see text] 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 [Formula: see text] is a thermal state [Formula: see text] for the time-like Killing field at Hawking temperature [Formula: see text], where [Formula: see text] 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 [Formula: see text]-KMS state in the double wedge [Formula: see text]. We construct the HHI state by Wick rotation in Killing time coordinates, using the notion of the Calderón projector for elliptic boundary value problems.