Hadamard state in Schwarzschild–de Sitter spacetime

Abstract

Abstract. We construct a state in the Schwarzschild-de Sitter spacetime which is invariant under the action of its group of symmetries. Our state is not defined in the whole Kruskal extension of this spacetime, but rather in a subset of the maximally extended conformal diagram. The construction is based on a careful use of the bulk-to-boundary technique. We will show that our state is Hadamard and that it is not a KMS state, differently from the case of states constructed in spacetimes containing only one event horizon.

Quantum Fields from Global Propagators on Asymptotically Minkowski and Extended de Sitter Spacetimes

“…One possible alternative is to define Λ ± by specifying its asymptotic data, in terms of which positivity can be hoped to be realized explicitly. In fact, this strategy has already been successfully applied indeed in the case of the conformal wave equation on a class of asymptotically flat spacetimes [51,52,24] (see also [12,15,16] for other classes of spacetimes), where one can consider as data at future null infinity the characteristic Cauchy data for a conformally rescaled metric. Recent advances also show that one can define Hadamard states for asymptotically static spacetimes using tools from scattering theory [28].…”

“…(3.35)]) one sees that the v dependence ofã ± can be essentially completely eliminated in that one can write the integrand as χ 0 (v) times a v independent symbol, with χ 0 ≡ 1 near 0 and of compact support, again modulo S −l (C; C ∞ (∂M )). In particular, the leading term of the asymptotic expansion ofã ± as γ → ±∞ is recovered by simply taking the Fourier 11 Note that components of b SN * S are not even Legendre in b S * M since the symplectic structure degenerates at ∂M in the b-normal directions, so b SN * S has dimension n − 2 if n is the dimension of M : both the boundary defining function ρ and its b-dual variable σ vanish on b SN * S. 12 Near the boundary M admits a product decomposition of the form [0, )ρ × ∂M , we can then take η+ supported in, say, ρ < /2, which makes the Mellin transform of η+u well defined. 13 Here we use L ∞ -based symbols, so a symbol a of order 0 satisfies |D α y D k v D N γ a| ≤ C αkN γ −N for all α, k, N .…”

Hadamard States for the Klein–Gordon Equation on Lorentzian Manifolds of Bounded Geometry

“…Specific examples of Hadamard states on spacetimes with special (asymptotic) symmetries include passive states for stationary spacetimes [SV], states constructed from data at null infinity on various classes of asymptotically flat or asymptotically de Sitter spacetimes [Mo,DMP1,BJ,VW] and on cosmological spacetimes 4 [DMP2,JS,BT]. Furthermore, a remarkable recent result by Sanders [Sa] proves the existence and Hadamard property of the so-called Hartle-Hawking-Israel state on spacetimes with a static bifurcate Killing horizon.…”

Hadamard Property of the in and out States for Klein–Gordon Fields on Asymptotically Static Spacetimes

“…It is known that in the special case of the conformal wave equation, one can study the wave front set of the two-point functions quite directly in the geometrical setup of conformal scattering on asymptotically flat spacetimes [Mo2,GW3] (cf. [DMP1,DMP2,BJ] for generalizations on the allowed classes of spacetimes). Furthermore, propagation estimates in b-Sobolev spaces of variable order were used recently to show a similar result in the case of the wave equation on asymptotically Minkowski spacetimes [VW], drawing on earlier developments by Vasy et al [BVW,HV,Va1,Va2].…”