Dappiaggi, Claudio and Moretti, Valter and Pinamonti, Nicola (2009) Rigorous construction and Hadamard property of the Unruh state in the full Schwarzschild spacetime. UNSPECIFIED. (Unpublished)

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Abstract

The discovery of the radiation properties of black holes prompted the search for a natural candidate quantum ground state for a massless scalar field theory on the Schwarzschild spacetime. Among the several available proposals in the literature, an important physical role is played by the so-called Unruh state which is supposed to be appropriate to capture the physics of a real spherically symmetric black hole formed by collapsing matter. One of the aims of this paper, referring to a massless Klein-Gordon field, is to rigorously construct that state globally, i.e. on the algebra of Weyl observables localized in the union of the static external region, the future event horizon and the non-static black hole region. The Unruh state is constructed following the traditional recipe that it is the vacuum state with respect to the affine parameter U of the geodesic forming the whole past horizon whereas it is the vacum state with respect to the Schwarzschild Killing time t on the past light infinity, interpreting these data within our algebraic formalism. Eventually, making use of the microlocal-analysis approach, we prove that the Unruh state built up following our procedure fulfills the so-called Hadamard condition everywhere it is defined and, hence, it is perturbatively stable, realizing the natural candidate with which one could study purely quantum phenomena such as the role of the back reaction of Hawking's radiation. The achieved results are obtained by means of a bulk-to-boundary reconstruction technique which exploits the Killing (horizon) structure and the conformal asymptotic structure of the underlying background, employing Hörmander's theorem on propagation of singularities, some recent results about passive state extended to our case, and a careful analysis of the remaining part of the wavefront set of the state. A crucial technical role is played by the recent results due to Dafermos and Rodnianski on the peeling behaviour of the solutions of Klein-Gordon equations in Schwarzschild spacetime.

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