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April 2011 Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime
Claudio Dappiaggi, Valter Moretti, Nicola Pinamonti
Adv. Theor. Math. Phys. 15(2): 355-447 (April 2011).


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 Schwarzschild spacetime, here considered in the Eddington–Finkelstein representation. 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 black hole formed by spherically symmetric collapsing matter. Within this respect, we shall consider a massless Klein–Gordon field and we shall rigorously and globally construct such state, that is on the algebra of Weyl observables localised in the union of the static external region, the future event horizon and the non-static black hole region. Eventually, out of a careful use of microlocal techniques, we prove that the built state fulfils, where defined, the so-called Hadamard condition; hence, it is perturbatively stable, in other words realizing the natural candidate with which one could study purely quantum phenomena such as the role of the back reaction of Hawking’s radiation.

From a geometrical point of view, we shall make a profitable use of a bulk-to-boundary reconstruction technique which carefully exploits the Killing horizon structure as well as the conformal asymptotic behaviour of the underlying background. From an analytical point of view, our tools will range from Hörmander’s theorem on propagation of singularities, results on the role of passive states, and a detailed use of the recently discovered peeling behaviour of the solutions of the wave equation in Schwarzschild spacetime.


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Claudio Dappiaggi. Valter Moretti. Nicola Pinamonti. "Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime." Adv. Theor. Math. Phys. 15 (2) 355 - 447, April 2011.


Published: April 2011
First available in Project Euclid: 25 May 2012

zbMATH: 1257.83008
MathSciNet: MR2924234

Rights: Copyright © 2011 International Press of Boston

Vol.15 • No. 2 • April 2011
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