Conformal Scattering and the Goursat Problem

Mason, Lionel; Nicolas, Jean-Philippe

doi:10.1142/s0219891604000123

Cited by 34 publications

(72 citation statements)

References 27 publications

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“…As a matter of fact we embedded Minkowski spacetime into an open region of Einstein static universe. Hence this amounts to select a preferred gauge factor ω according to the definitions of Section 1.1 contrary to the projection operator introduced in [4,12] which provides a smooth function over + for any possible choice of ω. Nonetheless we feel that, fixing ω in our analysis, does not lead to a loss of generality since we can ultimately interpret our results in terms of a general field theory constructed over + without the need for such a choice of the gauge factor.…”

Section: Discussionmentioning

confidence: 99%

“…Given any function F ∈ L 2 (C) let us consider it as a function over all R 4 by means of decomposition (12) and let us split it as F = F + + F − where + represents the contribution of the integral in the ρ-variable between 0 and infinity in (12) whereas the pedex − refers to that between minus infinity and 0.…”

Section: Dappiaggi Ann Henri Poincarémentioning

confidence: 99%

“…Hence it is manifest how null conformal boundaries can be exploited as a powerful tool to study either the asymptotic properties of radiation fields associated to massless wave equations, either the scattering properties of massless fields [12].…”

Section: Introductionmentioning

confidence: 99%

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Projecting Massive Scalar Fields to Null Infinity

Dappiaggi

2008

Ann. Henri Poincaré

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It is known that, in an asymptotically flat spacetime, null infinity cannot act as an initial-value surface for massive real scalar fields. Exploiting tools proper of harmonic analysis on hyperboloids and global norm estimates for the wave operator, we show that it is possible to circumvent such obstruction at least in Minkowski spacetime. Hence we project norm-finite solutions of the Klein-Gordon equation of motion in data on null infinity and, eventually, we interpret them in terms of boundary free field theory.Comment: 26 page

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Section: Discussionmentioning

confidence: 99%

Section: Dappiaggi Ann Henri Poincarémentioning

confidence: 99%

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Projecting Massive Scalar Fields to Null Infinity

Dappiaggi

2008

Ann. Henri Poincaré

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“…Our strategy is similar to that of Hörmander in [31] for the characteristic Cauchy problem for the wave equation (see also [41] for weaker assumptions on the metric). A characteristic Cauchy problem for the Dirac equation has been considered in [35] in a somewhat different setting.…”

Section: The Characteristic Cauchy Problemmentioning

confidence: 99%

Creation of fermions by rotating charged black holes

Häfner¹

2009

Mémoires Soc. Math. France

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:This work is devoted to the mathematical study of the Hawking effect for fermions in the setting of the collapse of a rotating charged star. We show that an observer who is located far away from the star and at rest with respect to the Boyer Lindquist coordinates observes the emergence of a thermal state when his proper time goes to infinity. We first introduce a model of the collapse of the star. We suppose that the space-time outside the star is given by the Kerr-Newman metric. The assumptions on the asymptotic behavior of the surface of the star are inspired by the asymptotic behavior of certain timelike geodesics in the Kerr-Newman metric. The Dirac equation is then written using coordinates and a Newman-Penrose tetrad which are adapted to the collapse. This coordinate system and tetrad are based on the so called simple null geodesics. The quantization of Dirac fields in a globally hyperbolic space-time is described. We formulate and prove a theorem about the Hawking effect in this setting. The proof of the theorem contains a minimal velocity estimate for Dirac fields that is slightly stronger than the usual ones and an existence and uniqueness result for solutions of a characteristic Cauchy problem for Dirac fields in the Kerr-Newman space-time. In an appendix we construct explicitly a Penrose compactification of block I of the Kerr-Newman space-time based on simple null geodesics. Résumé :Ce travail est dédiéà l'étude mathématique de l'effet Hawking pour des fermions dans le cadre de l'effondrement d'uneétoile chargée en rotation. On démontre qu'un observateur localisé loin de l'étoile et au repos par rapport aux coordonnées de BoyerLindquist observe l'émergence d'unétat thermal quand son temps propre tend vers l'infini. On introduit d'abord un modèle de l'effondrement de l'étoile. On suppose que l'espace-tempsà l'extérieur de l'étoile est donné par la métrique de Kerr-Newman. Les hypothèses sur le comportement asymptotique de la surface de l'étoile sont inspirées par le comportement asymptotique de certaines géodésiques de type temps dans la métrique de Kerr-Newman. L'équation de Dirac est alorsécrite en utilisant des coordonnées et une tétrade de Newman-Penrose adaptésà l'effondrement. Ce système de coordonnées et cette tétrade sont basés sur des géodésiques qu'on appelle des géodésiques simples isotropes. La quantification des champs de Dirac dans un espace-temps globalement hyperbolique est décrite. On formule un théorème sur l'effet Hawking dans ce cadre. La preuve du théorème contient une estimation de vitesse minimale pour les champs de Dirac légèrement plus forte que les estimations usuelles ainsi qu'un résultat d'existence et d'unicité pour les solutions d'un problème caractéristique pour les champs de Dirac dans l'espace-temps de Kerr-Newman. Dans un appendice, nous construisons explicitement la compactification de Penrose du bloc I de l'espace-temps de Kerr-Newman qui est basée sur les géodésiques simples isotropes.2000 Mathematics Subject Classification : 35P25, 35Q75, 58J45, 83C47, 8...

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“…This can also be used as a basis for reformulating the scattering theory of massless fields into a Goursat problem based on I (see [1,8,9,15,17]). The asymptotic series of the physical field in the physical space-time translates, in an unphysical conformally rescaled space-time, into the Taylor series of the field off the finite hypersurface I .…”

Section: Introductionmentioning

confidence: 99%

Regularity at space-like and null infinity

Mason¹,

Nicolas²

2008

J. Inst. Math. Jussieu

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We extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino-Schoen/Chrusciel-Delay. We combine conformal techniques and vector field methods: a naive adaptation of the 'Morawetz vector field' to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino-Schoen/Chrusciel-Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows larger classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino-Schoen/Chrusciel-Delay space-times.

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Conformal Scattering and the Goursat Problem

Cited by 34 publications

References 27 publications

Projecting Massive Scalar Fields to Null Infinity

Projecting Massive Scalar Fields to Null Infinity

Creation of fermions by rotating charged black holes

Regularity at space-like and null infinity

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