Problème de Cauchy caractéristique et scattering conforme en relativité générale

Abstract : This work presents two aspects of the characteristic Cauchy problem in general relativity. On the one hand, an integral formula for the characteristic Cauchy problem for the Dirac equation on a curved space-time is derived. This generalizes the work of Penrose in the 60's. The functional framework is adapted, so that the algebraic structures on spinors can be brought to distributions on spinors. This gives an integral formula which is simplified using the Geroch-Held-Penrose formalism. Penrose's formula on the Minkowski space-time is recovered for arbitrary spin. On the other hand, a conformal scattering theory for a conformally invariant nonlinear wave equation is established. Using a conformal rescaling, the space-time is completed with two null hypersurfaces representing respectively the past and future endpoints of null geodesics. The asymptotic behaviour of fields is then obtained by considering the traces of solutions of the rescaled equations on these hypersurfaces. The inversibility of these trace operators is obtained by solving a characteristic Cauchy problem and the conformal scattering operator is obtained by composing these trace operators.
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Contributor : Jérémie Joudioux <>
Submitted on : Tuesday, September 14, 2010 - 11:49:10 AM
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  • HAL Id : tel-00517339, version 1



Jérémie Joudioux. Problème de Cauchy caractéristique et scattering conforme en relativité générale. Mathématiques [math]. Université de Bretagne occidentale - Brest, 2010. Français. ⟨tel-00517339⟩



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