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Scattering theory for Dirac fields in various spacetimes of the General Relativity

Abstract : This thesis is devoted to the study of scattering theory for Dirac
fields in various spacetimes of General Relativity. The
time-dependent methods developed by Enss, Sigal, Soffer, Graf,
Derezi\'nski and Gérard are the backbone of this
work. These methods are based on propagation estimates, such as minimal
velocity estimates (obtained using a Mourre theory) that are a weak
form of the Huygens principle, and on the
study of natural and intuitive asymptotic observables such as the
asymptotic velocity operators. They turn out to be extremely
convenient when studying relativistic equations since they are
intimately related to the fundamental structure of General
Relativity: the light cone. We first test these methods studying the
asymptotic behaviour of Dirac fields perturbed by long-range potentials
in flat spacetime and we prove the existence and asymptotic
completeness of modified wave operators. We then consider more
complicated geometrical situations: the propagation of Dirac
fields in the exterior region of a Reissner-Nordström black hole
(spherically symmetric) and a Kerr-Newman black hole (in rotation) from
the point of view of observers static at infinity. The peculiarity of
such situations is that the observer perceives two asymptotic regions
(the horizon of the black hole and spacelike infinity) having very
different geometrical structures; this leads to the existence of two
distinct scattering channels. In the case of spherically symmetric
black holes, we can use a decomposition into spherical harmonics in
order to obtain a Dirac equation with potentials on one dimensional
flat spacetime. The main difficulty in the Kerr-Newman case comes from
the absence of spherical symmetry. In both cases, we prove the
existence and asymptotic completeness of (modified at infinity) wave operators by means of the
previous time-dependent methods.
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Contributor : Thierry Daude <>
Submitted on : Friday, March 17, 2006 - 5:46:50 PM
Last modification on : Friday, March 17, 2006 - 6:36:32 PM
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  • HAL Id : tel-00011974, version 1



Thierry Daude. Scattering theory for Dirac fields in various spacetimes of the General Relativity. Mathematics [math]. Université Sciences et Technologies - Bordeaux I, 2004. English. ⟨tel-00011974⟩



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