The equation for the rescaled metric ˆg and the commutations

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The Newman-Penrose tetrad and spin coefficients

The Goursat problem and the conformal scattering operator

Non-trivial solutions of the constraint system on C∞0 (0)

Neighbourhood of spacelike infinity

Table of contents :

1 Introduction 
1.1 Une courte introduction à la relativité générale
1.2 L’approche conforme de l’analyse asymptotique
1.2.1 Peeling
1.2.2 Diffusion conforme
1.3 Contenu de la thèse
2 Peeling for scalar fields on Kerr spacetime 
2.1 Introduction
2.2 Geometric setting
2.2.1 The Kerr metric
2.2.2 Star-Kerr and Kerr-star coordinates
2.2.3 Kruskal-Boyer-Lindquist coordinates
2.2.4 Penrose compactification
2.2.5 Neighbourhood of spacelike infinity
2.2.6 The Morawetz vector field
2.3 The linear scalar fields
2.3.1 Wave equation for rescaled metric ˆg
2.3.2 Commutation with vector fields
2.3.3 Approximate conservation laws
2.3.4 The energy density
2.3.5 The energy estimates and the peeling
2.3.6 Interpretation
2.4 The nonlinear case
2.4.1 The equation for the rescaled metric ˆg and the commutations .
2.4.2 Approximate conservation laws
2.4.3 The energies and peeling
2.4.4 Interpretation
2.5 Appendix
2.5.1 The divergence theorem
2.5.2 Proof of Lemma 2.3.1
2.5.3 The dominant energy condition
2.5.4 Two basic inequalities
3 Peeling for Dirac field on Kerr spacetime 
3.1 Introduction
3.2 Geometric setting
3.2.1 Spin structure for block I
3.2.2 The Newman-Penrose tetrad and spin coefficients
3.2.3 Neighbourhood of spacelike infinity
3.3 The Dirac field
3.3.1 Dirac’s and Weyl’s equations
3.3.2 Rescaling of the Weyl equation
3.3.3 Energy
3.4 Approach using partial derivatives
3.4.1 The conservation laws of Dirac’s system
3.4.2 Energy estimates and peeling
3.5 Approach using covariant derivatives
3.5.1 Curvature spinors
3.5.2 Commutation of covariant derivatives
3.5.3 Energy estimates and peeling
3.6 Interpretation
3.7 Appendix
3.7.1 Compacted spin coefficient formalism
3.7.2 Proof of Lemma 3.4.1
3.7.3 Proof of Lemma 3.5.2
3.7.4 Proof of Lemma 3.5.4
3.7.5 Some calculations
4 Conformal scattering on Minkowski spacetime 
4.1 Introduction
4.2 Geometric setting
4.2.1 The full conformal compactification
4.2.2 A partial conformal compactification
4.3 The spin n/2 zero rest-mass field
4.4 The Cauchy problem
4.5 Energies
4.6 The Goursat problem and the conformal scattering operator
4.7 Appendix
4.7.1 Spinor form of commutators
4.7.2 Non-trivial solutions of the constraint system on C∞0 (0)
4.7.3 Generalisation of L.Hörmander’s result
4.7.4 Detailed calculations for the Goursat problem

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