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E-raamat: Asymptotic Analysis in General Relativity

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This volume compiles notes from courses given at the summer school on asymptotic analysis in general relativity, held at the Institut Fourier in Grenoble, France. It provides an up-to-date panorama of modern techniques in the asymptotic analysis of classical and quantum fields in general relativity.

This volume compiles notes from four mini courses given at the summer school on asymptotic analysis in general relativity, held at the Institut Fourier in Grenoble, France. It contains an up-to-date panorama of modern techniques in the asymptotic analysis of classical and quantum fields in general relativity. Accessible to graduate students, these notes gather results that were not previously available in textbooks or monographs and will be of wider interest to researchers in general relativity. The topics of these mini courses are: the geometry of black hole spacetimes; an introduction to quantum field theory on curved spacetimes; conformal geometry and tractor calculus; and microlocal analysis for wave propagation.

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This book provides a panorama of modern techniques in the asymptotic analysis of classical and quantum fields in general relativity.
1 Introduction to Modern Methods for Classical and Quantum Fields in General Relativity
1(8)
Thierry Daude
Dietrich Hafner
Jean-Philippe Nicolas
1.1 Geometry of Black Hole Spacetimes
2(1)
1.2 Quantum Field Theory on Curved Spacetimes
3(1)
1.3 Conformal Geometry and Conformal Tractor Calculus
4(1)
1.4 A Minicourse in Microlocal Analysis and Wave Propagation
5(4)
References
6(3)
2 Geometry of Black Hole Spacetimes
9(77)
Lars Andersson
Thomas Backdahl
Pieter Blue
2.1 Introduction
9(4)
2.2 Background
13(15)
2.3 Black Holes
28(15)
2.4 Spin Geometry
43(10)
2.5 The Kerr Spacetime
53(2)
2.6 Monotonicity and Dispersion
55(10)
2.7 Symmetry Operators
65(5)
2.8 Conservation Laws for the Teukolsky System
70(5)
2.9 A Morawetz Estimate for the Maxwell Field on Schwarzschild
75(11)
References
79(7)
3 An Introduction to Conformal Geometry and Tractor Calculus, with a view to Applications in General Relativity
86(85)
Sean N. Curry
A. Rod Gover
3.1 Introduction
86(5)
3.2 Lecture 1: Riemannian Invariants and Invariant Operators
91(3)
3.3 Lecture 2: Conformal Transformations and Conformal Covariance
94(10)
3.4 Lecture 3: Prolongation and the Tractor Connection
104(16)
3.5 Lecture 4: The Tractor Curvature, Conformal Invariants and Invariant Operators
120(8)
3.6 Lecture 5: Conformal Compactification of Pseudo-Riemannian Manifolds
128(17)
3.7 Lecture 6: Conformal Hypersurfaces
145(6)
3.8 Lecture 7: Geometry of Conformal Infinity
151(3)
3.9 Lecture 8: Boundary Calculus and Asymptotic Analysis
154(17)
Appendix: Conformal Killing Vector Fields and Adjoint Tractors
160(8)
References
168(3)
4 An Introduction to Quantum Field Theory on Curved Spacetimes
171(48)
Christian Gerard
4.1 Introduction
171(4)
4.2 A Quick Introduction to Quantum Mechanics
175(3)
4.3 Notation
178(1)
4.4 CCR and CAR Algebras
179(4)
4.5 States on CCR/CAR Algebras
183(7)
4.6 Lorentzian Manifolds
190(2)
4.7 Klein--Gordon Fields on Lorentzian Manifolds
192(5)
4.8 Free Dirac Fields on Lorentzian Manifolds
197(4)
4.9 Microlocal Analysis of Klein--Gordon Quasi-Free States
201(8)
4.10 Construction of Hadamard States
209(10)
References
217(2)
5 A Minicourse on Microlocal Analysis for Wave Propagation
219
Andras Vasy
5.1 Introduction
219(2)
5.2 The Overview
221(9)
5.3 The Basics of Microlocal Analysis
230(55)
5.4 Propagation Phenomena
285(30)
5.5 Conformally Compact Spaces
315(34)
5.6 Microlocal Analysis in the b-Setting
349
References
371
Thierry Daudé is a Lecturer in Mathematics at Université de Cergy-Pontoise. Dietrich Häfner is a Professor of Mathematics at Université Grenoble Alpes. Jean-Philippe Nicolas is a Professor of Mathematics at Université de Bretagne Occidentale, Brest.
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