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Conformal Methods in General Relativity

Book Description

This book offers a systematic exposition of conformal methods and how they can be used to study the global properties of solutions to the equations of Einstein's theory of gravity. It shows that combining these ideas with differential geometry can elucidate the existence and stability of the basic solutions of the theory. Introducing the differential geometric, spinorial and PDE background required to gain a deep understanding of conformal methods, this text provides an accessible account of key results in mathematical relativity over the last thirty years, including the stability of de Sitter and Minkowski spacetimes. For graduate students and researchers, this self-contained account includes useful visual models to help the reader grasp abstract concepts and a list of further reading, making this the perfect reference companion on the topic.

Table of Contents

  1. Coverpage
  2. Halftitle page
  3. Series page
  4. Title page
  5. Copyright
  6. Dedication
  7. Contents
  8. Preface
  9. Acknowledgements
  10. List of Symbols
  11. 1 Introduction
    1. 1.1 On the Einstein field equations
    2. 1.2 Exact solutions
    3. 1.3 The Cauchy problem in general relativity
    4. 1.4 Conformal geometry and general relativity
    5. 1.5 Existence of asymptotically simple spacetimes
    6. 1.6 Perspectives
    7. 1.7 Structure of this book
  12. Part I Geometric tools
    1. 2 Differential geometry
      1. 2.1 Manifolds
      2. 2.2 Vectors and tensors on a manifold
      3. 2.3 Maps between manifolds
      4. 2.4 Connections, torsion and curvature
      5. 2.5 Metric tensors
      6. 2.6 Frame formalisms
      7. 2.7 Congruences and submanifolds
      8. 2.8 Further reading
    2. 3 Spacetime spinors
      1. 3.1 Algebra of 2-spinors
      2. 3.2 Calculus of spacetime spinors
      3. 3.3 Global considerations
      4. 3.4 Further reading
      5. Appendix: the Newman-Penrose formalism
    3. 4 Space spinors
      1. 4.1 Hermitian inner products and 2-spinors
      2. 4.2 The space spinor formalism
      3. 4.3 Calculus of space spinors
      4. 4.4 Further reading
    4. 5 Conformal geometry
      1. 5.1 Basic concepts of conformal geometry
      2. 5.2 Conformal transformation formulae
      3. 5.3 Weyl connections
      4. 5.4 Spinorial expressions
      5. 5.5 Conformal geodesics
      6. 5.6 Further reading
  13. Part II General relativity and conformal geometry
    1. 6 Conformal extensions of exact solutions
      1. 6.1 Preliminaries
      2. 6.2 The Minkowski spacetime
      3. 6.3 The de Sitter spacetime
      4. 6.4 The anti-de Sitter spacetime
      5. 6.5 Conformal extensions of static and stationary black hole spacetimes
      6. 6.6 Further reading
    2. 7 Asymptotic simplicity
      1. 7.1 Basic definitions
      2. 7.2 Other related definitions
      3. 7.3 Penrose’s proposal
      4. 7.4 Further reading
    3. 8 The conformal Einstein field equations
      1. 8.1 A singular equation for the conformal metric
      2. 8.2 The metric regular conformal field equations
      3. 8.3 Frame and spinorial formulation of the conformal field equations
      4. 8.4 The extended conformal Einstein field equations
      5. 8.5 Further reading
    4. 9 Matter models
      1. 9.1 General properties of the conformal treatment of matter models
      2. 9.2 The Maxwell field
      3. 9.3 The scalar field
      4. 9.4 Perfect fluids
      5. 9.5 Further reading
    5. 10 Asymptotics
      1. 10.1 Basic set up: general structure of the conformal boundary
      2. 10.2 Peeling properties
      3. 10.3 The Newman-Penrose gauge
      4. 10.4 Other aspects of asymptotics
      5. 10.5 Further reading
      6. Appendix: spin-weighted functions
  14. Part III Methods of the theory of partial differential equations
    1. 11 The conformal constraint equations
      1. 11.1 General setting and basic formulae
      2. 11.2 Basic notions of elliptic equations
      3. 11.3 The Hamiltonian and momentum constraints
      4. 11.4 The conformal constraint equations
      5. 11.5 The constraints on compact manifolds
      6. 11.6 Asymptotically Euclidean manifolds
      7. 11.7 Hyperboloidal manifolds
      8. 11.8 Other methods for solving the constraint equations
      9. 11.9 Further reading
      10. Appendix: some results of analysis
    2. 12 Methods of the theory of hyperbolic differential equations
      1. 12.1 Basic notions
      2. 12.2 Uniqueness and domains of dependence
      3. 12.3 Local existence results for symmetric hyperbolic systems
      4. 12.4 Local existence for boundary value problems
      5. 12.5 Local existence for characteristic initial value problems
      6. 12.6 Concluding remarks
      7. 12.7 Further reading
      8. Appendix
    3. 13 Hyperbolic reductions
      1. 13.1 A model problem: the Maxwell equations on a fixed background
      2. 13.2 Hyperbolic reductions using gauge source functions
      3. 13.3 The subsidiary equations for the standard conformal field equations
      4. 13.4 Hyperbolic reductions using conformal Gaussian systems
      5. 13.5 Other hyperbolic reduction strategies
      6. 13.6 Further reading
      7. Appendix A.1: the reduced Einstein field equations
      8. Appendix A.2: differential forms
    4. 14 Causality and the Cauchy problem in general relativity
      1. 14.1 Basic elements of Lorentzian causality
      2. 14.2 PDE causality versus Lorentzian causality
      3. 14.3 Cauchy developments and maximal Cauchy developments
      4. 14.4 Stability of solutions
      5. 14.5 Causality and conformal geometry
      6. 14.6 Further reading
  15. Part IV Applications
    1. 15 De Sitter-like spacetimes
      1. 15.1 The de Sitter spacetime as a solution to the conformal field equations
      2. 15.2 Perturbations of initial data for the de Sitter spacetime
      3. 15.3 Global existence and stability using gauge source functions
      4. 15.4 Global existence and stability using conformal Gaussian systems
      5. 15.5 Extensions
      6. 15.6 Further reading
    2. 16 Minkowski-like spacetimes
      1. 16.1 The Minkowski spacetime and the conformal field equations
      2. 16.2 Perturbations of hyperboloidal data for the Minkowski spacetime
      3. 16.3 A priori structure of the conformal boundary
      4. 16.4 The proof of the main existence and stability result
      5. 16.5 Extensions and further reading
    3. 17 Anti-de Sitter-like spacetimes
      1. 17.1 General properties of anti-de Sitter-like spacetimes
      2. 17.2 The formulation of an initial boundary value problem
      3. 17.3 Covariant formulation of the boundary conditions
      4. 17.4 Other approaches to the construction of anti-de Sitter-like spacetimes
      5. 17.5 Further reading
    4. 18 Characteristic problems for the conformal field equations
      1. 18.1 Geometric and gauge aspects of the standard characteristic initial value problem
      2. 18.2 The conformal evolution equations in the standard characteristic initial value problem
      3. 18.3 A local existence result for characteristic problems
      4. 18.4 The asymptotic characteristic problem on a cone
      5. 18.5 Further reading
    5. 19 Static solutions
      1. 19.1 The static field equations
      2. 19.2 Analyticity at infinity
      3. 19.3 A regularity condition
      4. 19.4 Multipole moments
      5. 19.5 Uniqueness of the conformal structure of static metrics
      6. 19.6 Characterisation of static initial data
      7. 19.7 Further reading
      8. Appendix 1: Hölder conditions
      9. Appendix 2: the Cauchy-Kowalewskaya theorem
    6. 20 Spatial infinity
      1. 20.1 Cauchy data for the conformal field equations near spatial infinity
      2. 20.2 Massless and purely radiative spacetimes
      3. 20.3 A regular initial value problem at spatial infinity
      4. 20.4 Spatial infinity and peeling
      5. 20.5 Existence of asymptotically simple spacetimes
      6. 20.6 Obstructions to the smoothness of null infinity
      7. 20.7 Further reading
      8. Appendix: properties of functions on the complex null cone
    7. 21 Perspectives
      1. 21.1 Stability of cosmological models
      2. 21.2 Stability of black hole spacetimes
      3. 21.3 Conformal methods and numerics
      4. 21.4 Computer algebra
      5. 21.5 Concluding remarks
  16. References
  17. Index