Partial differential equations in general relativity / Alan D. Rendall.

Format:

Book

Author/Creator:

Rendall, Alan D.

Series:

Oxford graduate texts in mathematics ; 16.

Oxford graduate texts in mathematics ; 16

Language:

English

Subjects (All):

General relativity (Physics)--Mathematics.

General relativity (Physics).

Differential equations, Partial.

Physical Description:

xvi, 279 pages : illustrations ; 24 cm.

Place of Publication:

Oxford ; New York : Oxford University Press, 2008.

Summary:

In recent years the theory of partial differential equations has played an ever more important role in research on general relativity and in this text Alan Rendall covers the key themes-matter models and symmetry classes; the most important classes of partial differential equations, including ordinary differential equations; and material on functional analysis-before discussing a variety of important examples in the field of mathematical relativity including asymptotically flat spacetimes, which are used to describe isolated systems, and spatially compact spacetimes, which are of importance in cosmology.

Including simple explanations of the key terms, many of the basic formulae and equations, a variety of background material, and an extensive list of further reading with each chapter, this text is written in a style accessible to both physics and mathematics graduates.

Contents:

1 Introduction 1

1.1 Physical background 1

1.2 Mathematical background 4

1.3 Structure of the book 6

2 General relativity 8

2.1 Basic concepts 8

2.1.1 Lorentzian algebra 8

2.1.2 Lorentzian geometry 11

2.1.3 Geodesic deviation and singularity theorems 16

2.1.4 Volume and integration 20

2.2 The Einstein equations 21

2.3 The 3+1 decomposition 22

2.4 Conformal rescalings 28

2.5 Covering spaces and foliations 29

2.6 Further reading 29

3 Matter models 30

3.1 Scalar fields 33

3.2 The Maxwell and Yang-Mills equations 39

3.3 Continuum mechanics 41

3.4 Kinetic theory 43

3.5 Other matter models 46

3.6 Further reading 49

4 Symmetry classes 50

4.1 Static and stationary models 52

4.2 Spatially homogeneous models 56

4.3 Surface symmetry 61

4.4 T2 symmetry 63

4.5 U(1) symmetry 68

4.6 Further reading 69

5 Ordinary differential equations 71

5.1 Existence and uniqueness 73

5.2 Dynamical systems 75

5.3 Formal power series solutions and asymptotic expansions 76

5.4 Linearization and the Hartman-Grobman theorem 79

5.5 Examples (Bianchi models) 80

5.5.1 The Wainwright-Hsu system 80

5.5.2 Models of Bianchi types II and VI0 83

5.6 Centre manifolds and the reduction theorem 85

5.7 Further examples 86

5.7.1 Bianchi types II and VI0 revisited 86

5.7.2 The massive scalar field 88

5.7.3 Bianchi type III Einstein-Vlasov 90

5.8 Bifurcation theory 93

5.9 Global existence for homogeneous spacetimes 94

5.10 An application to surface symmetry 98

5.11 Further reading 102

6 Functional analysis 103

6.1 Abstract function spaces 103

6.2 Distributions 106

6.3 Concrete function spaces 108

6.4 Littlewood-Paley theory 115

6.5 Pseudodifferential operators 116

6.6 Further reading 118

7 Elliptic equations 119

7.1 The concept of ellipticity 119

7.2 Boundary value problems 122

7.3 Douglis-Nirenberg ellipticity 123

7.4 Fredholm operators 123

7.5 The Einstein constraints 127

7.6 Further reading 131

8 Hyperbolic equations 132

8.1 The Cauchy problem 132

8.2 Examples of ill-posed problems 134

8.3 Symmetric hyperbolic systems 136

8.4 Strong hyperbolicity 150

8.5 Leray hyperbolicity 152

8.6 The analytic Cauchy problem 154

8.7 Initial boundary value problems 155

8.8 The null condition 158

8.9 Global difficulties 160

8.10 Comparison with parabolic equations 162

8.11 Fuchsian methods 164

8.12 Further reading 169

9 The Cauchy problem for the Einstein equations 170

9.1 Coordinate conditions 170

9.2 The local Cauchy problem 171

9.3 Inclusion of matter 177

9.4 Cosmic censorship 179

9.5 The BKL picture 182

9.6 Further reading 184

10 Global results 186

10.1 Gowdy spacetimes 186

10.2 Stability of de Sitter space 193

10.3 Stability of Minkowski space 199

10.4 Stability of the Milne model 203

10.5 Stability of the flat Bianchi type III model 203

10.6 The Newtonian limit 207

10.7 Further reading 211

11 The Einstein-Vlasov system 213

11.1 Other kinetic equations 213

11.2 Small data global existence 214

11.2.1 Schwarzschild coordinates 214

11.2.2 Maximal-isotropic and double null coordinates 220

11.3 Cosmological solutions 223

11.3.1 Einstein-Vlasov solutions with T2 symmetry 224

11.3.2 T2 symmetry and CMC time 232

11.3.3 Einstein-Vlasov solutions with surface symmetry 241

11.3.4 Spherical symmetry and CMC time 245

11.3.5 Strong cosmic censorship without full asymptotics 245

11.4 Isotropic singularities 248

11.5 Weak cosmic censorship and internal structure of black holes 249

11.6 Further reading 250

12 The Einstein-scalar field system 252

12.1 Asymptotically flat solutions 252

12.2 Weak null singularities 255

12.3 Price's law 258

12.4 Cosmological solutions 260

12.5 Further reading 262.

Notes:

Includes bibliographical references (pages [263]-275) and index.

ISBN:

9780199215409

0199215405

0199215413

9780199215416

OCLC:

226279733