The Cauchy Problem in General Relativity (ESI Lectures in Mathematics and Physics)

Preface......Page 7
Contents......Page 9
Historical overview......Page 15
Some global results, recent developments......Page 16
Purpose......Page 17
PDE theory......Page 20
The Fourier transform and Sobolev spaces......Page 22
Symmetric hyperbolic systems......Page 23
Linear and non-linear wave equations......Page 24
Geometry and global hyperbolicity......Page 26
Uniqueness......Page 27
General relativity......Page 28
Constraint equations and local existence......Page 29
Pathologies, strong cosmic censorship......Page 31
I Background from the theory of partial differential equations......Page 33
Measurability......Page 35
Dualities......Page 36
4 The Fourier transform......Page 40
Schwartz functions, the Fourier transform......Page 41
The Fourier inversion formula......Page 43
5 Sobolev spaces......Page 47
Mollifiers......Page 48
Weak differentiability, W^k,p spaces......Page 49
Temperate distributions, H^s spaces......Page 52
Dualities......Page 57
6 Sobolev embedding......Page 59
Basic inequalities......Page 61
Sobolev embedding......Page 63
Gagliardo–Nirenberg inequalities......Page 64
Grönwall's lemma......Page 71
The basic energy inequality......Page 72
Uniqueness......Page 77
Existence......Page 78
8 Linear wave equations......Page 83
Linear algebra......Page 85
Existence of solutions to linear wave equations......Page 87
Terminology......Page 90
Preliminaries......Page 93
Local existence......Page 97
Continuation criterion, smooth solutions......Page 101
Stability......Page 103
II Background in geometry, global hyperbolicity and uniqueness......Page 107
Manifolds......Page 109
Lorentz metrics......Page 114
Covariant differentiation......Page 115
Coordinate expressions for curvature......Page 119
Basic causality concepts......Page 121
Global hyperbolicity......Page 122
Technical observations......Page 123
Existence of a Cauchy hypersurface......Page 125
Basic constructions......Page 131
Smooth time functions......Page 134
Smooth temporal functions adapted to Cauchy hypersurfaces......Page 138
Auxiliary observations......Page 142
Preliminary technical observations......Page 145
Uniqueness of solutions to tensor wave equations......Page 149
Existence......Page 155
III General relativity......Page 159
Introduction, equations......Page 161
The constraint equations......Page 162
Constraint equations, non-linear scalar field case......Page 165
Gauge choice......Page 166
Initial data......Page 168
Existence of a globally hyperbolic development......Page 170
Two developments are extensions of a common development......Page 172
Sobolev spaces on manifolds......Page 178
Background solutions......Page 180
Cauchy stability in general relativity......Page 181
Background from set theory......Page 190
Existence of a maximal globally hyperbolic development......Page 191
IV Pathologies, strong cosmic censorship......Page 199
Purpose......Page 201
Strong cosmic censorship......Page 202
The cosmological case......Page 203
Genericity......Page 205
Construction of extensions in the unimodular case......Page 206
Sketch of the proof of existence of inequivalent extensions......Page 207
Outline......Page 208
Calculus of variations......Page 210
Constant mean curvature hypersurfaces......Page 214
Conditions ensuring maximality......Page 217
Conditions ensuring inextendibility......Page 218
Unimodular Lie groups......Page 220
Curvature......Page 223
The constraint equations......Page 224
Model metrics......Page 227
Constructing a spacetime......Page 228
Elementary properties of developments......Page 232
Causal geodesic completeness and incompleteness......Page 235
21 Closed universe recollapse......Page 239
Recollapse for an open set of initial data......Page 244
22 Asymptotic behaviour......Page 246
The Wainwright–Hsu variables......Page 247
Relation between the time coordinates......Page 248
Terminology, asymptotic behaviour......Page 249
Limit characterization of the Taub solutions......Page 250
Asymptotic behaviour, Bianchi type I and II......Page 252
Type VI_0 solutions......Page 253
Type VII_0 solutions......Page 254
Bianchi type VIII and IX......Page 255
Curvature blow up......Page 256
Bianchi type I......Page 257
Bianchi VII_0......Page 259
Bianchi type VI_0......Page 260
Bianchi type II, VIII and IX......Page 261
Construction of an embedding......Page 266
Basic properties of the extensions......Page 268
SCC, unimodular vacuum case......Page 272
25 Existence of inequivalent extensions......Page 274
V Appendices......Page 279
Conventions......Page 281
Different notions of measurability......Page 282
Separability......Page 284
Measurability......Page 285
Hilbert spaces......Page 286
Smooth functions with compact support......Page 287
Differentiability in the infinite dimensional case......Page 289
Identities concerning permutation symbols......Page 291
Connection coefficients......Page 293
Ricci curvature......Page 294
Proof of Lemma 20.2......Page 295
Proof of Lemma 22.7......Page 296
Bibliography......Page 299
Index......Page 305