The Cauchy problem in general relativity:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Zürich
European Mathematical Society
2009
|
| Schriftenreihe: | ESI lectures in mathematics and physics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XIII, 294 S. 240 mm x 170 mm |
| ISBN: | 9783037190531 |
Internformat
MARC
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| 100 | 1 | |a Ringström, Hans |d 1972- |e Verfasser |0 (DE-588)1077081626 |4 aut | |
| 245 | 1 | 0 | |a The Cauchy problem in general relativity |c Hans Ringström |
| 264 | 1 | |a Zürich |b European Mathematical Society |c 2009 | |
| 300 | |a XIII, 294 S. |c 240 mm x 170 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a ESI lectures in mathematics and physics | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Cauchy problem | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-017629420 | |
Datensatz im Suchindex
| _version_ | 1805092528266936320 |
|---|---|
| adam_text |
Contents
Preface
vii
1
Introduction
1
1.1
Historical overview
. 1
1.2
Some global results, recent developments
. 2
1.3
Purpose
. 3
2
Outline
6
2.1
PDE theory
. 6
2.1.1
The Fourier transform and Sobolev spaces
. 8
2.1.2
Symmetric hyperbolic systems
. 9
2.1.3
Linear and non-linear wave equations
. 10
2.2
Geometry, global
hyperbolicky
and uniqueness
. 12
2.2.1
Geometry and global hyperbolicity
. 12
2.2.2
Uniqueness
. 13
2.3
General relativity
. 14
2.3.1
Constraint equations and local existence
. 15
2.3.2
Cauchy stability
. 17
2.3.3
Existence of a maximal globally hyperbolic development
. . 17
2.4
Pathologies, strong cosmic censorship
. 17
Part I Background from the theory of partial differential equations
19
3
Functional analysis
21
3.1
Measurability
. 21
3.2
Dualities
. 22
4
The Fourier transform
26
4.1
Schwartz functions, the Fourier transform
. 27
4.2
The Fourier inversion formula
. 29
5
Sobolev spaces
33
5.1
Mollifiers
. 34
5.2
Weak differentiability, Wk'p spaces
. 35
5.3
Temperate distributions, H" spaces
. 38
5.4
Dualities
. 43
x
Contents
6
Sobolev embedding
45
6.1
Basic inequalities
. 47
6.2
Sobolev embedding
. 49
6.3
Gagliardo-Nirenberg inequalities
. 50
7
Symmetric hyperbolic systems
57
7.1 Grönwall's
lemma
. 57
7.2
The basic energy inequality
. 58
7.3
Uniqueness
. 63
7.4
Existence
. 64
8
Linear wave equations
69
8.1
Linear algebra
. 71
8.2
Existence of solutions to linear wave equations
. 73
9
Local existence, non-linear wave equations
76
9.1
Terminology
. 76
9.2
Preliminaries
. 79
9.3
Local existence
. 83
9.4
Continuation criterion, smooth solutions
. 87
9.5
Stability
. 89
Part II Background in geometry, global hyperbolicity and uniqueness
93
10
Basic
Lorentz
geometry
95
10.1
Manifolds
. 95
10.2
Lorentz
geometry
. 100
10.2.1
Lorentz
metrics
. 100
10.2.2
Covariant differentiation
. 101
10.2.3
Coordinate expressions for curvature
. 105
10.2.4
Basic causality concepts
. 107
10.2.5
Geodesies
. 108
10.2.6
Global hyperbolicity
. 108
10.2.7
Cauchy surfaces
. 109
10.2.8
Technical observations
. 109
11
Characterizations of global hyperbolicity
111
11.1
Existence of a Cauchy hypersurface
.
Ill
11.2
Basic constructions
. 117
11.3
Smooth time functions
. 120
11.4
Smooth temporal functions adapted to Cauchy hypersurfaces
. 124
11.5
Auxiliary observations
. 128
Contents
xi
12
Uniqueness of solutions to linear wave equations
131
12.1
Preliminary technical observations
. 131
12.2
Uniqueness of solutions to tensor wave equations
. 135
12.3
Existence
. 141
Part III General relativity
145
13
The constraint equations
147
13.1
Introduction, equations
. 147
13.2
The constraint equations
. 149
13.3
Constraint equations, non-linear scalar field case
. 151
14
Local existence
152
14.1
Gauge choice
. 152
14.2
Initial data
. 154
14.3
Existence of a globally hyperbolic development
. 156
14.4
Two developments are extensions of a common development
. 158
15
Cauchy stability
164
15.1
Sobolev spaces on manifolds
. 164
15.2
Background solutions
. 166
15.3
Cauchy stability in general relativity
. 167
16
Existence of a maximal globally hyperbolic development
176
16.1
Background from set theory
. 176
16.2
Existence of a maximal globally hyperbolic development
. 177
Part IV Pathologies, strong cosmic censorship
185
17
Preliminaries
187
17.1
Purpose
. 187
17.2
Strong cosmic censorship
. 188
17.2.1
The asymptotically flat case
. 189
17.2.2
The cosmological case
. 189
17.2.3
Genericky
. 191
17.3
Construction of extensions in the unimodular case
. 192
17.4
Sketch of the proof of existence of inequivalent extensions
. 193
17.5
Outline
. 194
18
Constant mean curvature
196
18.1
Calculus of variations
. 196
18.2
Constant mean curvature hypersurfaces
. 200
18.3
Conditions ensuring maximality
. 203
xii
Contents
18.4
Conditions ensuring inextendibility
. 204
19
Initial data
206
19.1
Unimodular Lie groups
. 206
19.2
Curvature
. 209
19.3
The constraint equations
. 210
20
Einstein's vacuum equations
213
20.1
Model metrics
. 213
20.2
Constructing a spacetime
. 214
20.3
Elementary properties of developments
. 218
20.4
Causal geodesic completeness and incompleteness
. 221
21
Closed universe recollapse
225
21.1
Recollapse for an open set of initial data
. 230
22
Asymptotic behaviour
232
22.1
The Wainwright-Hsu variables
. 233
22.2
Relation between the time coordinates
. 234
22.3
Terminology, asymptotic behaviour
. 235
22.4
Criteria ensuring curvature blow up
. 236
22.5
Limit characterization of the
Taub
solutions
. 236
22.6
Asymptotic behaviour,
Bianchi
type I and II
. 238
22.7
Type VI0 solutions
. 239
22.8
TypeVIIo solutions
. 240
22.9
Bianchi
type
VIII
and IX
. 241
22.10
Curvature blow up
. 242
23
LRS
Bianchi
class A solutions
243
23.1
Bianchi
type I
. 243
23.2
BianchiVIIo
. 245
23.3
Bianchi
type VI0
. 246
23.4
Bianchi
type II,
VIII
and IX
. 247
24
Existence of extensions
252
24.1
Construction of an embedding
. 252
24.2
Basic properties of the extensions
. 254
24.3
SCC, unimodular vacuum case
. 258
25
Existence of inequivalent extensions
260
Contents
xiii
Part V Appendices
265
Appendix A
267
A.1 Conventions
. 267
A.2 Different notions of measurability
. 268
A.3 Separability
. 270
A.4 Measurability
. 271
A.5 Hubert spaces
. 272
A.6 Smooth functions with compact support
. 273
A.7 Differentiability in the infinite dimensional case
. 275
Appendix
В
277
B.I Identities concerning permutation symbols
. 277
B.2 Proof of Lemma
20.1. 279
B.2.1 Connection coefficients
. 279
B.2.2 Commutators
. 280
B.2.3
Ricci
curvature
. 280
B.3 Proof of Lemma
20.2. 281
B.4 Proof of Lemma
22.7. 282
Bibliography
285
Index
291 |
| any_adam_object | 1 |
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| discipline | Physik Mathematik |
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| spelling | Ringström, Hans 1972- Verfasser (DE-588)1077081626 aut The Cauchy problem in general relativity Hans Ringström Zürich European Mathematical Society 2009 XIII, 294 S. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier ESI lectures in mathematics and physics Mathematik Cauchy problem General relativity (Physics) Mathematics Relativität (DE-588)4191596-3 gnd rswk-swf Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd rswk-swf Cauchy-Anfangswertproblem (DE-588)4147404-1 s Relativität (DE-588)4191596-3 s DE-604 Erscheint auch als Ringström, Hans, 1972- The Cauchy problem in general relativity Online-Ausgabe 978-3-03719-553-6 (DE-604)BV036751443 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3320235&prov=M&dok_var=1&dok_ext=htm Inhaltstext text/html http://www.ems-ph.org/books/book.php?proj_nr=99 Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017629420&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ringström, Hans 1972- The Cauchy problem in general relativity Mathematik Cauchy problem General relativity (Physics) Mathematics Relativität (DE-588)4191596-3 gnd Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd |
| subject_GND | (DE-588)4191596-3 (DE-588)4147404-1 |
| title | The Cauchy problem in general relativity |
| title_auth | The Cauchy problem in general relativity |
| title_exact_search | The Cauchy problem in general relativity |
| title_full | The Cauchy problem in general relativity Hans Ringström |
| title_fullStr | The Cauchy problem in general relativity Hans Ringström |
| title_full_unstemmed | The Cauchy problem in general relativity Hans Ringström |
| title_short | The Cauchy problem in general relativity |
| title_sort | the cauchy problem in general relativity |
| topic | Mathematik Cauchy problem General relativity (Physics) Mathematics Relativität (DE-588)4191596-3 gnd Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd |
| topic_facet | Mathematik Cauchy problem General relativity (Physics) Mathematics Relativität Cauchy-Anfangswertproblem |
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