On the topology and future stability of the universe:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Oxford University Press
2013
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Oxford mathematical monographs
Oxford science publications |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 718 S. Ill., graph. Darst. |
| ISBN: | 9780199680290 |
Internformat
MARC
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| 245 | 1 | 0 | |a On the topology and future stability of the universe |c Hans Ringström |
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Datensatz im Suchindex
| _version_ | 1804150617186238464 |
|---|---|
| adam_text | Titel: On the topology and future stability of the universe
Autor: Ringström, Hans
Jahr: 2013
CONTENTS
PART I PROLOGUE
1 Introduction..........................................3
1.1 General remarkson the limitsofobservations 6
1.2 The Standard modeis of the universe 8
1.3 Approximation by matter ofVlasov type 10
2 TheCauchyproblemingeneralrelativity....................... 14
2.1 The initial value problem in general relativity 15
2.2 Spaces of initial data and associated distance concepts 22
2.3 Minimal degreeofregularityensuringlocalexistence 27
2.4 On linearisations 28
3 The topology of the universe............................... 30
3.1 An example of how to characterise topology by geometry 31
3.2 Geometrisation of 3-manifolds 35
3.3 A vacuum conjecture 39
4 Notions ofproximitytospatialhomogeneity and isotropy............. 44
4.1 Almost EGStheorems 45
4.2 On the relation between Solutions with small spatial Variation
and spatially homogeneous Solutions 47
5 Observational support for the Standard model.................... 55
5.1 Using observations to determine the cosmological parameters 55
5.2 Distance measurements 63
5.3 Supernovae observations 66
5.4 Concluding remarks 67
6 Concluding remarks.................................... 68
6.1 On the technical formulation of stability 68
6.2 Notions of proximity to spatial homogeneity and isotropy 74
6.3 Models ofthe universe with arbitrary closed spatial topology 77
6.4 The cosmological principle 79
6.5 Symmetry assumption °1
PART II INTRODUCTORY MATERIAL
7 Mainresults......................................... °*
7.1 Vlasov matter 85
7.2 Scalarfield matter 90
7.3 Theequations 91
7.4 The constraint equations 91
7.5 Previous results 4
X | CONTENTS
7.6 Background Solution and intuition 96
7.7 Drawing global conclusions from local assumptions 102
7.8 Stabilityofspatiallyhomogeneous Solutions 107
7.9 Limitations on the global topology imposed by local observations 109
8 Outiine, general theory of the Einstein-Vlasov system...............117
8.1 Main goals and issues 117
8.2 Background 124
8.3 Function Spaces and estimates 126
8.4 Existence, uniqueness and stability 128
8.5 The Cauchy problem in general relativity 131
9 Outline, main results...................................135
9.1 Spatially homogeneous Solutions 136
9.2 Stability in the n-torus case 142
9.3 Estimates for the Vlasov matter, fiiture global existence and asymptotics 145
9.4 Proofofthe main results 147
10 References to the literature and outlook.......................150
10.1 Local existence 151
10.2 Generalisations 151
10.3 Potential improvements 156
10.4 References to the literature 159
PART III BACKGROUND AND BASIC CONSTRUCTIONS
11 Basic analysis estimates.................................165
11.1 Terminology concerning differentiation and weak derivatives 165
11.2 Weighted Sobolev Spaces 168
11.3 Sobolev spaces on the torus 171
11.4 Sobolev spaces for distribution functions 174
11.5 Sobolev spaces corresponding to a non-integer number of derivatives 178
11.6 Basic analysis estimates 180
11.7 Locallyx-compact support 187
12 Linear algebra.......................................188
12.1 Basic terminology and equalities 189
12.2 Momentum components 190
12.3 Metric-momentum-function spaces 192
12.4 Normals 194
12.5 Projections 195
13 Coordinates........................................199
13.1 The mass shell 199
13.2 Measurability 204
13.3 Measure on the mass shell 204
13.4 Energy and current density induced on a spacelike hypersurface 207
CONTENTS | XI
PART IV FUNCTION SPACES, ESTIMATES
14 Function Spaces for distribution hinctions I: local theory............. 213
14.1 Pathologies 215
14.2 Definition and basic properties 218
14.3 Patching together 227
14.4 Changing coordinates 228
14.5 Restrictions 231
15 Function spaces for distribution hinctions II: the manifold setting....... 233
15.1 Definition 233
15.2 Criteria ensuring membership 235
15.3 Distribution functions on hypersurfaces 236
15.4 Relations 242
15.5 The Vlasov equation 245
16 Main weighted estimate................................. 248
16.1 Basic weighted interpolation estimates 249
16.2 The main weighted estimate 251
17 Concepts of convergence................................ 260
17.1 Mixed interpolation estimates 261
17.2 Equivalence of different concepts of convergence for distribution functions 262
17.3 Weak convergence and strong boundedness imply strong boundedness
ofthelimit 268
17.4 Weak continuity 274
PART V LOCAL THEORY
18 Uniqueness......................................... 279
18.1 The divergence theorem in low regularity 280
18.2 The basic uniqueness lemma 281
18.3 A rough uniqueness result 289
18.4 Geometrie uniqueness 291
19 Local existence....................................... 294
19.1 Terminology 296
19.2 Uniqueness 302
19.3 Solving the Vlasov-type equation on a given background 309
19.4 Relation between different regularity notions 311
19.5 Boundedness estimates 321
19.6 Convergence estimates 328
19.7 Higher order time derivatives 330
19.8 Local existence 333
19.9 Continuation criterion, smooth Solutions 342
20 Stability........................................... 347
20.1 Terminology 347
20.2 Stability 348
xii | CONTENTS
PART VI THE CAUCHY PROBLEM IN GENERAL RELATIVITY
21 The Vlasov equation................................... 359
21.1 The initial value problem for the Vlasov equation 359
21.2 Preservation of regularity 360
22 The initial value problem................................ 369
22.1 Gauge choice 369
22.2 Equations with respect to local coordinates 373
22.3 Local existence 381
22.4 Two developmentsareextensionsofa common development 385
23 Existence ofa maximal globaUyhyperbolic development............. 394
23.1 Outline of the proof 394
23.2 Uniqueness of the MGHD 398
23.3 Elements ofset theory 398
23.4 Partialorderingofisometryclassesofdevelopments 399
23.5 Existence ofa maximal element 400
23.6 Constructing an extension of two developments 405
23.7 Propertiesofboundariesofgloballyhyperbolicregions 409
23.8 Properties of common extensions that are not Hausdorff 410
23.9 Null geodesics in 9 Ü 413
23.10 Existence ofa maximal globallyhyperbolic development 416
24 Cauchy stability...................................... 420
24.1 Terminology 420
24.2 Cauchy stability 421
PART VII SPATIAL HOMOGENEITY
25 Spatiallyhomogeneousmetrics, symmetry reductions............... 435
25.1 Spatiallyhomogeneousmetrics 435
25.2 Symmetry reductions 436
25.3 Bianchi initial data, symmetry reduced equations 439
25.4 Isometries 445
26 Criteria ensuring global existence........................... 447
26.1 Improvementofthecontinuationcriterion 447
26.2 Global existence 450
26.3 Spatially isotropic Solutions arising from initial data on S3 454
27 A potential with a positive non-degenerate local minimum............ 461
27.1 Improved asymptotics 461
27.2 Energy estimates for the Vlasov matter 464
27.3 Asymptotics of the Vlasov matter 472
27.4 Improved asymptotics in the absence of a scalar field 476
28 Approximating perfect fluids with matter of Vlasov type..............478
28.1 The spatially flat Standard fluid modeis 480
28.2 Future global and spatially flat Standard fluid modeis 483
28.3 Matter of Vlasov type 486
CONTENTS
28.4 Asymptotics of Standard Vlasov Solutions 493
28.5 Approximating fluids 497
28.6 Special Solutions 500
28.7 Definition of the approximating family 501
28.8 Approximating the dust 503
28.9 Approximating the radiation 504
28.10 Metrie, first approximation 505
28.11 Difference in the Hubble constants 508
28.12 Comparisonofthepressures 510
28.13 Estimates for the second derivatives 511
PART VIII FUTURE GLOBAL NONLINEAR STABILITY
29 Background material...................................515
29.1 Gauge source functions, initial data 516
29.2 Development of the data 520
29.3 Reformulations of the equations 525
29.4 Preliminarybootstrap assumptions 530
29.5 Energies 532
29.6 Main bootstrap assumptions 536
29.7 Algorithm for estimating the nonlinear terms 538
29.8 Christoffel Symbols 539
30 Estimating the Vlasov contribution to the stress energy tensor..........541
30.1 Statement of the general assumptions 542
30.2 Estimates for rational functions of the momenta 543
30.3 Zeroth order energy estimates for the Vlasov matter 549
30.4 Terminology, background estimates 553
30.5 Estimates for the higher order energies 557
30.6 Basic consequences of the higher order estimates 563
30.7 Estimates ofthe Vlasov contribution to the stress energy tensor 565
31 Global existence......................................568
31.1 Introduction 568
31.2 Statement ofthe assumptions 571
31.3 Estimates 572
31.4 Differential inequalities 577
31.5 Global existence 579
32 Asymptotics........................................592
32.1 Asymptotics for the Vlasov matter given assumptions concerning the metric 593
32.2 Asymptotics in the general case 605
32.3 Asymptotics in the case ofa vanishing scalar field 609
33 Proof ofthe stability results...............................620
33.1 Causal structure 620
33.2 Proof of Theorem 7.16 623
33.3 Stability of spatially homogeneous Solutions 630
XIV CONTENTS
34 Models, fitting the observations, with arbitrary closed spatial
topology.......................................... 634
34.1 Changing time coordinate 636
34.2 Applying the global existence result 640
34.3 Initial data and coordinates 647
34.4 The topology ofthe universe 656
34.5 Proof of Theorem 6.2 676
PART IX APPENDICES
A Examples of pathological behaviour of Solutions to nonlinear wave
equations.......................................... 681
AI Anormtooweakto guarantee local existence 681
A.2 A counterexample to local existence 683
B Quotients and universal covering spaces....................... 684
B.l Simple connectedness 684
C Spatially homogeneous and isotropic metrics.................... 687
D Auxiliary computations in low regularity....................... 689
E The curvature ofleft invariant metrics........................ 695
E. 1 Left invariant metrics on 3-dimensional Lie groups 695
E.2 Scalar curvature 698
F Comments concerning the Einstein-Boltzmann System.............. 700
F. 1 Estimates for the loss term in weighted Spaces 700
F.2 Non-negativity 705
References 707
Index 715
|
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| id | DE-604.BV041194413 |
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| indexdate | 2024-07-10T00:41:46Z |
| institution | BVB |
| isbn | 9780199680290 |
| language | English |
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| physical | XIV, 718 S. Ill., graph. Darst. |
| publishDate | 2013 |
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| publisher | Oxford University Press |
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| spelling | Ringström, Hans 1972- Verfasser (DE-588)1077081626 aut On the topology and future stability of the universe Hans Ringström 1. publ. Oxford Oxford University Press 2013 XIV, 718 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford mathematical monographs Oxford science publications Kosmologie (DE-588)4114294-9 gnd rswk-swf Vlasov-Gleichung (DE-588)4188465-6 gnd rswk-swf Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd rswk-swf Kosmologie (DE-588)4114294-9 s Cauchy-Anfangswertproblem (DE-588)4147404-1 s Vlasov-Gleichung (DE-588)4188465-6 s 1\p DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026169425&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ringström, Hans 1972- On the topology and future stability of the universe Kosmologie (DE-588)4114294-9 gnd Vlasov-Gleichung (DE-588)4188465-6 gnd Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd |
| subject_GND | (DE-588)4114294-9 (DE-588)4188465-6 (DE-588)4147404-1 |
| title | On the topology and future stability of the universe |
| title_auth | On the topology and future stability of the universe |
| title_exact_search | On the topology and future stability of the universe |
| title_full | On the topology and future stability of the universe Hans Ringström |
| title_fullStr | On the topology and future stability of the universe Hans Ringström |
| title_full_unstemmed | On the topology and future stability of the universe Hans Ringström |
| title_short | On the topology and future stability of the universe |
| title_sort | on the topology and future stability of the universe |
| topic | Kosmologie (DE-588)4114294-9 gnd Vlasov-Gleichung (DE-588)4188465-6 gnd Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd |
| topic_facet | Kosmologie Vlasov-Gleichung Cauchy-Anfangswertproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026169425&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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