Ancient dynamics of the Einstein equations and the tumbling universe:
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| Main Author: | |
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| Format: | Thesis Book |
| Language: | English |
| Published: |
2013
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| Subjects: | |
| Online Access: | Volltext Inhaltsverzeichnis |
| Item Description: | Nebentitel: Antike Dynamik der Einsteingleichungen und das taumelnde Universum |
| Physical Description: | VIII, 90 S. graph. Darst. |
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| 245 | 1 | 0 | |a Ancient dynamics of the Einstein equations and the tumbling universe |c Johannes Buchner |
| 246 | 1 | 3 | |a Antike Dynamik der Einsteingleichungen und das taumelnde Universum |
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Record in the Search Index
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| adam_text | CONTENTS
INTRODUCTION V
ACKNOWLEDGEMENTS VIII
CHAPTER 1. GENERAL RELATIVITY 1
1. THE EINSTEIN EQUATIONS 1
2. THE METRIC APPROACH 2
3. THE ORTHONORMAL FRAME APPROACH 2
4. EXAMPLE: THE KASNER SOLUTION 3
5. SYMMETRIES AND SPATIALLY HOMOGENOUS MODELS 3
CHAPTER 2. BIANCHI SPACETIMES - EXISTING RESULTS, CHALLENGES
AND TECHNIQUES 5
1. THE EQUATIONS OF WAINWRIGHT AND HSU 5
1.1. VACUUM MODELS OF BIANCHI CLASS A 6
1.2. THE KASNER MAP 7
1.3. EIGENVALUES IN TERMS OF THE KASNER PARAMETER U 10
2. BIANCHI VI _Y
G
12
3. EXISTING RESULTS IN BIANCHI IX AND DIFFICULTIES IN BIANCHI B 15
4. DYNAMICAL SYSTEMS TECHNIQUES 16
4.1. TOPOLOGICAL EQUIVALENCE 16
4.2. LINEARIZATION 16
4.3. COUNTER-EXAMPLES 17
CHAPTER 3. TAKENS LINEARIZATION THEOREM FOR PARTIALLY
HYPERBOLIC FIXED POINTS 18
1. STERNBERG NON-RESONANCE CONDITIONS 19
2. THE FORMULA FOR A(K)
AND (5{K) 20
3. THE OVERALL STRUCTURE OF THE PROOF 20
4. PROOF OF PROPOSITION 1 21
4.1. GENERAL IDEA OF THE PROOF 21
4.2. USAGE OF STERNBERG NON-RESONANCE CONDITIONS 22
4.3. CHOICES AND DEFINITIONS 22
4.4. LEMMA 2.2. 23
4.5. CONSTRUCTION OF THE REQUIRED COORDINATE SYSTEM 24
II
HTTP://D-NB.INFO/1048229599
5. PROOF OF PROPOSITION 2 25
5.1. GENERAL IDEA OF THE PROOF 25
5.2. JET BUNDLES 25
5.3. METRIC ON THE FIBRE BUNDLE 25
5.4. METRIC ON R
N
26
5.5. LEMMA 3.5 26
5.6. END OF THE PROOF OF PROPOSITION 2 27
5.7. PROPOSITION 2 27
6. PROOF OF PROPOSITION 3 28
6.1. SKETCH OF PROOF 28
6.2. UNDERSTANDING THE FORM OF A(K)
AND /3(K) 28
7. TAKENS LINEARIZATION THEOREM FOR VECTOR FIELDS 29
7.1. PARTIALLY HYPERBOLIC FIXED POINTS 29
7.2. STERNBERG NON-RESONANCE CONDITIONS 30
7.3. THE FORMULA FOR A(K)
AND F3{K) 30
7.4. THE STRUCTURE OF THE PROOF 31
7.5. PROPOSITION 1 31
7.6. PROPOSITION 2 31
8. CONCLUDING REMARKS 32
CHAPTER 4. C
1
- STABLE - MANIFOLDS FOR PERIODIC HETEROCLINIC
CHAINS IN BIANCHI IX 33
1. RESONANCES FOR PERIODIC CHAINS IN BIANCHI IX 37
1.1. INFINITE PERIODIC CONTINUED FRACTIONS 37
1.2. THE CASE OF BIANCHI IX 38
1.3. SNC FOR INFINITE PERIODIC HETEROCLINIC CHAINS 39
1.4. CONCLUSIONS FOR BIANCHI IX 40
1.5. UNIQUENESS OF THE RESONANCE 40
2. CONTINUED FRACTION EXPANSION FOR QUADRATIC IRRATIONALS 41
2.1. CONSTANT CONTINUED FRACTION 41
2.2. 2-PERIODIC CONTINUED FRACTION EXPANSION 42
2.3. 3-PERIODIC CONTINUED FRACTION EXPANSION 42
3. RESULTS ON ADMISSIBILITY OF PERIODIC HETEROCLINIC CHAINS IN
BIANCHIIX 44
3.1. CONSTANT CONTINUED FRACTION DEVELOPMENT 44
3.2. 2-PERIODIC CONTINUED FRACTION DEVELOPMENT 46
3.3. CONTINUED FRACTION DEVELOPMENT WITH HIGHER PERIODS 48
4. DETAILS ON THE PROOF FOR STABLE MANIFOLDS 52
4.1. APPLICATION OF TAKENS THEOREM 52
4.2. LOCAL PASSAGE 54
4.3. GLOBAL PASSAGE 56
4.4. THE RETURN MAP AND THE HYPERBOLIC STRUCTURE 58
111
4.5. (^-STABLE MANIFOLDS FOR C^-HYPERBOLIC SETS 60
4.6. GENERALIZED STABLE MANIFOLD THEOREM BY HIRSCH/PUGH 60
4.7. DIFFERENTIABILITY OF THE STABLE MANIFOLD 62
CHAPTER 5. TAKENS LINEARIZATION AND COMBINED LINEAR LOCAL
PASSAGE AT THE 18-CYCLE IN BIANCHI VL ^J
9
63
1. EIGENVALUES IN TERMS OF THE KASNER PARAMETER U 66
1.1. GENERAL FORMULAS FOR POINTS ON THE KASNER CIRCLE 66
1.2. EIGENVALUES AT THE 3-CYCLE 67
2. THE 3-CYCLE IN BIANCHI VF_
X
J
9
68
2.1. (NON-)RESONANCE AND TAKENS LINEARIZATION 68
2.2. COMBINED LINEAR LOCAL PASSAGES 69
3. THE 18 CYCLES IN BIANCHI VL^JG 71
3.1. POSSIBLE PASSAGES IN BIANCHI VI*_
X
J
9
71
3.2. THE CLASSIC 18-CYCLE 71
3.3. THE ADVANCED 18-CYCLE 72
3.4. (NON-)RESONANCE AND TAKENS LINEARIZATION AT THE 18-CYCLE 72
3.5. COMBINED LINEAR LOCAL PASSAGE AT THE 18-CYCLE 72
4. NUMERICAL SIMULATION 73
CONCLUSION AND OUTLOOK 74
BIBLIOGRAPHY 77
APPENDIX A. SYMBOLIC COMPUTATIONS WITH MATHEMATICA 80
1. RESULTS ON ADMISSIBILITY OF PERIODIC HETEROCLINIC CHAINS IN
BIANCHI IX 80
1.1. CONSTANT CONTINUED FRACTION EXPANSION 80
1.2. 2-PERIODIC CONTINUED FRACTION EXPANSION 82
1.3. 3-PERIODIC CONTINUED FRACTION EXPANSION 83
1.4. PRE-PERIODIC SEQUENCES 84
2. RESULTS ON NON-RESONANCE-CONDITIONS AND CLLP FOR
HETEROCLINIC CYCLES IN BIANCHI VI_
X
J
9
86
2.1. TAKENS LINEARIZATION AT THE BASE POINTS OF THE 3-CYCLE 86
2.2. TAKENS LINEARIZATION AT THE BASE POINTS OF THE 18-CYCLE 86
2.3. CLLP FOR THE CLASSIC 18-CYCLE 88
IV
|
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| spelling | Buchner, Johannes Verfasser aut Ancient dynamics of the Einstein equations and the tumbling universe Johannes Buchner Antike Dynamik der Einsteingleichungen und das taumelnde Universum 2013 VIII, 90 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Nebentitel: Antike Dynamik der Einsteingleichungen und das taumelnde Universum Berlin, Freie Univ., Diss., Nichtlineare Dynamik (DE-588)4126141-0 gnd rswk-swf Kosmologie (DE-588)4114294-9 gnd rswk-swf Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Nichtlineare Dynamik (DE-588)4126141-0 s Kosmologie (DE-588)4114294-9 s Allgemeine Relativitätstheorie (DE-588)4112491-1 s DE-604 Erscheint auch als Online-Ausgabe urn:nbn:de:kobv:188-fudissthesis000000095841-4 http://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000095841 kostenfrei Volltext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026984141&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Buchner, Johannes Ancient dynamics of the Einstein equations and the tumbling universe Nichtlineare Dynamik (DE-588)4126141-0 gnd Kosmologie (DE-588)4114294-9 gnd Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd |
| subject_GND | (DE-588)4126141-0 (DE-588)4114294-9 (DE-588)4112491-1 (DE-588)4113937-9 |
| title | Ancient dynamics of the Einstein equations and the tumbling universe |
| title_alt | Antike Dynamik der Einsteingleichungen und das taumelnde Universum |
| title_auth | Ancient dynamics of the Einstein equations and the tumbling universe |
| title_exact_search | Ancient dynamics of the Einstein equations and the tumbling universe |
| title_full | Ancient dynamics of the Einstein equations and the tumbling universe Johannes Buchner |
| title_fullStr | Ancient dynamics of the Einstein equations and the tumbling universe Johannes Buchner |
| title_full_unstemmed | Ancient dynamics of the Einstein equations and the tumbling universe Johannes Buchner |
| title_short | Ancient dynamics of the Einstein equations and the tumbling universe |
| title_sort | ancient dynamics of the einstein equations and the tumbling universe |
| topic | Nichtlineare Dynamik (DE-588)4126141-0 gnd Kosmologie (DE-588)4114294-9 gnd Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd |
| topic_facet | Nichtlineare Dynamik Kosmologie Allgemeine Relativitätstheorie Hochschulschrift |
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