The evolution problem in general relativity:
"The global aspects of the problem of evolution equations in general relativity are examined. Central to the work is a revisit of the proof of the global stability of Minkowski space, as presented by Christodoulou and Klainerman (1993). The focus, therefore, is on a new self-contained proof of...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2003
|
| Series: | Progress in mathematical physics
25 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "The global aspects of the problem of evolution equations in general relativity are examined. Central to the work is a revisit of the proof of the global stability of Minkowski space, as presented by Christodoulou and Klainerman (1993). The focus, therefore, is on a new self-contained proof of the main part of that result which concerns the full solution of the radiation problem in vacuum for arbitrary asymptotic flat initial data sets. While technical motivation is clearly and systematically provided for this proof, many important related concepts and results, some well established, others new, unfold along the way." "A comprehensive bibliography and index complete this important monograph, aimed at researchers and graduate students in mathematics, mathematical physics, and physics in the area of general relativity."--BOOK JACKET. |
| Physical Description: | XII, 385 S. |
| ISBN: | 0817642544 3764342544 |
Staff View
MARC
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| 100 | 1 | |a Klainerman, Sergiu |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The evolution problem in general relativity |c Sergiu Klainerman ; Francesco Nicolò |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2003 | |
| 300 | |a XII, 385 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Progress in mathematical physics |v 25 | |
| 520 | 1 | |a "The global aspects of the problem of evolution equations in general relativity are examined. Central to the work is a revisit of the proof of the global stability of Minkowski space, as presented by Christodoulou and Klainerman (1993). The focus, therefore, is on a new self-contained proof of the main part of that result which concerns the full solution of the radiation problem in vacuum for arbitrary asymptotic flat initial data sets. While technical motivation is clearly and systematically provided for this proof, many important related concepts and results, some well established, others new, unfold along the way." "A comprehensive bibliography and index complete this important monograph, aimed at researchers and graduate students in mathematics, mathematical physics, and physics in the area of general relativity."--BOOK JACKET. | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Evolution equations | |
| 650 | 4 | |a General relativity (Physics) | |
| 650 | 4 | |a Mathematical physics | |
| 650 | 0 | 7 | |a Evolutionsgleichung |0 (DE-588)4129061-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Allgemeine Relativitätstheorie |0 (DE-588)4112491-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Allgemeine Relativitätstheorie |0 (DE-588)4112491-1 |D s |
| 689 | 0 | 1 | |a Evolutionsgleichung |0 (DE-588)4129061-6 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Nicolò, Francesco |e Verfasser |4 aut | |
| 830 | 0 | |a Progress in mathematical physics |v 25 |w (DE-604)BV013823265 |9 25 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-009887172 | ||
Record in the Search Index
| _version_ | 1804129318377357312 |
|---|---|
| adam_text | Contents
Preface
χι
1
Introduction
1
1.1
Generalities about
Lorentz
manifolds
.................... 1
1.1.1
Lorentz
metric, vector and tensor fields,
covariant derivative, Lie derivative
................. 1
1.1.2
Riemann curvature tensor,
Ricci
tensor,
Bianchi
identities
.... 7
1.1.3
Isometries and
conformai isometries,
Killing and
conformai
Killing vector fields
......................... 9
1.2
The Einstein equations
........................... 12
1.2.1
The initial value problem, initial data sets and constraint equations
13
1.3
Local existence for Einstein s vacuum equations
.............. 14
1.3.1
Reduction to the nonlinear wave equations
............ 14
1.3.2
Local existence for the Einstein vacuum equations
using wave coordinates
....................... 17
1.3.3
General foliations of the Einstein spacetime
............ 19
1.3.4
Maximal foliations of Einstein spacetime
............. 21
1.3.5
A proof of local existence using the maximal foliation
...... 21
1.3.6
Maximal Cauchy developments
.................. 23
1.3.7
Hawking-Penrose singularities, the cosmic censorship
...... 23
1.3.8
The
С
-K
Theorem and the Main Theorem
............. 25
1.4
Appendix
.................................. 27
2
Analytic Methods in the Study of the Initial Value Problem
31
2.1
Local and global existence for systems of nonlinear wave equations
... 31
2.1.1
Local existence for nonlinear wave equations
........... 31
vi
Contents
2.1.2
Global
existence
for nonlinear wave equations
.......... 37
2.2
Weyl fields and
Bianchi
equations in Minkowski spacetime
........ 41
2.2.1
Asymptotic behavior of the Weyl fields in Minkowski spacetime
. 43
2.3
Global nonlinear stability of Minkowski spacetime
............ 52
2.4
Stractureof thework
............................ 53
3
Definitions and Results
55
3.1
Connection coefficients
........................... 55
3.1.1
Null second fundamental forms and torsion of a spacelike 2-surface
55
3.1.2
Null decomposition of the curvature tensor
............ 60
3.1.3
Null structure equations of a 2-surface
S
.............. 62
3.1.4
Integrable
5-foliations of the spacetime
.............. 62
3.1.5
Null structure equations of a double null foliation
......... 67
3.1.6
The Einstein equations relative to a double null foliation
..... 69
3.1.7
The characteristic initial value problem
f
or the Einstein equations
72
3.2
Bianchi
equations in an Einstein vacuum spacetime
............ 75
3.3
Canonical double null foliation of the spacetime
.............. 78
3.3.1
Canonical foliation of the initial hypersurface
........... 78
3.3.2
Foliations on the last slice
..................... 79
3.3.3
Canonical foliation of the last slice
................. 80
3.3.4
Initial layer foliation
........................ 82
3.4
Deformation tensors
............................. 84
3.4.1
Approximate Killing and
conformai
Killing vector fields
...... 84
3.4.2
Deformation tensors of the vector fields T, S, Ko
......... 85
3.4.3
Rotation deformation tensors
.................... 87
3.5
The definitions of the fundamental norms
................. 87
3.5.1
Q
integral norms
.......................... 88
3.5.2 72.
norms for the Riemann null components
............ 90
3.5.3
С
norms for the connection coefficients
.............. 93
3.5.4
Norms on the initial layer region
.................. 96
3.5.5
О
norms on the initial and final hypersurfaces
........... 96
3.5.6
ľ
norms for the rotation deformation tensors
........... 97
3.6
The initial data
............................... 98
3.6.1
Global initial data conditions
.................... 98
3.7
The Main Theorem
............................. 101
3.7.1
Estimates for the initial layer foliation
............... 102
3.7.2
Estimates for the
О
norms in
К
.................. 102
3.7.3
Estimates for the X> norms in
A3.................. 102
3.7.4
Estimates for the
Ό
norms on the initial hypersurface
....... 103
3.7.5
Estimates for the
О
norms and the V norms on the last slice
. . . 103
3.7.6
Estimates for the
Έ.
norms
..................... 104
3.7.7
Estimates for the
Q
integral norms
................. 105
3.7.8
Extension theorem
......................... 105
3.7.9
Proof of the Main Theorem
..................... 105
3.8
Appendix
.................................. 109
Contents
vii
3.8.1
Proof
of Proposition
3.1.1..................... 109
3.8.2
Derivation of the structure equations
................ 110
3.8.3
Some remarks on the definition of the adapted null frame
.....
Ill
3.8.4
Proof of Proposition
3.3.1 ..................... 112
Estimates for the Connection Coefficients
115
4.1
Preliminary results
............................. 115
4.1.1
Elliptic estimates for Hodge systems
................ 115
4.1.2
Global Sobolev inequalities
.................... 117
4.1.3
The initial layer foliation
...................... 125
4.1.4
Comparison estimates for the function
r (u, u
)........... 128
4.2
Proof of Theorem Ml
............................ 130
4.3
Proof of Theorem
4.2.1
and estimates for the zero and first derivatives of
the connection coefficents
.......................... 134
4.3.1
EstimateforC>Ś f(trx)andOjf(x)withJp€[2,4]
....... 134
4.3.2
Estimates for
г2~Цих
-tix) p,s and
r3~2/pTtix p,s,
with
p
є
[2,4]........................... 137
4.3.3
Estimates for r2~^rl(tix
-ΰχ)|ρ,5
with
ρ
є
[2,4]....... 137
4.3.4
Estimate for r2^
(Ωίτχ
-
) p,s with
p e
[2,4] ........ 137
4.3.5
Estimates for
Ο^(ίΐχ)
and
Ο^(χ)
with
p e
[2,4]....... 137
4.3.6
Estimates for |r2 ? (tr/.
-
δχ)
Р,ѕ
and
r3~2/pT xx p<s,
with
ρ
є
[2,4]........................... 139
4.3.7
Estimateforl^Mítrx-tř^U.swithpeP,
4] ....... 140
4.3.8
Estimate
for
|г ~рт_
(sitc^
+
±ì |PiS
with
ρ
є
[2,4]....... 140
4.3.9
Estimates for
ö£f
(η)
and C^fO?) with
ρ
є
[2,4]........ 140
4.3.10
Estimates for O^s(co) and O^s(m) with
ρ
є
[2,4]........ 146
4.3.11
Estimate for sup
|γ(Ω
-
|)|
.................... 147
4.3.12
Completion of the estimates for trx and tr^
............ 149
4.3.13
Estimates for 0f 5(tu) and e?fllS(ft)) with
ρ
є
[2,4]........ 151
4.3.14
Estimates for O^5(D4ty) and O^S(D3») with
ρ
€ [2,4]..... 153
4.3.15
Estimate for
Οιίω)
with
ρ
є
[2,4] ................ 155
4.3.16
Improved estimates under stronger assumptions on
Σο
and C^
. . 155
4.4
Proof of Theorem
4.2.2
and estimates for the second derivatives
of the connection coefficients
........................ 161
4.4.1
Estimates for Ofs(<») and Of-S(iu) with
ρ
€[2,4]........ 161
4.4.2
Estimate for
02(ω)
with
p e
[2,4] ................ 164
4.5
Proof of Theorem
4.2.3
and control of third derivatives
of the connection coefficients
........................ 168
4.6
Rotation tensor estimates
.......................... 172
4.6.1
Technical aspects
.......................... 172
4.6.2
Derivatives of the rotation deformation tensors
.......... 175
viii Contents
4.7
Proof of Theorem M2 and estimates for the T> norms
of the rotation deformation tensors
.....................177
4.8
Appendix
..................................183
4.8.1
Some commutation relations
.................... 183
4.8.2
Proof of Lemma
4.3.5 ....................... 186
4.8.3
Proof of Lemma
4.4.1 ....................... 188
4.8.4
Proof of Proposition
4.6.2..................... 188
4.8.5
Proof of Proposition
4.6.3..................... 190
4.8.6
Proof of the Oscillation Lemma
.................. 192
4.8.7
Proof of Lemma
4.1.7 ....................... 200
5
Estimates for the Riemann Curvature Tensor
203
5.1
Preliminary tools
.............................. 205
5.1.1
L2
estimates for the zero derivatives
................ 208
5.1.2
L2 estimates for the first derivatives
................ 213
5.1.3
Auxiliary
Ĺ1
norms for the zero and first derivatives
of the Riemann components
.................... 220
5.1.4
The asymptotic behavior
ofranda
................ 223
5.1.5
Asymptotic behavior of the null Riemann components
...... 226
5.2
Appendix
.................................. 226
5.2.1
Proof of Proposition
5.1.4..................... 226
5.2.2
Proof of Proposition
5.1.5..................... 230
6
The Error Estimates
241
6.1
Definitions and prerequisites
........................ 243
6.1.1
Estimates for the
T, S, K
deformation tensors
.......... 250
6.1.2
Estimates for the rotation deformation tensors
........... 257
6.2
The error terms
f i
.............................. 259
6.2.1 Esum& &oiJV(uJ)iyQ(tTW)ßYb{Kii,KY,Ki)......... 259
6.2.2
Esumate
of
fVi^Q(tTW)aßYs((Kbia^KyKs) .......... 268
6.2.3
Esumate
oí
fY(u!umvQ(£oW)ßY5(K ?
КУТ*)
........... 269
6.2.4
Estímate
of
/^^ö^oW^yaí^^^r»)
.......... 273
6.2.5
Estímate
of
/v,(iijoo(¿oW)0,rya((r)^¿ ¿ii)
.......... 275
6.3
The error
terms
¿ г
.............................. 276
6.3.1
Estímate
of
Д^
Oi Q(L20W)ßYs(KßKyTs)........... 277
6.3.2
Proof of
Lemma
6.3.1
and Lemma
6.3.2.............. 283
6.3.3
Estimate of
/vM
о^^і^^^^)..........
285
6.3.4
Esumate
of
f
Viiiu)Q(t20W)aßyS(}T)K^KyKs).......... 285
6.3.5
Estimate of f
Vf^mrQ(CoCTW)ßYSK^KyKs.......... 285
6.3.6
Estimate of
/^ ß(£0£rW)aj8Kä((K>^
Ќу Ќб)
........ 287
6.3.7
Estimate of
Д^
TAyQ(tsCTW)ßYS{KliKYK&)......... 287
Contents ix
6.3.8
Estimate of J^^QiCsLTW^si^n^KyK1)
......... 292
6.4
Appendix
.................................. 293
6.4.1
The third-order derivatives of the connection coefficients
..... 293
The Initial Hypersurface and the Last Slice
295
7.1
Initial hypersurface foliations
........................ 295
7.1.1
Some general properties of a foliation of
Σο
............ 295
7.1.2
The structure equations on
Σο
................... 296
7.1.3
The construction of the background foliation of
Σο
........ 298
7.1.4
The construction of the canonical foliation of
Σο
......... 299
7.1.5
Proof of Theorem M3
....................... 300
7.2
The initial hypersurface connection estimates
............... 300
7.2.1
Proof of Lemma
3.7.1 ....................... 303
7.3
The last slice foliation
............................ 304
7.3.1
Construction of the canonical foliation of
Ç%
........... 304
7.4
The last slice connection estimates
..................... 305
7.4.1
О
norms on the last slice
...................... 305
7.4.2
Implementation of Proposition
7.4.1................ 308
7.4.3
Implementation of Proposition
7.4.2................ 313
7.5
The last slice rotation deformation estimates
................ 315
7.6
The extension argument
........................... 320
7.7
Appendix
.................................. 323
7.7.1
Comparison between different foliations
.............. 323
7.7.2
Proof of the local existence part of Theorem M6
......... 327
7.7.3
Proof of Propositions
7.4.4,7.4.5 ................. 333
Conclusions
347
8.1
The spacetime null infinity
......................... 349
8.1.1
The existence of a global optical function
............. 349
8.1.2
The null-outgoing infinity J+
................... 351
8.1.3
The null-outgoing limit of the metric
............... 353
8.1.4
The null-outgoing infinite limit of the
5(λ
,
v)-orthonormal frame
354
8.2
The behavior of the curvature tensor at the null-outgoing infinity
..... 355
8.3
The behavior of the connection coefficients at the null-outgoing infinity
. 358
8.4
The null-outgoing infinity limit of the structure equations
......... 365
8.5
The
Bondi
mass
............................... 366
8.6
Asymptotic behavior of null-outgoing hypersurfaces
........... 370
References
375
Index
381
|
| any_adam_object | 1 |
| author | Klainerman, Sergiu Nicolò, Francesco |
| author_facet | Klainerman, Sergiu Nicolò, Francesco |
| author_role | aut aut |
| author_sort | Klainerman, Sergiu |
| author_variant | s k sk f n fn |
| building | Verbundindex |
| bvnumber | BV014522105 |
| callnumber-first | Q - Science |
| callnumber-label | QC173 |
| callnumber-raw | QC173.6 |
| callnumber-search | QC173.6 |
| callnumber-sort | QC 3173.6 |
| callnumber-subject | QC - Physics |
| classification_rvk | SK 370 SK 950 UH 8300 |
| ctrlnum | (OCoLC)49873259 (DE-599)BVBBV014522105 |
| dewey-full | 530.11 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.11 |
| dewey-search | 530.11 |
| dewey-sort | 3530.11 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV014522105 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:03:14Z |
| institution | BVB |
| isbn | 0817642544 3764342544 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009887172 |
| oclc_num | 49873259 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-703 DE-824 DE-384 DE-11 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-824 DE-384 DE-11 DE-188 |
| physical | XII, 385 S. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematical physics |
| series2 | Progress in mathematical physics |
| spelling | Klainerman, Sergiu Verfasser aut The evolution problem in general relativity Sergiu Klainerman ; Francesco Nicolò Boston [u.a.] Birkhäuser 2003 XII, 385 S. txt rdacontent n rdamedia nc rdacarrier Progress in mathematical physics 25 "The global aspects of the problem of evolution equations in general relativity are examined. Central to the work is a revisit of the proof of the global stability of Minkowski space, as presented by Christodoulou and Klainerman (1993). The focus, therefore, is on a new self-contained proof of the main part of that result which concerns the full solution of the radiation problem in vacuum for arbitrary asymptotic flat initial data sets. While technical motivation is clearly and systematically provided for this proof, many important related concepts and results, some well established, others new, unfold along the way." "A comprehensive bibliography and index complete this important monograph, aimed at researchers and graduate students in mathematics, mathematical physics, and physics in the area of general relativity."--BOOK JACKET. Mathematische Physik Evolution equations General relativity (Physics) Mathematical physics Evolutionsgleichung (DE-588)4129061-6 gnd rswk-swf Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd rswk-swf Allgemeine Relativitätstheorie (DE-588)4112491-1 s Evolutionsgleichung (DE-588)4129061-6 s DE-604 Nicolò, Francesco Verfasser aut Progress in mathematical physics 25 (DE-604)BV013823265 25 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009887172&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Klainerman, Sergiu Nicolò, Francesco The evolution problem in general relativity Progress in mathematical physics Mathematische Physik Evolution equations General relativity (Physics) Mathematical physics Evolutionsgleichung (DE-588)4129061-6 gnd Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd |
| subject_GND | (DE-588)4129061-6 (DE-588)4112491-1 |
| title | The evolution problem in general relativity |
| title_auth | The evolution problem in general relativity |
| title_exact_search | The evolution problem in general relativity |
| title_full | The evolution problem in general relativity Sergiu Klainerman ; Francesco Nicolò |
| title_fullStr | The evolution problem in general relativity Sergiu Klainerman ; Francesco Nicolò |
| title_full_unstemmed | The evolution problem in general relativity Sergiu Klainerman ; Francesco Nicolò |
| title_short | The evolution problem in general relativity |
| title_sort | the evolution problem in general relativity |
| topic | Mathematische Physik Evolution equations General relativity (Physics) Mathematical physics Evolutionsgleichung (DE-588)4129061-6 gnd Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd |
| topic_facet | Mathematische Physik Evolution equations General relativity (Physics) Mathematical physics Evolutionsgleichung Allgemeine Relativitätstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009887172&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013823265 |
| work_keys_str_mv | AT klainermansergiu theevolutionproblemingeneralrelativity AT nicolofrancesco theevolutionproblemingeneralrelativity |