General relativity and the Einstein equations:
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Oxford Univ. Press
2009
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| Ausgabe: | 1. publ. |
| Schriftenreihe: | Oxford mathematical monographs
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXIV, 785 S. graph. Darst. |
| ISBN: | 9780199230723 |
Internformat
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| 100 | 1 | |a Choquet-Bruhat, Yvonne |d 1923- |e Verfasser |0 (DE-588)128916206 |4 aut | |
| 245 | 1 | 0 | |a General relativity and the Einstein equations |c Yvonne Choquet-Bruhat |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford |b Oxford Univ. Press |c 2009 | |
| 300 | |a XXIV, 785 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Oxford mathematical monographs | |
| 650 | 0 | 7 | |a Allgemeine Relativitätstheorie |0 (DE-588)4112491-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Einstein-Feldgleichungen |0 (DE-588)4013941-4 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Einstein-Feldgleichungen |0 (DE-588)4013941-4 |D s |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016683902 | ||
Datensatz im Suchindex
| _version_ | 1804137937257889793 |
|---|---|
| adam_text | CONTENTS
I Lorentz
geometry
1
1
Introduction
1
2
Manifolds
1
3
Differentiable mappings
2
4
Vectors and tensors
2
4.1
Tangent and cotangent space
2
4.2
Vector fields
3
4.3
Tensors and tensor fields
5
5
Pseudo-Riemannian metrics
6
5.1
General properties
6
5.2
Riemannian and Lorentzian metrics
8
6
Riemannian connection
9
7
Geodesies
12
8
Curvature
13
9
Geodesic deviation
14
10
Maximum of length and conjugate points
15
11
Linearized
Ricci
and Einstein tensors
17
12
Second derivative of the
Ricci
tensor
17
II Special Relativity
19
1
Newton s mechanics
19
1.1
The Galileo-Newton spacetime
19
1.2
Newton s dynamics
-
the Galileo group
20
2
Maxwell s equations
20
3
Minkowski spacetime
21
3.1
Definition
21
3.2
Maxwell s equations on
Mą
22
4
Poincaré
group
23
5
Lorentz
group
24
5.1
General formulae
24
5.2
Transformation of electric and magnetic
vector fields (case
η
= 3) 25
5.3
Lorentz
contraction and dilatation
26
6
Special Relativity
26
6.1
Proper time
26
6.2
Proper frame and relative velocities
28
7
Dynamics of a pointlike mass
30
7.1
Newtonian law
30
Contents
7.2
Relativistic law
ЗО
7.3
Equivalence of mass and energy
32
8
Continuous matter
33
8.1
Case of dust (incoherent matter)
34
8.2
Perfect fluids
35
III General relativity and Einstein s equations
37
1
Introduction
37
2
Newton s gravity law
37
3
General relativity
39
3.1
Physical motivations
39
4
Observations and experiments
40
4.1
Deviation of light rays
40
4.2
Proper time, gravitational time delay
40
5
Einstein s equations
42
5.1
Vacuum case
42
5.2
Equations with sources
43
6
Field sources
45
6.1
Electromagnetic sources
45
6.2
Electromagnetic potential
47
6.3
Yang-Mills fields
47
6.4
Scalar fields
49
6.5
Wave maps
49
6.6
Energy conditions
51
7
Lagrangians
51
7.1
Einstein-Hilbert Lagrangian
51
7.2
Lagrangians and stress energy tensors
of sources
52
7.3
Coupled Lagrangian
53
8
Fluid sources
54
9
Einsteinian spacetimes
55
9.1
Definition
55
9.2
Regularity hypotheses
55
10
Newtonian approximation
57
10.1
Equations for potentials
57
10.2
Equations of motion
59
11
Gravitational waves
60
11.1
Minkowskian approximation
60
11.2
General linear waves
61
12
High-frequency gravitational waves
62
12.1
Phase and polarizations
64
12.2
Radiative coordinates
66
12.3
Energy conservation
68
13
Coupled electromagnetic and gravitational waves
68
Contents ix
13.1 Phase
and polarizations
69
13.2
Propagation equations
69
IV
Schwarzschild spacetime
and black holes
72
1
Introduction
72
2
Spherically symmetric spacetimes
72
3 Schwarzschild
metric
74
4
Other coordinates
75
4.1 Isotropie
coordinates
75
4.2
Wave coordinates
76
4.3
Painlevé-Gullstr
and-like coordinates
77
4.4
Regge-Wheeler coordinates
77
5 Schwarzschild
spacetime
78
6
The motion of the planets and perihelion precession
78
6.1
Equations
78
6.2
Results of observations
81
6.3
Escape velocity
81
7
Stability of circular orbits
83
8
Deflection of light rays
84
8.1
Theoretical prediction
84
8.2
Results of observation
85
8.3
Fermat s principle and light travel
parameter time
85
9
Red shift and time delay
86
10
Spherically symmetric interior solutions
87
10.1
Static solutions. Upper limit on mass
88
10.2
Matching with an exterior solution
91
10.3
Non-static solutions
91
11
The
Schwarzschild
black hole
92
11.1
The event horizon
92
11.2
The Eddington-Finkelstein extension
93
11.3
Eddington-Finkelstein white hole
94
11.4
Kruskal complete spacetime
94
11.5
Observations
96
12
Spherically symmetric gravitational collapse
96
12.1
Tolman
metric
98
12.2
Monotonically decreasing density
101
13
The
Reissner-Nordström
solution
103
14 Schwarzschild
spacetime in dimension
η
+ 1 104
14.1
Standard coordinates
104
14.2
Wave coordinates
104
V Cosmology
106
1
Introduction
106
2
Cosmological principle
107
Contents
3 Isotropie
and homogeneous Riemannian manifolds
108
3.1
Isotropy
108
3.2
Homogeneity
109
4
Robertson-Walker spacetimes 111
4.1
Space metrics
112
4.2
Robertson-Walker spacetime metrics
113
4.3
Robertson-Walker dynamics
113
4.4
Einstein static universe
115
4.5
Cosmological red shift and the Hubble constant
115
4.6 De
Sitter spacetime
118
4.7
Anti
de
Sitter (AdS) spacetime
120
5
Friedmann-Lemaitre models.
121
5.1
Equation of state
121
5.2
General properties
122
5.3 Friedmann
models
123
5.4
Some other models
124
5.5
Confrontation with observations
125
6
Homogeneous non-isotropic cosmologies
125
7
Bianchi
class I universes
128
7.1
Kasner solutions
128
7.2
Models with matter
131
8
Bianchi
type IX
132
9
The Kantowski-Sachs models
134
10 Taub
and
Taub
NUT spacetimes
135
10.1 Taub
spacetime
135
10.2 Taub
NUT spacetime
136
11
Locally homogeneous models
136
11.1
n-dimensional
compact manifolds
137
11.2
Compact 3-manifolds
139
12
Recent observations and conjectures
140
VI Local Cauchy problem
142
1
Introduction
142
2
Moving frame formulae
142
2.1
Frame and coframe
142
2.2
Metric
143
2.3
Connection
144
2.4
Curvature
145
3
η
+ 1
splitting adapted to space slices
146
3.1
Adapted frame and coframe
146
3.2
Structure coefficients
147
3.3
Splitting of the connection.
147
3.4
Extrinsic curvature
148
3.5
Splitting of the Riemann tensor
148
Contents
4
Constraints and evolution
149
4.1
Equations. Conservation of constraints
149
5
Hamiltonian and symplectic formulation
151
5.1
Hamilton equations
151
5.2
Symplectic formulation
154
6
Cauchy problem
155
6.1
Definitions
155
6.2
The analytic case
156
7
Wave gauges
157
7.1
Wave coordinates
158
7.2
Generalized wave coordinates
161
7.3
Damped wave coordinates
162
7.4
Globalization in space,
ê
wave gauges
162
7.5
Local in time existence in a wave gauge
164
8
Local existence for the full Einstein equations
166
8.1
Preservation of the wave gauges
166
8.2
Geometric local existence
168
8.3
Geometric uniqueness
168
8.4
Causality
170
9
Constraints in a wave gauge
172
10
Einstein equations with field sources
173
10.1
Maxwell constraints
174
10.2
Lorentz
gauge
175
10.3
Existence and uniqueness theorems
176
10.4
Wave equation for
F
177
VII
Constraints
179
1
Introduction
179
2
Linearization and stability
180
2.1
Linearization of the constraints map,
adjoint map
181
2.2
Linearization stability
183
3
CF (Conformally Formulated) constraints
186
3.1
Hamiltonian constraint
187
3.2
Momentum constraint
188
3.3
Scaling of the sources
189
3.4
Summary of results
194
3.5
Conformai
transformation of the CF constraints
195
3.6
The momentum constraint as an elliptic system
197
4
Case
η
= 2 200
5
Solutions on compact manifolds
200
6
Solution of the momentum constraint
201
7
Lichnerowicz equation
204
7.1
The Yamabe classification
204
Contents
7.2
Non-existence and uniqueness
210
7.3
Existence theorems
211
8
System of constraints
217
8.1
Constant mean curvature
τ,
sources with
York-scaled momentum
217
8.2
Solutions with
τ φ.
constant or Jq
φ
0 218
9
Solutions on asymptotically Euclidean Manifolds
221
10
Momentum constraint
222
11
Solution of the Lichnerowicz equation
223
11.1
Uniqueness theorem
223
11.2
Generalized Brill-Cantor theorem
223
11.3
Existence theorems
226
12
Solutions of the system of constraints
229
12.1
Decoupled system
229
12.2
Coupled system
230
13
Gluing solutions of the constraint equations
232
13.1
Connected sum gluing
233
13.2
Exterior (Corvino-Schoen) gluing
235
VIII
Other hyperbolic-elliptic well-posed systems
238
1
Introduction
238
2
Leray-Ohya non-hyperbolicity of
^Лц
= 0 238
3
Wave equation for
К
240
3.1
Hyperbolic system
240
3.2
Hyperbolic-elliptic system
242
4
Fourth-order non-strict and strict hyperbolic systems
for
g
243
5
First-order hyperbolic systems
243
5.1
FOSH systems
243
6
Bianchi-Einstein equations
244
6.1
Wave equation for the Riemann tensor
245
6.2
Case
η
= 3,
FOS
system
246
6.3
Cauchy-adapted frame
247
6.4
FOSH system for
и
=
{E, H, D, B, g,
ÜT, f)
250
6.5
Elliptic-hyperbolic system
250
7
Bel-Robinson tensor and energy
254
7.1
The Bel tensor
254
7.2
The Bel-Robinson tensor and energy
255
8
Bel-Robinson energy in a strip
256
IX Relativistic fluids
259
1
Introduction
259
2
Case of dust
260
2.1
Equations
260
Contents
2.2 Motion
of isolated bodies (Choquet-Bruhat and
Friedrichs 2006) 262
3
Charged dust
263
3.1
Equations
263
3.2
Existence and uniqueness theorem in wave and
Lorentz
gauges
264
3.3
Motion of isolated bodies
265
4
Perfect fluid,
Euler
equations
265
5
Energy properties
266
6
Particle number conservation
267
7
Thermodynamics
268
7.1
Definitions. Conservation of entropy
268
7.2
Equations of state
268
8
Wave fronts, propagation speeds, shocks
270
8.1
General definitions
270
8.2
Case of perfect fluids
272
8.3
Shocks
273
9
Stationary motion
274
10
Dynamic velocity for barotropic fluids
274
10.1
Fluid index and equations
274
10.2
Vorticity tensor and Helmholtz equations
276
10.3
Irrotational flows
277
11
General perfect fluids
278
12
Hyperbolic Leray system
279
12.1
Hyperbolicity of the
Euler
equations.
279
12.2
Reduced Einstein-Euler entropy system
280
12.3
Cauchy problem for the Einstein-Euler
entropy system
281
12.4
Motion of isolated bodies
282
13
First-order symmetric hyperbolic system
282
14
Equations in a flow adapted frame
284
14.1
η
+ 1
splitting in a time adapted frame
285
14.2
Bianchi
equations (case
η
= 3) 287
14.3
Vacuum case
287
14.4
Perfect fluid
288
14.5
Conclusion
290
15
Charged fluids
290
15.1
Equations
290
15.2
Fluids with zero conductivity
291
16
Fluids with finite conductivity
293
17
Magnetohydrodynamics
294
17.1
Equations
294
17.2
Wave fronts
295
18
Yang-Mills fluids
296
Contents
19 Dissipative
fluids
297
19.1
Viscous fluids
297
19.2
The heat equation
300
X Relativistic kinetic theory
301
1
Introduction
301
2
Distribution function
302
2.1
Definition
302
2.2
Interpretation
303
2.3
Moments of the distribution function
304
3
Vlasov
equations
307
3.1
Liouville-
Vlasov
equation.
307
3.2
Maxwell-
Vlasov
equation
310
3.3
Yang-Mills-Vlasov equation
311
3.4
Particles of a given rest mass
311
3.5
Conservation of moments
312
4
Cauchy problem for the Liouville-
Vlasov
equation
313
4.1
General solution
313
4.2
Distribution function on a Robertson-Walker
space time
313
4.3
Energy estimates
314
4.4
Existence theorem
322
4.5
Stress energy tensor of a distribution function
323
5
The Einstein-Vlasov system
324
5.1
Constraints
324
5.2
Cauchy problem for the Einstein equations
325
5.3
Cauchy problem for the coupled system
325
6
The Einstein-Maxwell-Vlasov system
326
7
Boltzmann equation. Definitions
328
8
Moments and conservation laws
329
9
Einstein-Boltzmann system
331
10
Thermodynamics
331
10.1
Entropy and
Я
theorem
331
10.2 Maxwell-Jüttner
equilibrium distribution
333
10.3
Dissipative fluids
334
11
Extended thermodynamics
334
11.1
The phenomenological
14
moments theory
335
11.2
Extended thermodynamics of moments
338
11.3
Maximum characteristic velocity
339
XI Progressive waves
341
1
Introduction
341
2
Quasilinear systems
342
3
Quasilinear first-order systems
343
3.1
Phase and polarization
343
3.2
Propagation equations
344
Contents
4 Progressive
waves in relativistic fluids
348
4.1
Equations
348
4.2 Progressive
waves
348
4.3
Phases and polarizations
349
4.4
Polarization and propagation of acoustic waves
351
4.5
Polarization and propagation of matter waves
354
4.6
Gauge gravitational waves
354
5 Quasilinear quasidiagonal
second-order systems
354
5.1
Definitions
354
5.2
Hyperquasilinear systems with
ƒ
quadratic
in
Du 356
5.3
The null condition
358
6
Non
quasidiagonal
second-order systems
359
6.1
Phase and polarization
360
6.2
Propagation equations
360
7
Yang-Mills-scalar equations
361
7.1
Fields and equations
361
7.2
Phase and polarization
362
7.3
Propagation
363
8
Strong gravitational waves
364
8.1
Einstein equations
364
8.2
Phase and polarization
365
8.3
Propagation and back reaction
366
8.4
Example
368
XII
Global hyperbolicity and causality
371
1
Introduction
371
2
Global existence of Lorentzian metrics
371
3
Time orientation
374
4
Futures and pasts
375
4.1
Paths and curves
375
4.2
Chronology and causality
375
5
Causal structure of Minkowski spacetime
377
6
Causal structures on general spacetimes
378
7
Geodesic coordinates, normal neighbourhoods
382
8
Topology on a space of paths
387
8.1
Rectifiable paths
387
8.2
Topology on sets of rectifiable paths
388
9
Global hyperbolicity
389
9.1
Definition and first criterion
389
9.2
Maximum of proper length
390
9.3
Images in V of subsets of
V{x,
y)
391
10
Strong and stable causalities
391
10.1
Strong causality
392
10.2
Stable causality
392
Contents
11
Cauchy surface
393
11.1
Domain of dependence. Cauchy horizon
393
11.2
Cauchy surface
394
11.3
Global existence of solutions of linear
wave equations
397
11.4
Sufficient condition from analysis for global
hyperbolicity
397
12
Globally hyperbolic Einsteinian spacetimes
399
12.1
Existence
399
12.2
Global uniqueness
399
12.3
Examples
400
13
Strong cosmic censorship
400
XIII
Singularities
402
1
Introduction
402
2
Criteria for completeness or incompleteness
404
2.1
A completeness criterion
404
2.2
An incompleteness criterion
406
3
Congruence of timelike curves
408
3.1
Definitions
408
3.2
Geodesic deviation
410
3.3
Raychauduri equation
411
3.4
Null geodesic congruence
412
4
First singularity theorem
412
4.1
Conjugate points
412
4.2
Incompleteness theorem
414
5
Trapped surfaces and singularities
414
5.1
Trapped surfaces
414
5.2
Singularities linked to trapped surfaces
418
6
Black holes
418
6.1
Definitions
418
6.2
The Hawking area theorem
420
6.3
The Riemannian Penrose inequality. Case
η
= 3 420
7
Weak cosmic censorship conjectures
421
7.1
Naked singularity
421
7.2
Weak cosmic censorship
422
8
Spherically symmetric Einstein scalar equations
423
8.1
Spherically symmetric spacetimes
423
8.2
Einstein equations in adapted frame
424
8.3
Reduction to one integro-differential equation
425
8.4
Bondi
mass
427
8.5
Global existence for small data
427
8.6
Existence of a global generalized solution
for large data
429
Contents xvii
8.7
Structure
of generalized solutions
432
8.8
Formation of a black hole. Cosmic censorship
433
8.9
Numerical results
434
8.10
Instability of naked singularities
434
9
Cosmological singularities. BKL conjecture
435
10
AVTD behaviour
441
10.1
Definitions
441
10.2 Fuchs
theorem
441
11
Case of 1-parameter spatial isometry
443
11.1
Equations
443
11.2
VTD solutions of the
2 + 1
Einstein evolution
equations
445
11.3
The polarized case
446
11.4
The unpolarized case
449
XIV
Stationary spacetimes and black holes
451
1
Introduction
451
2
Spacetimes with 1-parameter isometry group
452
2.1
Connection and Riemann tensor
454
2.2
Curvature tensor
454
2.3
Ricci
tensor
455
3
Stationary spacetimes
455
3.1
General case
455
3.2
Static spacetimes
457
4
Gravitational solitons
458
4.1
Elementary proofs
458
4.2
Case
η
= 3,
Komar
mass
460
5
Electrovac solitons
465
6
The Einstein-Yang-Mills case
466
7
Stationary black holes
466
7.1
Definitions
466
7.2
Axisymmetry
467
8
The rigidity theorem for black holes
469
9
The Kerr metric and black hole
471
9.1
Kerr metric in Boyer-Linquist coordinates
471
9.2
The Kerr-Schild spacetime
472
10
Uniqueness of stationary black holes (dimension
3+1) 474
10.1
Static black holes
475
10.2
Axisymmetric black holes
475
10.3
Uniqueness of the Kerr black hole
475
11
Further results
476
11.1
Multi
black hole solutions
476
11.2
The Emparan-Reali black rings
478
Contents
XV
Global
existence
theorems asymptotically Euclidean
data
482
1
Introduction
482
2
Global existence for small data via the Penrose map
483
2.1
Yang-Mills and associated equations
484
2.2
Quasi-linear wave equations
484
2.3
Cases
η
= 3,
the null condition
487
2.4
Wave maps
488
3
H.
Friedrich conformai
system,
η +
1 = 4 488
3.1
Equations
488
3.2
Friedrich
hyperbolic system
490
4
Einstein s equations in higher dimensions
491
4.1
Conformai
mapping
491
4.2
Transformed equations
492
4.3
Local Cauchy problem in
iž™ 1 1, n
> 5
and odd
494
4.4
Global Cauchy problem
496
4.5
Conclusion
497
5
Christodoulou-Klainerman theorem
497
5.1
CK main theorem
497
5.2
Local existence
499
5.3
Global existence
500
6
The Klainerman-Nicolo theorem
504
7
The Linblad-Rodnianski theorem
505
7.1
The Einstein equations in wave coordinates
506
7.2
Initial data
507
7.3
Unknowns and norms
507
7.4
LR theorem
508
XVI
Global existence theorems the cosmological
case
510
1
Introduction
510
2
Gowdy cosmological models
511
3
S1 invariant Einsteinian universes, equations
514
3.1
Introduction
514
3.2
Definition
514
3.3
Equations
515
3.4
Twist potential
515
3.5
Wave map system
516
3.6
Three-dimensional Einstein equations
517
3.7 Teichmüller
parameters.
520
4
S1 invariant Einstein universes, Cauchy problem
521
4.1
Cauchy data
521
4.2
Construction of A when
F
is known
522
Contenis
xix
4.3
Local
in time existence theorem
522
4.4
Global existence theorem
522
4.5
Future complete existence
526
4.6
Einstein-Maxwell-Higgs system
526
4.7
Conclusion
527
5
Andersson-Moncrief theorem
528
5.1
CMC gauge, elliptic system for
N
and
К
529
5.2
SH gauge, elliptic system for
g
and
β
529
5.3
The
Bianchi
equations
530
5.4
Existence theorems
531
5.5
Global existence theorem
532
6
Einstein non-linear scalar field system
533
APPENDICES
I Sobolev spaces on Riemannian manifolds
534
1
Definitions
534
2
Embedding and multiplication properties
535
2.1
Open subsets of Rn
535
2.2
Riemannian manifolds
536
3
Weighted Sobolev spaces
537
3.1
Definitions
537
3.2
Embedding and multiplication properties
538
II Second-order elliptic systems on Riemannian
manifolds
542
1
Linear elliptic systems
542
2
Linear elliptic systems on compact
M
544
2.1
General second-order systems
544
2.2
Poisson
operator
551
2.3
Conformai
Laplace operator
552
3
Asymptotically Euclidean manifolds
553
3.1
Definitions
553
3.2
Second-order linear elliptic systems
554
4
Special systems
560
4.1
Poisson
operator
560
4.2
Conformai
Laplace operator
561
5
Equation
ΔΊφ
=
f (x,
φ),
compact
M
562
6
ΑΊφ
= ƒ
(χ, φ)
on (M,
7)
asymptotically Euclidean
567
ΠΙ
Quasi-diagonal, quasi-linear, second-order hyperbolic
systems
571
1
Introduction
571
2
Wave equation on (V,g)
571
2.1
Definitions
572
Contents
IV
V
2.2
Stress energy tensor. Energy momentum vector
572
2.3
Energy density
574
2.4
Energy equality on a compact domain
575
2.5
Energy inequality in a compact causal
domain
578
2.6
Uniqueness theorem and causality
580
2.7
Case of a strip
581
2.8
Estimate of
и
582
2.9
Cauchy problem
583
2.10
Generalizations
589
! Quasidiagonal linear
systems
590
3.1
Definitions
590
3.2
Stress energy tensor
590
3.3
Energy inequality in a compact causal domain
591
3.4
Case of a strip Vt
592
3.5
Existence, uniqueness, causality and continuity
594
3.6
Higher order estimates
594
3.7
Other hypotheses on
g
600
3.8
Cauchy data in local spaces
601
і
Quasilinear
systems
604
4.1
Semilinear
systems
604
4.2
Further results for
semilinear
equations
607
4.3
Quasilinear
systems
608
5
Global
I existence
613
5.1
Semilinear
systems
613
5.2
Quasilinear
equations
616
General hyperbolic systems
617
1
Introduction
617
2
Leray
hyperbolic systems
617
2.1
Case of one equation
617
2.2
Leray hyperbolic systems
622
3
Leray-Ohya hyperbolic systems
624
4
First-order symmetric hyperbolic systems
625
4.1
FOSH systems on Rn+1
625
4.2
FOSH systems on a sliced manifold
627
Cauchy-Kovalevski and
Fuchs
theorems
631
1
Introduction
631
2
Cauchy-Kovalevski theorem
631
2.1
Linear system
631
2.2
Non-linear system
634
3 Fuchs
theorem
634
3.1
Definitions
634
3.2
Theorem
636
Contents
3.3
Equivalence
with an integral equation
637
3.4
Equivalence with another mapping
638
3.5
Convergence of iterations
641
3.6
Global in space theorem
642
VI Conformai
methods
643
1
Introduction
643
2
Conformai
metrics. Confomorphisms
643
2.1
Connections of
conformai
metrics
643
2.2
Riemann tensors of
conformai
metrics
644
2.3
Ricci
tensors of
conformai
metrics
644
3
The Weyl tensor
645
4
Conformai
transformations of field equations
646
4.1
Maxwell and Yang-Mills equations
646
5
Invariance
of wave equations
647
6
Penrose transform
647
7
Einstein spaces with cosmological constant
650
7.1
Conformai
transformation of
De
Sitter
spacetime
650
7.2
Conformai
transformation of anti-De Sitter
spacetime
650
8
Asymptotically simple spacetimes
650
8.1
Conformai compactifications
650
8.2
Black holes
652
VII
Kaluza-Klein theories
653
1
Introduction
653
2
Isometries
653
3
Kaluza-Klein metrics
654
3.1
Metric in adapted frame
654
3.2
Structure coefficients
655
3.3
Kaluza-Klein connection
656
4
Curvature tensor
657
5
Ricci
tensor and K-K equations
659
6
Equations in
conformai
spacetime metric
660
RELATED PAPERS
Causality of classical supergravity
665
Lecture Notes in Physics
1986,
E. Flaherty
ed.
Springer.
61-84
Gravitation with gauss bonnet terms
689
Australian National University Publications,
1988
R.
Bartnik
ed. 53-72
xxii Contents
Interaction
of gravitational and fluid waves
709
In collaboration with A. Greco, Cericolo
nat.
di
Paleruno 1994
Serie
II
n° 45,
III.
123.
Positive-energy theorems
723
Relativity, Group and Topology II, B. Dewitt and R.
Stora ed.
742-786.
REFERENCES
771
INDEX
781
|
| adam_txt |
CONTENTS
I Lorentz
geometry
1
1
Introduction
1
2
Manifolds
1
3
Differentiable mappings
2
4
Vectors and tensors
2
4.1
Tangent and cotangent space
2
4.2
Vector fields
3
4.3
Tensors and tensor fields
5
5
Pseudo-Riemannian metrics
6
5.1
General properties
6
5.2
Riemannian and Lorentzian metrics
8
6
Riemannian connection
9
7
Geodesies
12
8
Curvature
13
9
Geodesic deviation
14
10
Maximum of length and conjugate points
15
11
Linearized
Ricci
and Einstein tensors
17
12
Second derivative of the
Ricci
tensor
17
II Special Relativity
19
1
Newton's mechanics
19
1.1
The Galileo-Newton spacetime
19
1.2
Newton's dynamics
-
the Galileo group
20
2
Maxwell's equations
20
3
Minkowski spacetime
21
3.1
Definition
21
3.2
Maxwell's equations on
Mą
22
4
Poincaré
group
23
5
Lorentz
group
24
5.1
General formulae
24
5.2
Transformation of electric and magnetic
vector fields (case
η
= 3) 25
5.3
Lorentz
contraction and dilatation
26
6
Special Relativity
26
6.1
Proper time
26
6.2
Proper frame and relative velocities
28
7
Dynamics of a pointlike mass
30
7.1
Newtonian law
30
Contents
7.2
Relativistic law
ЗО
7.3
Equivalence of mass and energy
32
8
Continuous matter
33
8.1
Case of dust (incoherent matter)
34
8.2
Perfect fluids
35
III General relativity and Einstein's equations
37
1
Introduction
37
2
Newton's gravity law
37
3
General relativity
39
3.1
Physical motivations
39
4
Observations and experiments
40
4.1
Deviation of light rays
40
4.2
Proper time, gravitational time delay
40
5
Einstein's equations
42
5.1
Vacuum case
42
5.2
Equations with sources
43
6
Field sources
45
6.1
Electromagnetic sources
45
6.2
Electromagnetic potential
47
6.3
Yang-Mills fields
47
6.4
Scalar fields
49
6.5
Wave maps
49
6.6
Energy conditions
51
7
Lagrangians
51
7.1
Einstein-Hilbert Lagrangian
51
7.2
Lagrangians and stress energy tensors
of sources
52
7.3
Coupled Lagrangian
53
8
Fluid sources
54
9
Einsteinian spacetimes
55
9.1
Definition
55
9.2
Regularity hypotheses
55
10
Newtonian approximation
57
10.1
Equations for potentials
57
10.2
Equations of motion
59
11
Gravitational waves
60
11.1
Minkowskian approximation
60
11.2
General linear waves
61
12
High-frequency gravitational waves
62
12.1
Phase and polarizations
64
12.2
Radiative coordinates
66
12.3
Energy conservation
68
13
Coupled electromagnetic and gravitational waves
68
Contents ix
13.1 Phase
and polarizations
69
13.2
Propagation equations
69
IV
Schwarzschild spacetime
and black holes
72
1
Introduction
72
2
Spherically symmetric spacetimes
72
3 Schwarzschild
metric
74
4
Other coordinates
75
4.1 Isotropie
coordinates
75
4.2
Wave coordinates
76
4.3
Painlevé-Gullstr
and-like coordinates
77
4.4
Regge-Wheeler coordinates
77
5 Schwarzschild
spacetime
78
6
The motion of the planets and perihelion precession
78
6.1
Equations
78
6.2
Results of observations
81
6.3
Escape velocity
81
7
Stability of circular orbits
83
8
Deflection of light rays
84
8.1
Theoretical prediction
84
8.2
Results of observation
85
8.3
Fermat's principle and light travel
parameter time
85
9
Red shift and time delay
86
10
Spherically symmetric interior solutions
87
10.1
Static solutions. Upper limit on mass
88
10.2
Matching with an exterior solution
91
10.3
Non-static solutions
91
11
The
Schwarzschild
black hole
92
11.1
The event horizon
92
11.2
The Eddington-Finkelstein extension
93
11.3
Eddington-Finkelstein white hole
94
11.4
Kruskal complete spacetime
94
11.5
Observations
96
12
Spherically symmetric gravitational collapse
96
12.1
Tolman
metric
98
12.2
Monotonically decreasing density
101
13
The
Reissner-Nordström
solution
103
14 Schwarzschild
spacetime in dimension
η
+ 1 104
14.1
Standard coordinates
104
14.2
Wave coordinates
104
V Cosmology
106
1
Introduction
106
2
Cosmological principle
107
Contents
3 Isotropie
and homogeneous Riemannian manifolds
108
3.1
Isotropy
108
3.2
Homogeneity
109
4
Robertson-Walker spacetimes 111
4.1
Space metrics
112
4.2
Robertson-Walker spacetime metrics
113
4.3
Robertson-Walker dynamics
113
4.4
Einstein static universe
115
4.5
Cosmological red shift and the Hubble constant
115
4.6 De
Sitter spacetime
118
4.7
Anti
de
Sitter (AdS) spacetime
120
5
Friedmann-Lemaitre models.
121
5.1
Equation of state
121
5.2
General properties
122
5.3 Friedmann
models
123
5.4
Some other models
124
5.5
Confrontation with observations
125
6
Homogeneous non-isotropic cosmologies
125
7
Bianchi
class I universes
128
7.1
Kasner solutions
128
7.2
Models with matter
131
8
Bianchi
type IX
132
9
The Kantowski-Sachs models
134
10 Taub
and
Taub
NUT spacetimes
135
10.1 Taub
spacetime
135
10.2 Taub
NUT spacetime
136
11
Locally homogeneous models
136
11.1
n-dimensional
compact manifolds
137
11.2
Compact 3-manifolds
139
12
Recent observations and conjectures
140
VI Local Cauchy problem
142
1
Introduction
142
2
Moving frame formulae
142
2.1
Frame and coframe
142
2.2
Metric
143
2.3
Connection
144
2.4
Curvature
145
3
η
+ 1
splitting adapted to space slices
146
3.1
Adapted frame and coframe
146
3.2
Structure coefficients
147
3.3
Splitting of the connection.
147
3.4
Extrinsic curvature
148
3.5
Splitting of the Riemann tensor
148
Contents
4
Constraints and evolution
149
4.1
Equations. Conservation of constraints
149
5
Hamiltonian and symplectic formulation
151
5.1
Hamilton equations
151
5.2
Symplectic formulation
154
6
Cauchy problem
155
6.1
Definitions
155
6.2
The analytic case
156
7
Wave gauges
157
7.1
Wave coordinates
158
7.2
Generalized wave coordinates
161
7.3
Damped wave coordinates
162
7.4
Globalization in space,
ê
wave gauges
162
7.5
Local in time existence in a wave gauge
164
8
Local existence for the full Einstein equations
166
8.1
Preservation of the wave gauges
166
8.2
Geometric local existence
168
8.3
Geometric uniqueness
168
8.4
Causality
170
9
Constraints in a wave gauge
172
10
Einstein equations with field sources
173
10.1
Maxwell constraints
174
10.2
Lorentz
gauge
175
10.3
Existence and uniqueness theorems
176
10.4
Wave equation for
F
177
VII
Constraints
179
1
Introduction
179
2
Linearization and stability
180
2.1
Linearization of the constraints map,
adjoint map
181
2.2
Linearization stability
183
3
CF (Conformally Formulated) constraints
186
3.1
Hamiltonian constraint
187
3.2
Momentum constraint
188
3.3
Scaling of the sources
189
3.4
Summary of results
194
3.5
Conformai
transformation of the CF constraints
195
3.6
The momentum constraint as an elliptic system
197
4
Case
η
= 2 200
5
Solutions on compact manifolds
200
6
Solution of the momentum constraint
201
7
Lichnerowicz equation
204
7.1
The Yamabe classification
204
Contents
7.2
Non-existence and uniqueness
210
7.3
Existence theorems
211
8
System of constraints
217
8.1
Constant mean curvature
τ,
sources with
York-scaled momentum
217
8.2
Solutions with
τ φ.
constant or Jq
φ
0 218
9
Solutions on asymptotically Euclidean Manifolds
221
10
Momentum constraint
222
11
Solution of the Lichnerowicz equation
223
11.1
Uniqueness theorem
223
11.2
Generalized Brill-Cantor theorem
223
11.3
Existence theorems
226
12
Solutions of the system of constraints
229
12.1
Decoupled system
229
12.2
Coupled system
230
13
Gluing solutions of the constraint equations
232
13.1
Connected sum gluing
233
13.2
Exterior (Corvino-Schoen) gluing
235
VIII
Other hyperbolic-elliptic well-posed systems
238
1
Introduction
238
2
Leray-Ohya non-hyperbolicity of
^Лц
= 0 238
3
Wave equation for
К
240
3.1
Hyperbolic system
240
3.2
Hyperbolic-elliptic system
242
4
Fourth-order non-strict and strict hyperbolic systems
for
g
243
5
First-order hyperbolic systems
243
5.1
FOSH systems
243
6
Bianchi-Einstein equations
244
6.1
Wave equation for the Riemann tensor
245
6.2
Case
η
= 3,
FOS
system
246
6.3
Cauchy-adapted frame
247
6.4
FOSH system for
и
=
{E, H, D, B, g,
ÜT, f)
250
6.5
Elliptic-hyperbolic system
250
7
Bel-Robinson tensor and energy
254
7.1
The Bel tensor
254
7.2
The Bel-Robinson tensor and energy
255
8
Bel-Robinson energy in a strip
256
IX Relativistic fluids
259
1
Introduction
259
2
Case of dust
260
2.1
Equations
260
Contents
2.2 Motion
of isolated bodies (Choquet-Bruhat and
Friedrichs 2006) 262
3
Charged dust
263
3.1
Equations
263
3.2
Existence and uniqueness theorem in wave and
Lorentz
gauges
264
3.3
Motion of isolated bodies
265
4
Perfect fluid,
Euler
equations
265
5
Energy properties
266
6
Particle number conservation
267
7
Thermodynamics
268
7.1
Definitions. Conservation of entropy
268
7.2
Equations of state
268
8
Wave fronts, propagation speeds, shocks
270
8.1
General definitions
270
8.2
Case of perfect fluids
272
8.3
Shocks
273
9
Stationary motion
274
10
Dynamic velocity for barotropic fluids
274
10.1
Fluid index and equations
274
10.2
Vorticity tensor and Helmholtz equations
276
10.3
Irrotational flows
277
11
General perfect fluids
278
12
Hyperbolic Leray system
279
12.1
Hyperbolicity of the
Euler
equations.
279
12.2
Reduced Einstein-Euler entropy system
280
12.3
Cauchy problem for the Einstein-Euler
entropy system
281
12.4
Motion of isolated bodies
282
13
First-order symmetric hyperbolic system
282
14
Equations in a flow adapted frame
284
14.1
η
+ 1
splitting in a time adapted frame
285
14.2
Bianchi
equations (case
η
= 3) 287
14.3
Vacuum case
287
14.4
Perfect fluid
288
14.5
Conclusion
290
15
Charged fluids
290
15.1
Equations
290
15.2
Fluids with zero conductivity
291
16
Fluids with finite conductivity
293
17
Magnetohydrodynamics
294
17.1
Equations
294
17.2
Wave fronts
295
18
Yang-Mills fluids
296
Contents
19 Dissipative
fluids
297
19.1
Viscous fluids
297
19.2
The heat equation
300
X Relativistic kinetic theory
301
1
Introduction
301
2
Distribution function
302
2.1
Definition
302
2.2
Interpretation
303
2.3
Moments of the distribution function
304
3
Vlasov
equations
307
3.1
Liouville-
Vlasov
equation.
307
3.2
Maxwell-
Vlasov
equation
310
3.3
Yang-Mills-Vlasov equation
311
3.4
Particles of a given rest mass
311
3.5
Conservation of moments
312
4
Cauchy problem for the Liouville-
Vlasov
equation
313
4.1
General solution
313
4.2
Distribution function on a Robertson-Walker
space time
313
4.3
Energy estimates
314
4.4
Existence theorem
322
4.5
Stress energy tensor of a distribution function
323
5
The Einstein-Vlasov system
324
5.1
Constraints
324
5.2
Cauchy problem for the Einstein equations
325
5.3
Cauchy problem for the coupled system
325
6
The Einstein-Maxwell-Vlasov system
326
7
Boltzmann equation. Definitions
328
8
Moments and conservation laws
329
9
Einstein-Boltzmann system
331
10
Thermodynamics
331
10.1
Entropy and
Я
theorem
331
10.2 Maxwell-Jüttner
equilibrium distribution
333
10.3
Dissipative fluids
334
11
Extended thermodynamics
334
11.1
The phenomenological
14
moments theory
335
11.2
Extended thermodynamics of moments
338
11.3
Maximum characteristic velocity
339
XI Progressive waves
341
1
Introduction
341
2
Quasilinear systems
342
3
Quasilinear first-order systems
343
3.1
Phase and polarization
343
3.2
Propagation equations
344
Contents
4 Progressive
waves in relativistic fluids
348
4.1
Equations
348
4.2 Progressive
waves
348
4.3
Phases and polarizations
349
4.4
Polarization and propagation of acoustic waves
351
4.5
Polarization and propagation of matter waves
354
4.6
"Gauge" gravitational waves
354
5 Quasilinear quasidiagonal
second-order systems
354
5.1
Definitions
354
5.2
Hyperquasilinear systems with
ƒ
quadratic
in
Du 356
5.3
The null condition
358
6
Non
quasidiagonal
second-order systems
359
6.1
Phase and polarization
360
6.2
Propagation equations
360
7
Yang-Mills-scalar equations
361
7.1
Fields and equations
361
7.2
Phase and polarization
362
7.3
Propagation
363
8
Strong gravitational waves
364
8.1
Einstein equations
364
8.2
Phase and polarization
365
8.3
Propagation and back reaction
366
8.4
Example
368
XII
Global hyperbolicity and causality
371
1
Introduction
371
2
Global existence of Lorentzian metrics
371
3
Time orientation
374
4
Futures and pasts
375
4.1
Paths and curves
375
4.2
Chronology and causality
375
5
Causal structure of Minkowski spacetime
377
6
Causal structures on general spacetimes
378
7
Geodesic coordinates, normal neighbourhoods
382
8
Topology on a space of paths
387
8.1
Rectifiable paths
387
8.2
Topology on sets of rectifiable paths
388
9
Global hyperbolicity
389
9.1
Definition and first criterion
389
9.2
Maximum of proper length
390
9.3
Images in V of subsets of
V{x,
y)
391
10
Strong and stable causalities
391
10.1
Strong causality
392
10.2
Stable causality
392
Contents
11
Cauchy surface
393
11.1
Domain of dependence. Cauchy horizon
393
11.2
Cauchy surface
394
11.3
Global existence of solutions of linear
wave equations
397
11.4
Sufficient condition from analysis for global
hyperbolicity
397
12
Globally hyperbolic Einsteinian spacetimes
399
12.1
Existence
399
12.2
Global uniqueness
399
12.3
Examples
400
13
Strong cosmic censorship
400
XIII
Singularities
402
1
Introduction
402
2
Criteria for completeness or incompleteness
404
2.1
A completeness criterion
404
2.2
An incompleteness criterion
406
3
Congruence of timelike curves
408
3.1
Definitions
408
3.2
Geodesic deviation
410
3.3
Raychauduri equation
411
3.4
Null geodesic congruence
412
4
First singularity theorem
412
4.1
Conjugate points
412
4.2
Incompleteness theorem
414
5
Trapped surfaces and singularities
414
5.1
Trapped surfaces
414
5.2
Singularities linked to trapped surfaces
418
6
Black holes
418
6.1
Definitions
418
6.2
The Hawking area theorem
420
6.3
The Riemannian Penrose inequality. Case
η
= 3 420
7
Weak cosmic censorship conjectures
421
7.1
Naked singularity
421
7.2
Weak cosmic censorship
422
8
Spherically symmetric Einstein scalar equations
423
8.1
Spherically symmetric spacetimes
423
8.2
Einstein equations in adapted frame
424
8.3
Reduction to one integro-differential equation
425
8.4
Bondi
mass
427
8.5
Global existence for small data
427
8.6
Existence of a global generalized solution
for large data
429
Contents xvii
8.7
Structure
of generalized solutions
432
8.8
Formation of a black hole. Cosmic censorship
433
8.9
Numerical results
434
8.10
Instability of naked singularities
434
9
Cosmological singularities. BKL conjecture
435
10
AVTD behaviour
441
10.1
Definitions
441
10.2 Fuchs
theorem
441
11
Case of 1-parameter spatial isometry
443
11.1
Equations
443
11.2
VTD solutions of the
2 + 1
Einstein evolution
equations
445
11.3
The polarized case
446
11.4
The unpolarized case
449
XIV
Stationary spacetimes and black holes
451
1
Introduction
451
2
Spacetimes with 1-parameter isometry group
452
2.1
Connection and Riemann tensor
454
2.2
Curvature tensor
454
2.3
Ricci
tensor
455
3
Stationary spacetimes
455
3.1
General case
455
3.2
Static spacetimes
457
4
Gravitational solitons
458
4.1
Elementary proofs
458
4.2
Case
η
= 3,
Komar
mass
460
5
Electrovac solitons
465
6
The Einstein-Yang-Mills case
466
7
Stationary black holes
466
7.1
Definitions
466
7.2
Axisymmetry
467
8
The rigidity theorem for black holes
469
9
The Kerr metric and black hole
471
9.1
Kerr metric in Boyer-Linquist coordinates
471
9.2
The Kerr-Schild spacetime
472
10
Uniqueness of stationary black holes (dimension
3+1) 474
10.1
Static black holes
475
10.2
Axisymmetric black holes
475
10.3
Uniqueness of the Kerr black hole
475
11
Further results
476
11.1
Multi
black hole solutions
476
11.2
The Emparan-Reali "black rings"
478
Contents
XV
Global
existence
theorems asymptotically Euclidean
data
482
1
Introduction
482
2
Global existence for small data via the Penrose map
483
2.1
Yang-Mills and associated equations
484
2.2
Quasi-linear wave equations
484
2.3
Cases
η
= 3,
the null condition
487
2.4
Wave maps
488
3
H.
Friedrich conformai
system,
η +
1 = 4 488
3.1
Equations
488
3.2
Friedrich
hyperbolic system
490
4
Einstein's equations in higher dimensions
491
4.1
Conformai
mapping
491
4.2
Transformed equations
492
4.3
Local Cauchy problem in
iž™"1"1, n
> 5
and odd
494
4.4
Global Cauchy problem
496
4.5
Conclusion
497
5
Christodoulou-Klainerman theorem
497
5.1
CK main theorem
497
5.2
Local existence
499
5.3
Global existence
500
6
The Klainerman-Nicolo theorem
504
7
The Linblad-Rodnianski theorem
505
7.1
The Einstein equations in wave coordinates
506
7.2
Initial data
507
7.3
Unknowns and norms
507
7.4
LR theorem
508
XVI
Global existence theorems the cosmological
case
510
1
Introduction
510
2
Gowdy cosmological models
511
3
S1 invariant Einsteinian universes, equations
514
3.1
Introduction
514
3.2
Definition
514
3.3
Equations
515
3.4
Twist potential
515
3.5
Wave map system
516
3.6
Three-dimensional Einstein equations
517
3.7 Teichmüller
parameters.
520
4
S1 invariant Einstein universes, Cauchy problem
521
4.1
Cauchy data
521
4.2
Construction of A when
F
is known
522
Contenis
xix
4.3
Local
in time existence theorem
522
4.4
Global existence theorem
522
4.5
Future complete existence
526
4.6
Einstein-Maxwell-Higgs system
526
4.7
Conclusion
527
5
Andersson-Moncrief theorem
528
5.1
CMC gauge, elliptic system for
N
and
К
529
5.2
SH gauge, elliptic system for
g
and
β
529
5.3
The
Bianchi
equations
530
5.4
Existence theorems
531
5.5
Global existence theorem
532
6
Einstein non-linear scalar field system
533
APPENDICES
I Sobolev spaces on Riemannian manifolds
534
1
Definitions
534
2
Embedding and multiplication properties
535
2.1
Open subsets of Rn
535
2.2
Riemannian manifolds
536
3
Weighted Sobolev spaces
537
3.1
Definitions
537
3.2
Embedding and multiplication properties
538
II Second-order elliptic systems on Riemannian
manifolds
542
1
Linear elliptic systems
542
2
Linear elliptic systems on compact
M
544
2.1
General second-order systems
544
2.2
Poisson
operator
551
2.3
Conformai
Laplace operator
552
3
Asymptotically Euclidean manifolds
553
3.1
Definitions
553
3.2
Second-order linear elliptic systems
554
4
Special systems
560
4.1
Poisson
operator
560
4.2
Conformai
Laplace operator
561
5
Equation
ΔΊφ
=
f (x,
φ),
compact
M
562
6
ΑΊφ
= ƒ
(χ, φ)
on (M,
7)
asymptotically Euclidean
567
ΠΙ
Quasi-diagonal, quasi-linear, second-order hyperbolic
systems
571
1
Introduction
571
2
Wave equation on (V,g)
571
2.1
Definitions
572
Contents
IV
V
2.2
Stress energy tensor. Energy momentum vector
572
2.3
Energy density
574
2.4
Energy equality on a compact domain
575
2.5
Energy inequality in a compact causal
domain
578
2.6
Uniqueness theorem and causality
580
2.7
Case of a strip
581
2.8
Estimate of
и
582
2.9
Cauchy problem
583
2.10
Generalizations
589
! Quasidiagonal linear
systems
590
3.1
Definitions
590
3.2
Stress energy tensor
590
3.3
Energy inequality in a compact causal domain
591
3.4
Case of a strip Vt
592
3.5
Existence, uniqueness, causality and continuity
594
3.6
Higher order estimates
594
3.7
Other hypotheses on
g
600
3.8
Cauchy data in local spaces
601
і
Quasilinear
systems
604
4.1
Semilinear
systems
604
4.2
Further results for
semilinear
equations
607
4.3
Quasilinear
systems
608
5
Global
I existence
613
5.1
Semilinear
systems
613
5.2
Quasilinear
equations
616
General hyperbolic systems
617
1
Introduction
617
2
Leray
hyperbolic systems
617
2.1
Case of one equation
617
2.2
Leray hyperbolic systems
622
3
Leray-Ohya hyperbolic systems
624
4
First-order symmetric hyperbolic systems
625
4.1
FOSH systems on Rn+1
625
4.2
FOSH systems on a sliced manifold
627
Cauchy-Kovalevski and
Fuchs
theorems
631
1
Introduction
631
2
Cauchy-Kovalevski theorem
631
2.1
Linear system
631
2.2
Non-linear system
634
3 Fuchs
theorem
634
3.1
Definitions
634
3.2
Theorem
636
Contents
3.3
Equivalence
with an integral equation
637
3.4
Equivalence with another mapping
638
3.5
Convergence of iterations
641
3.6
Global in space theorem
642
VI Conformai
methods
643
1
Introduction
643
2
Conformai
metrics. Confomorphisms
643
2.1
Connections of
conformai
metrics
643
2.2
Riemann tensors of
conformai
metrics
644
2.3
Ricci
tensors of
conformai
metrics
644
3
The Weyl tensor
645
4
Conformai
transformations of field equations
646
4.1
Maxwell and Yang-Mills equations
646
5
Invariance
of wave equations
647
6
Penrose transform
647
7
Einstein spaces with cosmological constant
650
7.1
Conformai
transformation of
De
Sitter
spacetime
650
7.2
Conformai
transformation of anti-De Sitter
spacetime
650
8
Asymptotically simple spacetimes
650
8.1
Conformai compactifications
650
8.2
Black holes
652
VII
Kaluza-Klein theories
653
1
Introduction
653
2
Isometries
653
3
Kaluza-Klein metrics
654
3.1
Metric in adapted frame
654
3.2
Structure coefficients
655
3.3
Kaluza-Klein connection
656
4
Curvature tensor
657
5
Ricci
tensor and K-K equations
659
6
Equations in
conformai
spacetime metric
660
RELATED PAPERS
Causality of classical supergravity
665
Lecture Notes in Physics
1986,
E. Flaherty
ed.
Springer.
61-84
Gravitation with gauss bonnet terms
689
Australian National University Publications,
1988
R.
Bartnik
ed. 53-72
xxii Contents
Interaction
of gravitational and fluid waves
709
In collaboration with A. Greco, Cericolo
nat.
di
Paleruno 1994
Serie
II
n° 45,
III.
123.
Positive-energy theorems
723
Relativity, Group and Topology II, B. Dewitt and R.
Stora ed.
742-786.
REFERENCES
771
INDEX
781 |
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| author | Choquet-Bruhat, Yvonne 1923- |
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| illustrated | Illustrated |
| index_date | 2024-07-02T21:45:03Z |
| indexdate | 2024-07-09T21:20:13Z |
| institution | BVB |
| isbn | 9780199230723 |
| language | English |
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| spelling | Choquet-Bruhat, Yvonne 1923- Verfasser (DE-588)128916206 aut General relativity and the Einstein equations Yvonne Choquet-Bruhat 1. publ. Oxford Oxford Univ. Press 2009 XXIV, 785 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Oxford mathematical monographs Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd rswk-swf Einstein-Feldgleichungen (DE-588)4013941-4 gnd rswk-swf Allgemeine Relativitätstheorie (DE-588)4112491-1 s Einstein-Feldgleichungen (DE-588)4013941-4 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016683902&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Choquet-Bruhat, Yvonne 1923- General relativity and the Einstein equations Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd Einstein-Feldgleichungen (DE-588)4013941-4 gnd |
| subject_GND | (DE-588)4112491-1 (DE-588)4013941-4 |
| title | General relativity and the Einstein equations |
| title_auth | General relativity and the Einstein equations |
| title_exact_search | General relativity and the Einstein equations |
| title_exact_search_txtP | General relativity and the Einstein equations |
| title_full | General relativity and the Einstein equations Yvonne Choquet-Bruhat |
| title_fullStr | General relativity and the Einstein equations Yvonne Choquet-Bruhat |
| title_full_unstemmed | General relativity and the Einstein equations Yvonne Choquet-Bruhat |
| title_short | General relativity and the Einstein equations |
| title_sort | general relativity and the einstein equations |
| topic | Allgemeine Relativitätstheorie (DE-588)4112491-1 gnd Einstein-Feldgleichungen (DE-588)4013941-4 gnd |
| topic_facet | Allgemeine Relativitätstheorie Einstein-Feldgleichungen |
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