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Tables of Contents for Exact Solutions of Einstein's Field Equations
Chapter/Section Title
Page #
Page Count
Preface
xix
List of tables
xxiii
Notation
xxvii
Introduction
1
8
What are exact solutions, and why study them?
1
2
The development of the subject
3
1
The contents and arrangement of this book
4
3
Using this book as a catalogue
7
2
Part I: General methods
9
148
Differential geometry without a metric
9
21
Introduction
9
1
Differentiable manifolds
10
2
Tangent vectors
12
1
One-forms
13
2
Tensors
15
2
Exterior products and p-forms
17
1
The exterior derivative
18
3
The Lie derivative
21
2
The covariant derivative
23
2
The curvature tensor
25
2
Fibre bundles
27
3
Some topics in Riemannian geometry
30
18
Introduction
30
1
The metric tensor and tetrads
30
4
Calculation of curvature from the metric
34
1
Bivectors
35
2
Decomposition of the curvature tensor
37
3
Spinors
40
3
Conformal transformations
43
2
Discontinuities and junction conditions
45
3
The Petrov classification
48
9
The eigenvalue problem
48
1
The Petrov types
49
4
Principal null directions and determination of the Petrov types
53
4
Classification of the Ricci tensor and the energy-momentum tensor
57
11
The algebraic types of the Ricci tensor
57
3
The energy-momentum tensor
60
3
The energy conditions
63
1
The Rainich conditions
64
1
Perfect fluids
65
3
Vector fields
68
7
Vector fields and their invariant classification
68
4
Timelike unit vector fields
70
1
Geodesic null vector fields
70
2
Vector fields and the curvature tensor
72
3
Timelike unit vector fields
72
2
Null vector fields
74
1
The Newman--Penrose and related formalisms
75
16
The spin coefficients and their transformation laws
75
3
The Ricci equations
78
3
The Bianchi identities
81
3
The GHP calculus
84
2
Geodesic null congruences
86
1
The Goldberg--Sachs theorem and its generalizations
87
4
Continuous groups of transformations; isometry and homothety groups
91
21
Lie groups and Lie algebras
91
4
Enumeration of distinct group structures
95
2
Transformation groups
97
1
Groups of motions
98
3
Spaces of constant curvature
101
3
Orbits of isometry groups
104
6
Simply-transitive groups
105
1
Multiply-transitive groups
106
4
Homothety groups
110
2
Invariants and the characterization of geometries
112
17
Scalar invariants and covariants
113
3
The Cartan equivalence method for space-times
116
4
Calculating the Cartan scalars
120
5
Determination of the Petrov and Segre types
120
4
The remaining steps
124
1
Extensions and applications of the Cartan method
125
1
Limits of families of space-times
126
3
Generation techniques
129
28
Introduction
129
1
Lie symmetries of Einstein's equations
129
5
Point transformations and their generators
129
2
How to find the Lie point symmetries of a given differential equation
131
1
How to use Lie point symmetries: similarity reduction
132
2
Symmetries more general than Lie symmetries
134
3
Contact and Lie--Backlund symmetries
134
1
Generalized and potential symmetries
134
3
Prolongation
137
8
Integral manifolds of differential forms
137
3
Isovectors, similarity solutions and conservation laws
140
1
Prolongation structures
141
4
Solutions of the linearized equations
145
1
Backlund transformations
146
2
Riemann--Hilbert problems
148
1
Harmonic maps
148
3
Variational Backlund transformations
151
1
Hirota's method
152
1
Generation methods including perfect fluids
152
5
Methods using the existence of Killing vectors
152
3
Conformal transformations
155
2
Part II: Solutions with groups of motions
157
250
Classification of solutions with isometries or homotheties
157
14
The possible space-times with isometries
157
2
Isotropy and the curvature tensor
159
3
The possible space-times with proper homothetic motions
162
5
Summary of solutions with homotheties
167
4
Homogeneous space-times
171
12
The possible metrics
171
3
Homogeneous vacuum and null Einstein-Maxwell space-times
174
1
Homogeneous non-null electromagnetic fields
175
2
Homogeneous perfect fluid solutions
177
3
Other homogeneous solutions
180
1
Summary
181
2
Hypersurface-homogeneous space-times
183
27
The possible metrics
183
5
Metrics with a G6 on V3
183
1
Metrics with a G4 on V3
183
4
Metrics with a G3 on V3
187
1
Formulations of the field equations
188
6
Vacuum, A-term and Einstein--Maxwell solutions
194
10
Solutions with multiply-transitive groups
194
2
Vacuum spaces with a G3 on V3
196
3
Einstein spaces with a G3 on V3
199
2
Einstein--Maxwell solutions with a G3 on V3
201
3
Perfect fluid solutions homogeneous on T3
204
3
Summary of all metrics with Gr on V3
207
3
Spatially-homogeneous perfect fluid cosmologies
210
16
Introduction
210
1
Robertson--Walker cosmologies
211
3
Cosmologies with a G4 on S3
214
4
Cosmologies with a G3 on S3
218
8
Groups G3 on non-null orbits V2. Spherical and plane symmetry
226
21
Metric, Killing vectors, and Ricci tensor
226
2
Some implications of the existence of an isotropy group I1
228
1
Spherical and plane symmetry
229
1
Vacuum, Einstein--Maxwell and pure radiation fields
230
5
Timelike orbits
230
1
Spacelike orbits
231
1
Generalized Birkhoff theorem
232
1
Spherically- and plane-symmetric fields
233
2
Dust solutions
235
2
Perfect fluid solutions with plane, spherical or pseudospherical symmetry
237
6
Some basic properties
237
1
Static solutions
238
1
Solutions without shear and expansion
238
1
Expanding solutions without shear
239
1
Solutions with nonvanishing shear
240
3
Plane-symmetric perfect fluid solutions
243
4
Static solutions
243
1
Non-static solutions
244
3
Spherically-symmetric perfect fluid solutions
247
17
Static solutions
247
4
Field equations and first integrals
247
3
Solutions
250
1
Non-static solutions
251
13
The basic equations
251
2
Expanding solutions without shear
253
7
Solutions with non-vanishing shear
260
4
Groups G2 and G1 on non-null orbits
264
11
Groups G2 on non-null orbits
264
4
Subdivisions of the groups G2
264
1
Groups G2I on non-null orbits
265
2
G2II on non-null orbits
267
1
Boost-rotation-symmetric space-times
268
3
Group G1 on non-null orbits
271
4
Stationary gravitational fields
275
17
The projection formalism
275
2
The Ricci tensor on Σ3
277
1
Conformal transformation of Σ3 and the field equations
278
1
Vacuum and Einstein--Maxwell equations for stationary fields
279
2
Geodesic eigenrays
281
2
Static fields
283
4
Definitions
283
1
Vacuum solutions
284
1
Electrostatic and magnetostatic Einstein--Maxwell fields
284
2
Perfect fluid solutions
286
1
The conformastationary solutions
287
2
Conformastationary vacuum solutions
287
1
Conformastationary Einstein--Maxwell fields
288
1
Multipole moments
289
3
Stationary axisymmetric fields: basic concepts and field equations
292
12
The Killing vectors
292
1
Orthogonal surfaces
293
3
The metric and the projection formalism
296
2
The field equations for stationary axisymmetric Einstein--Maxwell fields
298
1
Various forms of the field equations for stationary axisymmetric vacuum fields
299
3
Field equations for rotating fluids
302
2
Stationary axisymmetric vacuum solutions
304
15
Introduction
304
1
Static axisymmetric vacuum solutions (Weyl's class)
304
5
The class of solutions U = U(ω) (Papapetrou's class)
309
1
The class of solutions S = S(A)
310
1
The Kerr solution and the Tomimatsu--Sato class
311
2
Other solutions
313
3
Solutions with factor structure
316
3
Non-empty stationary axisymmetric solutions
319
22
Einstein--Maxwell fields
319
11
Electrostatic and magnetostatic solutions
319
3
Type D solutions: A general metric and its limits
322
3
The Kerr--Newman solution
325
3
Complexification and the Newman--Janis `complex trick'
328
1
Other solutions
329
1
Perfect fluid solutions
330
11
Line element and general properties
330
1
The general dust solution
331
2
Rigidly rotating perfect fluid solutions
333
4
Perfect fluid solutions with differential rotation
337
4
Groups G2I on spacelike orbits: cylindrical symmetry
341
17
General remarks
341
1
Stationary cylindrically-symmetric fields
342
8
Vacuum fields
350
4
Einstein--Maxwell and pure radiation fields
354
4
Inhomogeneous perfect fluid solutions with symmetry
358
17
Solutions with a maximal H3 on S3
359
2
Solutions with a maximal H3 on T3
361
1
Solutions with a G2 on S2
362
12
Diagonal metrics
363
9
Non-diagonal solutions with orthogonal transitivity
372
1
Solutions without orthogonal transitivity
373
1
Solutions with a G1 or a H2
374
1
Groups on null orbits. Plane waves
375
12
Introduction
375
1
Groups G3 on N3
376
1
Groups G2 on N2
377
2
Null Killing vectors (G1 on N1)
379
4
Non-twisting null Killing vector
380
2
Twisting null Killing vector
382
1
The plane-fronted gravitational waves with parallel rays (pp-waves)
383
4
Collision of plane waves
387
20
General features of the collision problem
387
2
The vacuum field equations
389
3
Vacuum solutions with collinear polarization
392
2
Vacuum solutions with non-collinear polarization
394
3
Einstein--Maxwell fields
397
6
Stiff perfect fluids and pure radiation
403
4
Stiff perfect fluids
403
2
Pure radiation (null dust)
405
2
Part III: Algebraically special solutions
407
111
The various classes of algebraically special solutions. Some algebraically general solutions
407
9
Solutions of Petrov type II, D. III or N
407
5
Petrov type D solutions
412
1
Conformally flat solutions
413
1
Algebraically general vacuum solutions with geodesic and non-twisting rays
413
3
The line element for metrics with κ = σ = 0 = R11 = R14 = R44, Θ + iω ≠ 0
416
6
The line element in the case with twisting rays (ω ≠ 0)
416
4
The choice of the null tetrad
416
2
The coordinate frame
418
2
Admissible tetrad and coordinate transformations
420
1
The line element in the case with non-twisting rays (ω = 0)
420
2
Robinson--Trautman solutions
422
15
Robinson--Trautman vacuum solutions
422
5
The field equations and their solutions
422
2
Special cases and explicit solutions
424
3
Robinson--Trautman Einstein--Maxwell fields
427
8
Line element and field equations
427
2
Solutions of type III, N and O
429
1
Solutions of type D
429
2
Type II solutions
431
4
Robinson--Trautman pure radiation fields
435
1
Robinson--Trautman solutions with a cosmological constant Λ
436
1
Twisting vacuum solutions
437
18
Twisting vacuum solutions -- the field equations
437
5
The structure of the field equations
437
1
The integration of the main equations
438
2
The remaining field equations
440
1
Coordinate freedom and transformation properties
441
1
Some general classes of solutions
442
9
Characterization of the known classes of solutions
442
3
The case ∂ζ = ∂ζ(G2 - ∂ζ) ≠ 0
445
1
The case ∂ζI = ∂ζ(G2 - ∂ζG) ≠ 0, L,u = 0
446
1
The case I = 0
447
2
The case I = 0 = L,u
449
1
Solutions independent of ζ and ζ
450
1
Solutions of type N (Ψ2 = 0 = Ψ3)
451
1
Solutions of type III (Ψ2 = 0, Ψ3 ≠ 0)
452
1
Solutions of type D (3Ψ2Ψ4 = 2Ψ23, Ψ2 ≠ 0)
452
2
Solutions of type II
454
1
Twisting Einstein--Maxwell and pure radiation fields
455
15
The structure of the Einstein-Maxwell field equations
455
1
Determination of the radial dependence of the metric and the Maxwell field
456
2
The remaining field equations
458
1
Charged vacuum metrics
459
1
A class of radiative Einstein--Maxwell fields (Φ02 ≠ 0)
460
1
Remarks concerning solutions of the different Petrov types
461
2
Pure radiation fields
463
7
The field equations
463
1
Generating pure radiation fields from vacuum by changing P
464
2
Generating pure radiation fields from vacuum by changing m
466
1
Some special classes of pure radiation fields
467
3
Non-diverging solutions (Kundt's class)
470
15
Introduction
470
1
The line element for metrics with Θ + iω = 0
470
2
The Ricci tensor components
472
1
The structure of the vacuum and Einstein--Maxwell equation
473
3
Vacuum solutions
476
4
Solutions of types III and N
476
2
Solutions of types D and II
478
2
Einstein--Maxwell null fields and pure radiation fields
480
1
Einstein--Maxwell non-null fields
481
2
Solutions including a cosmological constant Λ
483
2
Kerr--Schild metrics
485
21
General properties of Kerr--Schild metrics
485
7
The origin of the Kerr--Schild--Trautman ansatz
485
1
The Ricci tensor, Riemann tensor and Petrov type
485
2
Field equations and the energy-momentum tensor
487
1
A geometrical interpretation of the Kerr--Schild ansatz
487
2
The Newman--Penrose formalism for shearfree and geodesic Kerr--Schild metrics
489
3
Kerr--Schild vacuum fields
492
1
The case ρ = - (Θ + iω) ≠ 0
492
1
The case ρ = - (Θ + iω) = 0
493
1
Kerr--Schild Einstein--Maxwell fields
493
4
The case ρ = - (Θ + iω) ≠ 0
493
2
The case ρ = - (Θ + iω) = 0
495
2
Kerr--Schild pure radiation fields
497
2
The case ρ ≠ 0, σ = 0
497
2
The case σ ≠ 0
499
1
The case ρ = σ = 0
499
1
Generalizations of the Kerr--Schild ansatz
499
7
General properties and results
499
2
Non-flat vacuum to vacuum
501
1
Vacuum to electrovac
502
1
Perfect fluid to perfect fluid
503
3
Algebraically special perfect fluid solutions
506
12
Generalized Robinson--Trautman solutions
506
4
Solutions with a geodesic, shearfree, non-expanding multiple null eigenvector
510
2
Type D solutions
512
3
Solutions with κ = ν = 0
513
1
Solutions with κ ≠ 0, ν ≠ 0
513
2
Type III and type N solutions
515
3
Part IV: Special methods
518
87
Application of generation techniques to general relativity
518
35
Methods using harmonic maps (potential space symmetries)
518
11
Electrovacuum fields with one Killing vector
518
3
The group SU(2,1)
521
4
Complex invariance transformations
525
1
Stationary axisymmetric vacuum fields
526
3
Prolongation structure for the Ernst equation
529
3
The linearized equations, the Kinnersley--Chitre B group and the Hoenselaers--Kinnersley--Xanthopoulos transformations
532
6
The field equations
532
2
Infinitesimal transformations and transformations preserving Minkowski space
534
1
The Hoenselaers--Kinnersley--Xanthopoulos transformation
535
3
Backlund transformations
538
5
The Belinski--Zakharov technique
543
4
The Riemann--Hilbert problem
547
2
Some general remarks
547
1
The Neugebauer--Meinel rotating disc solution
548
1
Other approaches
549
1
Einstein--Maxwell fields
550
1
The case of two space-like Killing vectors
550
3
Special vector and tensor fields
553
18
Space-times that admit constant vector and tensor fields
553
3
Constant vector fields
553
1
Constant tensor fields
554
2
Complex recurrent, conformally recurrent, recurrent and symmetric spaces
556
3
The definitions
556
1
Space-times of Petrov type D
557
1
Space-times of type N
557
1
Space-times of type O
558
1
Killing tensors of order two and Killing--Yano tensors
559
5
The basic definitions
559
1
First integrals, separability and Killing or Killing--Yano tensors
560
1
Theorems on Killing and Killing--Yano tensors in four-dimensional space-times
561
3
Collineations and conformal motions
564
7
The basic definitions
564
1
Proper curvature collineations
565
1
General theorems on conformal motions
565
2
Non-conformally flat solutions admitting proper conformal motions
567
4
Solutions with special subspaces
571
9
The basic formulae
571
2
Solutions with flat three-dimensional slices
573
4
Vacuum solutions
573
1
Perfect fluid and dust solutions
573
4
Perfect fluid solutions with conformally flat slices
577
2
Solutions with other intrinsic symmetries
579
1
Local isometric embedding of four-dimensional Riemannian manifolds
580
25
The why of embedding
580
1
The basic formulae governing embedding
581
2
Some theorems on local isometric embedding
583
4
General theorems
583
1
Vector and tensor fields and embedding class
584
2
Groups of motions and embedding class
586
1
Exact solutions of embedding class one
587
9
The Gauss and Codazzi equations and the possible types of Ωab
587
1
Conformally flat perfect fluid solutions of embedding class one
588
3
Type D perfect fluid solutions of embedding class one
591
3
Pure radiation field solutions of embedding class one
594
2
Exact solutions of embedding class two
596
7
The Gauss--Codazzi--Ricci equations
596
2
Vacuum solutions of embedding class two
598
1
Conformally flat solutions
599
4
Exact solutions of embedding class p > 2
603
2
Part V: Tables
605
10
The interconnections between the main classification schemes
605
10
Introduction
605
1
The connection between Petrov types and groups of motions
606
3
Tables
609
6
References
615
75
Index
690