Exact Solutions of Einstein's Field Equations

Contents......Page 9
Preface......Page 21
List of Tables......Page 25
Metric and tetrads......Page 29
Connection and curvature......Page 30
Symmetries......Page 31
1.1 What are exact solutions, and why study them?......Page 33
1.2 The development of the subject......Page 35
1.3 The contents and arrangement of this book......Page 36
1.4 Using this book as a catalogue......Page 39
2.1 Introduction......Page 41
2.2 Differentiable manifolds......Page 42
2.3 Tangent vectors......Page 44
2.4 One-forms......Page 45
2.5 Tensors......Page 47
2.6 Exterior products and p-forms......Page 49
2.7 The exterior derivative......Page 50
2.8 The Lie derivative......Page 53
2.9 The covariant derivative......Page 55
2.10 The curvature tensor......Page 57
2.11 Fibre bundles......Page 59
3.2 The metric tensor and tetrads......Page 62
3.3 Calculation of curvature from the metric......Page 66
3.4 Bivectors......Page 67
3.5 Decomposition of the curvature tensor......Page 69
3.6 Spinors......Page 72
3.7 Conformal transformations......Page 75
3.8 Discontinuities and junction conditions......Page 77
4.1 The eigenvalue problem......Page 80
4.2 The Petrov types......Page 81
4.3 Principal null directions and determination of the Petrov types......Page 85
5.1 The algebraic types of the Ricci tensor......Page 89
5.2 The energy-momentum tensor......Page 92
5.3 The energy conditions......Page 95
5.4 The Rainich conditions......Page 96
5.5 Perfect fluids......Page 97
6.1 Vector fields and their invariant classification......Page 100
6.2.1 Timelike unit vector fields......Page 104
6.2.2 Null vector fields......Page 106
7.1 The spin coefficients and their transformation laws......Page 107
7.2 The Ricci equations......Page 110
7.3 The Bianchi identities......Page 113
7.4 The GHP calculus......Page 116
7.5 Geodesic null congruences......Page 118
7.6 The Goldberg–Sachs theorem and its generalizations......Page 119
8.1 Lie groups and Lie algebras......Page 123
8.2 Enumeration of distinct group structures......Page 127
8.3 Transformation groups......Page 129
8.4 Groups of motions......Page 130
8.5 Spaces of constant curvature......Page 133
8.6 Orbits of isometry groups......Page 136
8.6.1 Simply-transitive groups......Page 137
8.6.2 Multiply-transitive groups......Page 138
8.7 Homothety groups......Page 142
9 Invariants and the characterization of geometries......Page 144
9.1 Scalar invariants and covariants......Page 145
9.2 The Cartan equivalence method for space-times......Page 148
9.3.1 Determination of the Petrov and Segre types......Page 152
9.3.2 The remaining steps......Page 156
9.4 Extensions and applications of the Cartan method......Page 157
9.5 Limits of families of space-times......Page 158
10.2.1 Point transformations and their generators......Page 161
10.2.2 How to find the Lie point symmetries of a given differential equation......Page 163
10.2.3 How to use Lie point symmetries: similarity reduction......Page 164
10.3.2 Generalize and potential symmetries......Page 166
10.4.1 Integral manifolds of differential forms......Page 169
10.4.2 Isovectors, similarity solutions and conservation laws......Page 172
10.4.3 Prolongation structures......Page 173
10.5 Solutions of the linearized equations......Page 177
10.6 Bäcklund transformations......Page 178
10.8 Harmonic maps......Page 180
10.9 Variational Bäcklund transformations......Page 183
10.11.1 Methods using the existence of Killing vectors......Page 184
10.11.2 Conformal transformations......Page 187
11.1 The possible space-times with isometries......Page 189
11.2 Isotropy and the curvature tensor......Page 191
11.3 The possible space-times with proper homothetic motions......Page 194
11.4 Summary of solutions with homotheties......Page 199
12.1 The possible metrics......Page 203
12.2 Homogeneous vacuum and null Einstein-Maxwell space-times......Page 206
12.3 Homogeneous non-null electromagnetic fields......Page 207
12.4 Homogeneous perfect fluid solutions......Page 209
12.5 Other homogeneous solutions......Page 212
12.6 Summary......Page 213
13.1.2 Metrics with a G4 on V3......Page 215
Spatial rotation isotropy......Page 216
Null rotation isotropy......Page 218
13.1.3 Metrics with a G3 on V3......Page 219
13.2 Formulations of the field equations......Page 220
13.3.1 Solutions with multiply-transitive groups......Page 226
13.3.2 Vacuum spaces with a G3 on V3......Page 228
13.3.3 Einstein spaces with a G3 on V3......Page 231
13.3.4 Einstein–Maxwell solutions with a G3 on V3......Page 233
13.4 Perfect fluid solutions homogeneous on T3......Page 236
13.5 Summary of all metrics with Gr on V3......Page 239
14.1 Introduction......Page 242
14.2 Robertson–Walker cosmologies......Page 243
14.3 Cosmologies with a G4 on S3......Page 246
14.4 Cosmologies with a G3 on S3......Page 250
15.1 Metric, Killing vectors, and Ricci tensor......Page 258
15.2 Some implications of the existence of an isotropy group I1......Page 260
15.3 Spherical and plane symmetry......Page 261
15.4.1 Timelike orbits......Page 262
15.4.2 Spacelike orbits......Page 263
15.4.3 Generalize Birkhoff theorem......Page 264
15.4.4 Spherically- and plane-symmetric fields......Page 265
15.5 Dust solutions......Page 267
15.6.1 Some basic properties......Page 269
15.6.3 Solutions without shear and expansion......Page 270
15.6.4 Expanding solutions without shear......Page 271
Solutions with shear but without acceleration......Page 272
15.7.1 Static solutions......Page 275
15.7.2 Non-static solutions......Page 276
16.1.1 Field equations and first integrals......Page 279
16.1.2 Solutions......Page 282
16.2.1 The basic equations......Page 283
Some basic properties......Page 285
Known classes of solutions of…......Page 286
The Kustaanheimo–Qvist class of solutions......Page 288
Solutions with a homogeneous distribution of matter Mu = Mu(t)......Page 290
Solutions with an equation of state p = p (Mu)......Page 291
16.2.3 Solutions with non-vanishing shear......Page 292
Solutions with shear, acceleration and expansion......Page 293
17.1.1 Subdivisions of the groups G2......Page 296
17.1.2 Groups G2I on non-null orbits......Page 297
17.1.3 G2II on non-null orbits......Page 299
17.2 Boost-rotation-symmetric space-times......Page 300
17.3 Group G1 on non-null orbits......Page 303
18.1 The projection formalism......Page 307
18.2 The Ricci tensor on…......Page 309
18.3 Conformal transformation of…......Page 310
18.4 Vacuum and Einstein–Maxwell equations for stationary fields......Page 311
18.5 Geodesic eigenrays......Page 313
18.6.1 Definitions......Page 315
18.6.3 Electrostatic and magnetostatic Einstein–Maxwell fields......Page 316
18.6.4 Perfect fluid solutions......Page 318
18.7.1 Conformastationary vacuum solutions......Page 319
18.7.2 Conformastationary Einstein–Maxwell fields......Page 320
18.8 Multipole moments......Page 321
19.1 The Killing vectors......Page 324
19.2 Orthogonal surfaces......Page 325
19.3 The metric and the projection formalism......Page 328
19.4 The field equations for stationary axisymmetric Einstein–Maxwell fields......Page 330
19.5 Various forms of the field equations for stationary axisymmetric vacuum fields......Page 331
19.6 Field equations for rotating fluids......Page 334
20.2 Static axisymmetric vacuum solutions (Weyl’s class)......Page 336
20.3 The class of solutions U = U (Omega) (Papapetrou’s class)......Page 341
20.4 The class of solutions S = S(A )......Page 342
20.5 The Kerr solution and the Tomimatsu–Sato class......Page 343
20.6 Other solutions......Page 345
20.7 Solutions with factor structure......Page 348
21.1.1 Electrostatic and magnetostatic solutions......Page 351
21.1.2 Type D solutions: A general metric and its limits......Page 354
21.1.3 The Kerr–Newman solution......Page 357
21.1.4 Complexification and the Newman–Janis ‘complex trick’......Page 360
21.1.5 Other solutions......Page 361
21.2.1 Line element and general properties......Page 362
21.2.2 The general dust solution......Page 363
21.2.3 Rigidly rotating perfect fluid solutions......Page 365
21.2.4 Perfect fluid solutions with differential rotation......Page 369
22.1 General remarks......Page 373
22.2 Stationary cylindrically-symmetric fields......Page 374
…term solutions......Page 375
Einstein–Maxwell fields......Page 376
Static perfect fluid solutions......Page 378
Stationary perfect fluids......Page 381
22.3 Vacuum fields......Page 382
22.4 Einstein–Maxwell and pure radiation fields......Page 386
23 Inhomogeneous perfect fluid solutions with symmetry......Page 390
23.1 Solutions with a maximal H3 on S3......Page 391
23.2 Solutions with a maximal H3 on T3......Page 393
23.3 Solutions with a G2 on S2......Page 394
23.3.1 Diagonal metrics......Page 395
23.3.2 Non-diagonal solutions with orthogonal transitivity......Page 404
23.3.3 Solutions without orthogonal transitivity......Page 405
23.4 Solutions with a G1 or a H2......Page 406
24.1 Introduction......Page 407
24.2 Groups G3 on N3......Page 408
24.3 Groups G2 on N2......Page 409
24.4 Null Killing vectors (G1 on N1)......Page 411
24.4.1 Non-twistingnull Killing vector......Page 412
24.4.2 Twisting null Killing vector......Page 414
24.5 The plane-fronted gravitational waves with parallel rays (pp-waves)......Page 415
25.1 General features of the collision problem......Page 419
25.2 The vacuum field equations......Page 421
25.3 Vacuum solutions with collinear polarization......Page 424
25.4 Vacuum solutions with non-collinear polarization......Page 426
25.5 Einstein–Maxwell fields......Page 429
25.6.1 Stiff perfect fluids......Page 435
25.6.2 Pure radiation (null dust)......Page 437
26.1 Solutions of Petrov type II, D, III or N......Page 439
26.2 Petrov type D solutions......Page 444
26.4 Algebraically general vacuum solutions with geodesic and non-twisting rays......Page 445
27.1.1 The choice of the null tetrad......Page 448
27.1.2 The coordinate frame......Page 450
27.2 The line element in the case with non-twisting rays (Omaga = 0)......Page 452
28.1.1 The field equations and their solutions......Page 454
Type N solutions,…......Page 456
Type D solutions,…......Page 457
28.2.1 Line element and field equations......Page 459
28.2.3 Solutions of type D......Page 461
28.2.4 Type II solutions......Page 463
28.3 Robinson–Trautman pure radiation fields......Page 467
28.4 Robinson–Trautman solutions with a cosmological constant…......Page 468
29.1.1 The structure of the field equations......Page 469
29.1.2 The integration of the main equations......Page 470
29.1.3 The remaining field equations......Page 472
29.1.4 Coordinate freedom and transformation properties......Page 473
29.2.1 Characterization of the known classes of solutions......Page 474
29.2.2 The case…......Page 477
29.2.3 The case…......Page 478
29.2.4 The case I = 0......Page 479
29.2.5 The case I = 0 = L,u......Page 481
29.2.6 Solutions independent of…......Page 482
29.3 Solutions of type N…......Page 483
29.5 Solutions of type D…......Page 484
29.6 Solutions of type II......Page 486
30.1 The structure of the Einstein–Maxwell field equations......Page 487
30.2 Determination of the radial dependence of the metric and the Maxwell field......Page 488
30.3 The remaining field equations......Page 490
30.4 Charged vacuum metrics......Page 491
30.5 A class of radiative Einstein–Maxwell fields…......Page 492
30.6 Remarks concerning solutions of the different Petrov types......Page 493
30.7.1 The field equations......Page 495
30.7.2 Generating pure radiation fields from vacuum by changing P......Page 496
30.7.3 Generating pure radiation fields from vacuum by changing m......Page 498
30.7.4 Some special classes of pure radiation fields......Page 499
31.2 The line element for metrics with…......Page 502
31.3 The Ricci tensor components......Page 504
31.4 The structure of the vacuum and Einstein–Maxwell equation......Page 505
31.5.1 Solutions of types III and N......Page 508
31.5.2 Solutions of types D and II......Page 510
31.6 Einstein–Maxwell null fields and pure radiation fields......Page 512
31.7 Einstein–Maxwell non-null fields......Page 513
31.8 Solutions including a cosmological constant…......Page 515
32.1.2 The Ricci tensor, Riemann tensor and Petrov type......Page 517
32.1.4 A geometrical interpretation of the Kerr–Schild ansatz......Page 519
32.1.5 The Newman–Penrose formalism for shearfree and geodesic Kerr–Schild metrics......Page 521
32.2.1 The case…......Page 524
32.3.1 The case…......Page 525
32.3.2 The case…......Page 527
32.4.1 The case…......Page 529
32.5.1 General properties and results......Page 531
32.5.2 Non-flat vacuum to vacuum......Page 533
32.5.3 Vacuum to electrovac......Page 534
32.5.4 Perfect fluid to perfect fluid......Page 535
33.1 Generalized Robinson–Trautman solutions......Page 538
33.2 Solutions with a geodesic, shearfree, non-expanding multiple null eigenvector......Page 542
33.3 Type D solutions......Page 544
33.3.2 Solutions with…......Page 545
33.4 Type III and type N solutions......Page 547
34.1.1 Electrovacuum fields with one Killing vector......Page 550
34.1.2 The group SU (2,1)......Page 553
34.1.3 Complex invariance transformations......Page 557
34.1.4 Stationary axisymmetric vacuum fields......Page 558
34.2 Prolongation structure for the Ernst equation......Page 561
34.3.1 The field equations......Page 564
34.3.2 Infinitesimal transformations and transformations preserving Minkowski space......Page 566
34.3.3 The Hoenselaers–Kinnersley–Xanthopoulos transformation......Page 567
34.4 Bäcklund transformations......Page 570
34.5 The Belinski–Zakharov technique......Page 575
34.6.1 Some general remarks......Page 579
34.7 Other approaches......Page 581
34.9 The case of two space-like Killing vectors......Page 582
35.1.1 Constant vector fields......Page 585
35.1.2 Constant tensor fields......Page 586
35.2.1 The definitions......Page 588
35.2.3 Space-times of type N......Page 589
35.2.4 Space-times of type O......Page 590
35.3.1 The basic definitions......Page 591
35.3.2 Frist integrals, separability and Killing or Killing–Yano tensors......Page 592
Theorems and results on Killing tensors K......Page 593
35.4.1 The basic definitions......Page 596
35.4.3 General theorems on conformal motions......Page 597
35.4.4 Non-conformally flat solutions admitting proper conformal motions......Page 599
36.1 The basic formulae......Page 603
36.2.2 Perfect fluid and dust solutions......Page 605
36.3 Perfect fluid solutions with conformally flat slices......Page 609
36.4 Solutions with other intrinsic symmetries......Page 611
37.1 The why of embedding......Page 612
37.2 The basic formulae governing embedding......Page 613
37.3.1 General theorems......Page 615
37.3.2 Vector and tensor fields and embedding class......Page 616
37.3.3 Groups of motions and embedding class......Page 618
37.4.1 The Gauss and Codazzi equations and the possible types of…......Page 619
37.4.2 Conformally flat perfect fluid solutions of embedding class one......Page 620
37.4.3 Type D perfect fluid solutions of embedding class one......Page 623
37.4.4 Pure radiation field solutions of embedding class one......Page 626
37.5.1 The Gauss–Codazzi–Ricci equations......Page 628
37.5.2 Vacuum solutions of embedding class two......Page 630
37.5.3 Conformally flat solutions......Page 631
Conformally flat solutions with a perfect fluid......Page 632
37.6 Exact solutions of embedding class p > 2......Page 635
38.1 Introduction......Page 637
38.2 The connection between Petrov types and groups of motions......Page 638
38.3 Tables......Page 641
References......Page 647
Index......Page 722