Exact Space-Times in Einstein’s General Relativity

Cover......Page 1
Half-title......Page 3
Series-title......Page 5
Title......Page 7
Copyright......Page 8
Contents......Page 9
Preface......Page 17
1 Introduction......Page 21
2.1 Local geometry......Page 25
2.1.1 Curvature......Page 26
2.1.2 Algebraic classification......Page 28
2.1.3 Geodesics and geometrical optics......Page 30
2.1.4 Symmetries......Page 35
2.1.5 Continuity......Page 36
2.2.1 Electromagnetic field......Page 37
2.2.2 Perfect fluid and pure radiation......Page 39
2.3 Global structure......Page 40
2.3.1 Singularities......Page 41
2.3.2 Horizons......Page 43
2.3.3 Causal and conformal relations......Page 45
3.1 Coordinate representations......Page 47
3.2 An aside on the Einstein static universe......Page 49
3.3 Conformal structure of Minkowski space......Page 51
3.4 A simple model for a cosmic string......Page 55
3.4.1 A spinning cosmic string......Page 56
3.5 Coordinates adapted to uniform acceleration......Page 57
3.6 Other representations of Minkowski space......Page 60
4.1 Global representation......Page 61
4.2 Conformal structure......Page 63
4.3 Spherical coordinates and horizons......Page 64
4.4.1 Conformally flat coordinates......Page 68
4.4.2 Coordinates with a negative spatial curvature......Page 70
4.4.3 Constant curvature coordinates......Page 71
4.4.4 Bianchi III and Kantowski–Sachs coordinates......Page 73
4.5 Remarks and references......Page 74
5.1 Global representation......Page 75
5.2 Conformal structure......Page 78
5.3 Other standard coordinates......Page 81
5.4 Remarks and references......Page 86
6 Friedmann–Lemaitre–Robertson–Walker space-times......Page 87
6.1 Geometry and standard coordinates......Page 88
6.2 Matter content and dynamics......Page 91
6.3 Explicit solutions......Page 94
6.3.1 Vacuum FLRW space-times......Page 95
6.3.2 FLRW space-times without......Page 96
6.3.3 FLRW space-times with......Page 97
6.4.1 FLRW space-times without......Page 100
6.4.2 FLRW space-times with......Page 105
6.5 Some principal properties and references......Page 109
7.1 The Bertotti–Robinson solution......Page 115
7.2 Other direct-product space-times......Page 118
7.2.1 Nariai and anti-Nariai space-times......Page 120
7.2.2 Plebanski´–Hacyan space-times......Page 121
7.3 The Melvin solution......Page 122
8 Schwarzschild space-time......Page 126
8.1 Schwarzschild coordinates......Page 127
8.2 Eddington–Finkelstein coordinates and black holes......Page 130
8.2.1 Outgoing coordinates and white holes......Page 134
8.3 Kruskal–Szekeres coordinates......Page 136
8.4 Conformal structure......Page 139
8.4.1 The case when m < 0......Page 142
8.5 Interior solutions......Page 143
8.6 Exterior solutions......Page 145
9.1 Other A-metrics......Page 147
9.1.1 The AII-metrics…......Page 148
9.1.2 The AIII-metrics…......Page 153
9.2 The Reissner–Nordstrom solution......Page 156
9.2.1 Singularity and horizons......Page 157
9.2.2 Conformal structure......Page 162
9.2.3 (Hyper-)extreme Reissner–Nordstrom space-times......Page 166
9.3 The Schwarzschild–Melvin solution......Page 169
9.4 The Schwarzschild–de Sitter solution......Page 170
9.4.1 Black holes in de Sitter space-time......Page 171
9.4.2 Extreme holes in de Sitter space-time......Page 173
9.4.3 Black holes in anti-de Sitter space-time......Page 175
9.5 Vaidya’s radiating Schwarzschild solution......Page 176
9.5.1 Radiation from a star......Page 179
9.5.2 The Vaidya solution for incoming radiation......Page 181
9.6 Further generalisations......Page 185
10.1 The Weyl metric......Page 189
10.1.1 Flat solutions within a Weyl metric......Page 191
10.1.2 The Weyl solutions......Page 192
10.2 Static cylindrically symmetric space-times......Page 193
10.3 The Schwarzschild solution......Page 198
10.4 The Zipoy–Voorhees solution......Page 199
10.5 The Curzon–Chazy solution......Page 201
10.6 The Curzon–Chazy two-particle solution......Page 204
10.7 Further comments on the Weyl metrics......Page 206
10.8 Axially symmetric electrovacuum space-times......Page 208
10.8.1 Equilibrium configurations with distinct sources......Page 211
11.1 The Kerr solution......Page 214
11.1.1 Horizons......Page 216
11.1.2 Curvature singularity and central disc......Page 218
11.1.3 Ergoregions......Page 221
11.1.4 Conformal diagrams......Page 223
11.1.5 Further comments on the Kerr solution......Page 226
11.2 The Kerr–Newman solution......Page 228
11.3 The Kerr–Newman–(anti-)de Sitter solution......Page 229
12 Taub–NUT space-time......Page 233
12.1 The NUT solution......Page 234
12.1.1 The NUT regions in Weyl coordinates......Page 236
12.1.2 The Taub region......Page 237
12.1.3 The singularity on the axis and a gravitomagnetic interpretation......Page 238
12.1.4 The occurrence of closed timelike curves......Page 241
12.1.5 Conformal structure......Page 242
12.2 Misner’s construction of a singularity-free solution......Page 245
12.2.1 Conformal structure in Misner’s interpretation......Page 247
12.3 The wider family of NUT solutions......Page 249
12.3.1 The case…......Page 250
The existence of closed timelike curves......Page 251
The conformal structure......Page 252
12.3.2 The case…......Page 253
12.4 Further generalisations of the Taub–NUT solutions......Page 255
13.1 Cylindrically symmetric solutions......Page 258
13.1.1 The Lewis family of vacuum solutions......Page 259
The Weyl class of Lewis metrics......Page 260
The Lewis class of Lewis metrics......Page 261
13.1.2 The van Stockum solutions......Page 262
13.1.3 Other cylindrically symmetric solutions......Page 265
13.2.1 The Ernst equation......Page 266
13.2.2 Some exact solutions......Page 269
13.3.1 The Ernst electrovacuum equations......Page 272
13.3.2 Some exact solutions......Page 274
14 Accelerating black holes......Page 278
14.1 The C-metric......Page 279
14.1.1 The C-metric in spherical-type coordinates......Page 280
14.1.2 Conformal diagrams......Page 282
14.1.3 Some geometrical and physical properties......Page 285
14.1.4 The Minkowski limit......Page 288
14.1.5 Global structure of the C-metric......Page 293
14.1.6 Boost-rotation symmetric form......Page 297
14.1.7 Weyl and other coordinates in the static region......Page 298
14.2 Accelerating charged black holes......Page 301
14.2.1 The Ernst solution......Page 302
14.3 rotating black holesAcceleratingand......Page 303
14.4 The inclusion of a cosmological constant......Page 305
15.1 Boost-rotation symmetric space-times......Page 311
15.2 Bonnor–Swaminarayan solutions......Page 313
15.2.2 Four particles with semi-infinite strings......Page 315
15.2.3 Two particles and semi-infinite strings......Page 316
15.2.5 Two particles and a finite string......Page 317
15.2.6 Four particles and two finite strings......Page 318
15.3.1 Two particles in an external field......Page 319
15.3.2 Monopole-dipole particles......Page 320
15.3.3 Multipole particles......Page 321
16 Plebanski–Demianski solutions......Page 324
16.1 The Plebanski–Demianski metric......Page 325
16.2 A general metric for expanding solutions......Page 328
16.3 Generalised black holes......Page 331
16.3.1 The Kerr–Newman–NUT–(anti-)de Sitter solution......Page 334
16.3.2 Accelerating Kerr–Newman–(anti-)de Sitter black holes......Page 335
16.4 Non-expanding solutions......Page 336
16.4.1 The B-metrics......Page 338
16.5 An alternative extension......Page 339
16.6 The general family of type D space-times......Page 340
17 Plane and pp-waves......Page 343
17.1 The class of pp-wave space-times......Page 344
17.2 Focusing properties......Page 345
17.3 Linear superposition and beams of pure radiation......Page 347
17.4 Shock, sandwich and impulsive waves......Page 348
17.5 Plane wave space-times......Page 350
17.5.1 Shock, impulsive and sandwich plane waves......Page 351
17.5.2 Focusing properties of sandwich plane waves......Page 352
17.6 Some related topics......Page 354
18.1 The general class of Kundt space-times......Page 356
18.2 Vacuum and pure radiation Kundt space-times......Page 357
18.3 Type III, N and O solutions......Page 358
18.3.1 Type III solutions......Page 360
18.3.2 solutionsTypeN......Page 361
18.3.3 Conformally flat solutions......Page 364
18.4 Geometry of the wave surfaces......Page 365
18.4.1 Kundt waves in a Minkowski background......Page 367
18.4.2 Kundt waves in an (anti-)de Sitter background......Page 369
18.4.3 “pp-waves” in an (anti-)de Sitter background......Page 371
18.4.4 Siklos waves in an anti-de Sitter background......Page 372
18.5 “Spinning” null matter and “gyratons”......Page 373
18.6 Type D (electro)vacuum solutions......Page 376
18.7 Type II solutions......Page 379
19.1 Robinson–Trautman vacuum space-times......Page 381
19.2 Type N solutions......Page 383
19.2.1 Minkowski background coordinates......Page 385
19.2.2 (Anti-)de Sitter background coordinates......Page 388
The de Sitter background......Page 389
The anti-de Sitter background......Page 391
19.2.3 Particular type N sandwich waves......Page 392
Sandwich type N waves......Page 393
Character of singularities and global structure......Page 396
19.3 Type D solutions......Page 397
19.4 Type II solutions......Page 399
19.5 Robinson–Trautman space-times with pure radiation......Page 404
19.5.1 Type D solutions......Page 405
19.5.2 Kinnersley’s rocket......Page 406
19.5.3 Type II solutions......Page 408
19.5.4 Bonnor’s rocket......Page 409
19.6 Comments on further solutions......Page 410
20 Impulsive waves......Page 412
20.1 Methods of construction......Page 413
20.2 Non-expanding impulsive waves in Minkowski space......Page 414
20.3 Non-expanding impulsive waves in (anti-)de Sitter space......Page 418
20.4 Expanding impulsive waves in Minkowski space......Page 421
20.5 Expanding impulsive waves in (anti-)de Sitter space......Page 427
20.6 Other impulsive wave space-times......Page 430
21 Colliding plane waves......Page 432
21.1 Initial data......Page 433
21.2 The interaction region......Page 435
21.3 The Khan–Penrose solution......Page 438
21.4 The degenerate Ferrari–Ibanez solution......Page 442
21.5.2 The Chandrasekhar–Xanthopoulos (Kerr) solution......Page 447
21.5.3 The Ferrari–Ibanez (Taub–NUT) solution......Page 450
21.5.4 Further type D solutions......Page 451
21.6 The Bell–Szekeres (Bertotti–Robinson) solution......Page 453
21.7 Properties of other colliding plane wave space-times......Page 454
22 A final miscellany......Page 458
22.1 Homogeneous Bianchi models......Page 459
22.2 Kantowski–Sachs space-times......Page 461
22.3 Cylindrical gravitational waves......Page 462
22.4 Vacuum cosmologies......Page 466
22.5 Majumdar–Papapetrou solutions......Page 468
22.6 G¨odel’s rotating universe......Page 470
22.7 Spherically symmetric Lemaˆıtre–Tolman solutions......Page 474
22.8 Szekeres space-times without symmetry......Page 478
A.1 2-spaces of constant positive curvature......Page 482
A.2 2-spaces of zero curvature......Page 483
A.3 2-spaces of constant negative curvature......Page 484
A.4 Other representations......Page 487
Appendix B: 3-spaces of constant curvature......Page 490
B.1 The 3-sphere......Page 492
B.2 The hyperbolic 3-space......Page 494
References......Page 497
Index......Page 540