More documents

Recommendations

Info

14.4 Cosmologies with a G 3 on S 3 223and for 4 3
224 14 Spatially-homogeneous perfectfluid cosmologiesgive the only such explicitly known solutions. In principle solutions withother G 3 or H 3 on S 3 were also given in Rosquist and Jantzen (1985), butthe explicit parameter values required were not found.The matter is also rotating, shearing and expanding in the G 3 VII 0solution discussed by Demiański and Grishchuk (1972), who imposed thecondition that the orbits of the group are flat, but this is only knownup to a differential equation. Some other solutions with flat slices, ofPetrov type I , and with shear and rotation, similar to (36.22), are givenin Wolf (1986b).A number of stiff fluid solutions obtained by using Theorem 10.2 areknown as special cases of solutions with maximal groups H 3 and G 2 (forwhich see Chapter 23). The G 3 VI h stiff fluid solution corresponding to(13.57) is obtained by settingm 2 =3+n 2 +2(b 2 − a 2 ) (14.44)in (23.2) (Wainwright et al. 1979). It includes LRS and non-LRS G 3 III(n 2 = 1) and G 3 V (n = 0) solutions, in particular (14.18b): the G 3 Vsolutions can be generated from (13.59). The non-tilting case was alsofound, in another form, by Collins (1971), and by Ruban (1977, 1978).Setting nm = ab in (23.3) gives a tilted fluid with a G 3 VI 0 (Wainwrightet al. 1979), which is a limit of (23.2) with (14.44) as (13.58) is of (13.57).The non-tilted case is the G 3 VI 0 (n α α = 0) solution found by Ellis andMacCallum (1969).Similarly for Bianchi type VII h , one needsm 2 − 11 4 +2(a2 − b 2 ) = 0 (14.45)in (23.4). The non-tilted case was found by Barrow (1978).Putting p = q in (23.6) with F = u −2p gives a tilted stiff fluid solutionwith a G 3 VI −4 on u/t = const (Collins 1991), and the case a = 1 2of thestiff fluid solution (23.8) admits a G 3 VI −1/9 .The Petrov type D fluid solution with a group G 3 VI h on surfacese ax /t = const given in comoving coordinates by Allnutt (1980) asds 2 = dx 2 +e −2ax (t 1+n dy 2 + t 1−n dz 2 − dt 2 ) (14.46)can be interpreted as a stiff fluid with density κ 0 µ =(1− n 2 )e 2ax /4t 2 anda cosmological constant Λ = −3a 2 .Forn = 0, where the group is a G 3 V ,this is (15.75) with b = k =0,a>0, c = 1. The group orbits are S 3 orT 3 depending on the sign of t 2 a 2 − e 2ax . This and some analogous G 3 VI hsolutions have an inheriting conformal Killing vector (Czapor and Coley1995, Vera 1998a).
Page 2:
Exact Solutions of Einstein’s Fie
Page 7 and 8:
CAMBRIDGE UNIVERSITY PRESSCambridge
Page 10 and 11:
Contentsix8 Continuous groups of tr
Page 12 and 13:
Contentsxi15 Groups G 3 on non-null
Page 14 and 15:
Contentsxiii21.1.4 Complexification
Page 16 and 17:
Contentsxv29.2 Some general classes
Page 18 and 19:
Contentsxvii34.1.3 Complex invarian
Page 20 and 21:
PrefaceWhen, in 1975, two of the au
Page 22:
Prefacexxifor tolerating our incess
Page 25 and 26:
xxivList of Tables13.2 Subgroups G
Page 28 and 29:
NotationAll symbols are explained i
Page 30:
NotationxxixCurvature 2-forms: Θ a
Page 33 and 34:
2 1 Introductionother fields and ma
Page 35 and 36:
4 1 Introductionin physical applica
Page 37 and 38:
6 1 Introductionfluids, scalar, Dir
Page 39 and 40:
8 1 Introductionsince it is in prin
Page 41 and 42:
10 2 Differential geometry without
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
3Some topics in Riemannian geometry
Page 63 and 64:
32 3 Some topics in Riemannian geom
Page 65 and 66:
The Outbreak of the Napoleonic Wars
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
4The Petrov classificationThere are
Page 81 and 82:
50 4 The Petrov classificationTable
Page 83 and 84:
52 4 The Petrov classificationforms
Page 85 and 86:
54 4 The Petrov classificationand s
Page 87 and 88:
56 4 The Petrov classificationI✠I
Page 89 and 90:
58 5 Classification of the Ricci te
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
6Vector fields6.1 Vector fields and
Page 101 and 102:
70 6 Vector fields6.1.1 Timelike un
Page 103 and 104:
72 6 Vector fields6.2 Vector fields
Page 105 and 106:
74 6 Vector fieldsTheorem 6.4 For s
Page 107 and 108:
76 7 The Newman-Penrose and related
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
92 8 Continuous groups of transform
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
100 8 Continuous groups of transfor
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
9Invariants and the characterizatio
Page 145 and 146:
114 9 Invariants and the characteri
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
130 10 Generation techniquesarbitra
Page 163 and 164:
132 10 Generation techniquesEinstei
Page 166 and 167:
10.3 Symmetries more general than L
Page 168 and 169:
10.4 Prolongation 137Other examples
Page 170 and 171:
10.4 Prolongation 139If there were
Page 172 and 173:
Stokes’s theorem we have10.4 Prol
Page 174 and 175:
10.4 Prolongation 143The terms with
Page 176 and 177:
10.5 Solutions of the linearized eq
Page 178 and 179:
10.6 Bäcklund transformations 147T
Page 180 and 181:
10.8 Harmonic maps 149with a Lagran
Page 182 and 183:
10.9 Variational Bäcklund transfor
Page 184 and 185:
10.11 Generation methods includingp
Page 186 and 187:
10.11 Generation methods includingp
Page 188 and 189:
Part IISolutions with groups of mot
Page 190 and 191:
11.2 Isotropy and the curvature ten
Page 192 and 193:
11.2 Isotropy and the curvature ten
Page 194 and 195:
11.3 The possible space-times with
Page 196 and 197:
Table 11.2. Solutions with proper h
Page 198 and 199:
11.4 Summary of solutions with homo
Page 200 and 201:
p − 1 )(y∂ y + z∂z) Table 11.
Page 202 and 203:
12Homogeneous space-times12.1 The p
Page 204 and 205: 12.1 The possible metrics 173deriva
Page 206 and 207: 12.3 Homogeneous non-null electroma
Page 208 and 209: 12.4 Homogeneous perfect fluid solu
Page 210 and 211: 12.4 Homogeneous perfect fluid solu
Page 212 and 213: 12.6 Summary 181Table 12.1.Homogene
Page 214 and 215: 13Hypersurface-homogeneous space-ti
Page 216 and 217: 13.1 The possible metrics 185The ex
Page 218 and 219: 13.1 The possible metrics 187Summar
Page 220 and 221: 13.2 Formulations of the field equa
Page 226 and 227: 13.3 Vacuum, Λ-term and Einstein-M
Page 236 and 237: 13.4 Perfect fluid solutions homoge
Page 238 and 239: 13.5 Summary of all metrics with G
Page 240 and 241: Table 13.4. Solutions given explici
Page 242: 14.2 Robertson-Walker cosmologies 2
Page 245 and 246: 214 14 Spatially-homogeneous perfec
Page 253: 222 14 Spatially-homogeneous perfec
Page 257 and 258: 15Groups G 3 on non-null orbits V 2
Page 259 and 260: 228 15 Groups G 3 on non-null orbit
Page 279 and 280: 248 16 Spherically-symmetric perfec
Page 295 and 296: 17Groups G 2 and G 1 on non-null or
Page 297 and 298: 266 17 Groups G 2 and G 1 on non-nu
Page 305 and 306:
274 17 Groups G 2 and G 1 on non-nu
Page 307 and 308:
276 18 Stationary gravitational fie
Page 309 and 310:
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
Page 323 and 324:
19Stationary axisymmetric fields: b
Page 325 and 326:
294 19 Stationary axisymmetric fiel
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
20Stationary axisymmetric vacuumsol
Page 337 and 338:
306 20 Stationary axisymmetric vacu
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
320 21 Non-empty stationary axisymm
Page 353 and 354:
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
Page 361 and 362:
Page 363 and 364:
Page 365 and 366:
Page 367 and 368:
Page 369 and 370:
Page 371 and 372:
Page 373 and 374:
342 22 Groups G 2 I on spacelike or
Page 375 and 376:
Page 377 and 378:
Page 379 and 380:
Page 381 and 382:
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Page 389 and 390:
23Inhomogeneous perfect fluid solut
Page 391 and 392:
360 23 Inhomogeneous perfect fluid
Page 393 and 394:
Page 395 and 396:
Page 397 and 398:
Page 399 and 400:
Page 401 and 402:
Page 403 and 404:
Page 405 and 406:
Page 407 and 408:
376 24 Groups on null orbits. Plane
Page 409 and 410:
Page 411 and 412:
Page 413 and 414:
Page 415 and 416:
Page 417 and 418:
Page 419 and 420:
388 25 Collision of plane wavesIVII
Page 421 and 422:
390 25 Collision of plane waveswhic
Page 423 and 424:
392 25 Collision of plane wavesone
Page 425 and 426:
394 25 Collision of plane wavesErez
Page 427 and 428:
396 25 Collision of plane wavesfron
Page 429 and 430:
398 25 Collision of plane waves2V u
Page 431 and 432:
400 25 Collision of plane wavesThe
Page 433 and 434:
402 25 Collision of plane waves(a,
Page 435 and 436:
404 25 Collision of plane wavesAssu
Page 437 and 438:
406 25 Collision of plane wavessolu
Page 439 and 440:
408 26 The various classes of algeb
Page 441 and 442:
Page 443 and 444:
Page 445 and 446:
Page 447 and 448:
27The line element for metrics with
Page 449 and 450:
418 27 The line element for κ = σ
Page 451 and 452:
420 27 The line element for κ = σ
Page 453 and 454:
28Robinson-Trautman solutions28.1 R
Page 455 and 456:
424 28 Robinson-Trautman solutionsT
Page 457 and 458:
426 28 Robinson-Trautman solutionsr
Page 459 and 460:
428 28 Robinson-Trautman solutionsW
Page 461 and 462:
430 28 Robinson-Trautman solutionsf
Page 463 and 464:
432 28 Robinson-Trautman solutionsE
Page 465 and 466:
434 28 Robinson-Trautman solutionsh
Page 467 and 468:
436 28 Robinson-Trautman solutionsw
Page 469 and 470:
438 29 Twistingvacuum solutionsthen
Page 471 and 472:
440 29 Twistingvacuum solutions29.1
Page 473 and 474:
442 29 Twistingvacuum solutionsThe
Page 475 and 476:
444 29 Twistingvacuum solutionsTabl
Page 477 and 478:
446 29 Twistingvacuum solutions29.2
Page 479 and 480:
Page 481 and 482:
450 29 Twistingvacuum solutionsFor
Page 483 and 484:
Page 485 and 486:
454 29 Twistingvacuum solutionscove
Page 487 and 488:
456 30 TwistingEinstein-Maxwell and
Page 489 and 490:
Page 491 and 492:
Page 493 and 494:
Page 495 and 496:
Page 497 and 498:
Page 499 and 500:
Page 501 and 502:
31Non-diverging solutions (Kundt’
Page 503 and 504:
472 31 Non-diverging solutions (Kun
Page 505 and 506:
Page 507 and 508:
Page 509 and 510:
Page 511 and 512:
Page 513 and 514:
Page 515 and 516:
Page 517 and 518:
486 32 Kerr-Schild metricsThese rel
Page 519 and 520:
488 32 Kerr-Schild metricsay ( u )
Page 521 and 522:
490 32 Kerr-Schild metricsTheorem 3
Page 523 and 524:
492 32 Kerr-Schild metricsTable 32.
Page 525 and 526:
494 32 Kerr-Schild metricsHence k i
Page 527 and 528:
496 32 Kerr-Schild metricsThe only
Page 529 and 530:
498 32 Kerr-Schild metricssticking
Page 531 and 532:
500 32 Kerr-Schild metricswhere bot
Page 533 and 534:
502 32 Kerr-Schild metricsIf we now
Page 535 and 536:
504 32 Kerr-Schild metrics(Martín
Page 537 and 538:
33Algebraically special perfect flu
Page 539 and 540:
508 33 Algebraically special perfec
Page 541 and 542:
Page 543 and 544:
Page 545 and 546:
Page 547 and 548:
Page 549 and 550:
Part IVSpecial methods34Application
Page 551 and 552:
520 34 Application of generation te
Page 553 and 554:
Page 555 and 556:
Page 557 and 558:
Page 559 and 560:
Page 561 and 562:
Page 563 and 564:
Page 565 and 566:
Page 567 and 568:
Page 569 and 570:
Page 571 and 572:
Page 573 and 574:
Page 575 and 576:
Page 577 and 578:
Page 579 and 580:
Page 581 and 582:
Page 583 and 584:
Page 585 and 586:
554 35 Special vector and tensor fi
Page 587 and 588:
Page 589 and 590:
Page 591 and 592:
Page 593 and 594:
Page 595 and 596:
Page 597 and 598:
Page 599 and 600:
Page 601 and 602:
Page 603 and 604:
572 36 Solutions with special subsp
Page 605 and 606:
Page 607 and 608:
Page 609 and 610:
Page 611 and 612:
37Local isometric embedding offour-
Page 613 and 614:
582 37 Embeddingof four-dimensional
Page 615 and 616:
Page 617 and 618:
Page 619 and 620:
Page 621 and 622:
Page 623 and 624:
Page 625 and 626:
Page 627 and 628:
Page 629 and 630:
Page 631 and 632:
Page 633 and 634:
Page 635 and 636:
Page 637 and 638:
606 38 The interconnections between
Page 639 and 640:
Page 641 and 642:
Page 643 and 644:
Page 645 and 646:
Table 38.10. Algebraically special
Page 647 and 648:
616 ReferencesAlencar, P.S.C. and L
Page 649 and 650:
618 ReferencesBasu, A., Ganguly, S.
Page 651 and 652:
620 ReferencesBirkhoff, G.D. (1923)
Page 653 and 654:
622 ReferencesBradley, M.and Karlhe
Page 655 and 656:
624 ReferencesCarminati, J.(1981).A
Page 657 and 658:
626 ReferencesCharach, Ch.and Malin
Page 659 and 660:
628 ReferencesCollinson, C.D. (1964
Page 661 and 662:
630 ReferencesDas, K.C. and Chaudhu
Page 663 and 664:
632 ReferencesDemiański, M.and New
Page 665 and 666:
634 ReferencesEhlers, J.(1961).Beit
Page 667 and 668:
636 ReferencesFernandez-Jambrina, L
Page 669 and 670:
638 ReferencesGarcía D., A. and Br
Page 671 and 672:
640 ReferencesGowdy, R.H. (1975). C
Page 673 and 674:
642 ReferencesHajj-Boutros, J.(1985
Page 675 and 676:
644 ReferencesHarrison, B.K. (1978)
Page 677 and 678:
646 ReferencesHerlt, E.(1972).Über
Page 679 and 680:
648 ReferencesHoenselaers, C.(1992)
Page 681 and 682:
650 ReferencesIvanov, B.Y. (1999).
Page 683 and 684:
652 ReferencesKerns, R.M. and Wild,
Page 685 and 686:
654 ReferencesKolassis, C.and Griff
Page 687 and 688:
656 ReferencesKramer, D.and Neugeba
Page 689 and 690:
658 ReferencesKyriakopoulos, E.(198
Page 691 and 692:
660 ReferencesLi, W.and Ernst, F.J.
Page 693 and 694:
662 ReferencesLun, A.W.C., McIntosh
Page 695 and 696:
664 ReferencesMarder, L.(1969).Grav
Page 697 and 698:
666 ReferencesMehra, A.L. (1966). R
Page 699 and 700:
668 ReferencesNeugebauer, G.and Mei
Page 701 and 702:
670 ReferencesPant, D.N. (1994). Va
Page 703 and 704:
672 ReferencesPiper, M.S.(1997).Com
Page 705 and 706:
674 ReferencesRobertson, H.P. (1929
Page 707 and 708:
676 ReferencesSchmidt, B.G. (1996).
Page 709 and 710:
678 ReferencesSingleton, D.B. (1990
Page 711 and 712:
680 ReferencesStewart, B.W., Witten
Page 713 and 714:
682 ReferencesTeixeira, A.F.F., Wol
Page 715 and 716:
684 ReferencesVaidya, P.C. and Pate
Page 717 and 718:
686 References65th birthday, ed. M.
Page 719 and 720:
688 ReferencesXanthopoulos, B.C. (1
Page 721 and 722:
Indexacceleration, 70affine colline
Page 723 and 724:
692 Indexdust solutionsFriedmann un
Page 725 and 726:
694 IndexHarrison solutions, 272Har
Page 727 and 728:
696 IndexNUT solution, 310, 449, 45
Page 729 and 730:
698 IndexRiemann tensor, 25, 34cova
Page 731 and 732:
700 Indextype D solutions (contd)Ro
show all

Contents

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?