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Exact Solutions of Einstein’sField EquationsSecond EditionHANS STEPHANIFriedrich-Schiller-Universität, JenaDIETRICH KRAMERFriedrich-Schiller-Universität, JenaMALCOLM MACCALLUMQueen Mary, University of LondonCORNELIUS HOENSELAERSLoughborough UniversityEDUARD HERLTFriedrich-Schiller-Universität, Jena
CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São PauloCambridge University PressThe Edinburgh Building, Cambridge CB2 2RU, United KingdomPublished in the United States by Cambridge University Press, New Yorkwww.cambridge.orgInformation on this title: www.cambridge.org/9780521461368© H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, E. Herlt 2003This book is in copyright. Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University Press.First published in print format 2003ISBN-13ISBN-10978-0-511-06548-4 eBook (NetLibrary)0-511-06548-5 eBook (NetLibrary)ISBN-13 978-0-521-46136-8 hardbackISBN-10 0-521-46136-7 hardbackCambridge University Press has no responsibility for the persistence or accuracy ofURLs for external or third-party internet websites referred to in this book, and does notguarantee that any content on such websites is, or will remain, accurate or appropriate.
Page 2: Exact Solutions of Einstein’s Fie
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
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22 2 Differential geometry without
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3Some topics in Riemannian geometry
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32 3 Some topics in Riemannian geom
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The Outbreak of the Napoleonic Wars
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4The Petrov classificationThere are
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50 4 The Petrov classificationTable
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52 4 The Petrov classificationforms
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54 4 The Petrov classificationand s
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56 4 The Petrov classificationI✠I
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58 5 Classification of the Ricci te
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6Vector fields6.1 Vector fields and
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70 6 Vector fields6.1.1 Timelike un
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72 6 Vector fields6.2 Vector fields
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74 6 Vector fieldsTheorem 6.4 For s
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76 7 The Newman-Penrose and related
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92 8 Continuous groups of transform
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100 8 Continuous groups of transfor
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9Invariants and the characterizatio
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114 9 Invariants and the characteri
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130 10 Generation techniquesarbitra
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132 10 Generation techniquesEinstei
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10.3 Symmetries more general than L
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10.4 Prolongation 137Other examples
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10.4 Prolongation 139If there were
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Stokes’s theorem we have10.4 Prol
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10.4 Prolongation 143The terms with
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10.5 Solutions of the linearized eq
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10.6 Bäcklund transformations 147T
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10.8 Harmonic maps 149with a Lagran
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10.9 Variational Bäcklund transfor
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10.11 Generation methods includingp
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10.11 Generation methods includingp
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Part IISolutions with groups of mot
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11.2 Isotropy and the curvature ten
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11.2 Isotropy and the curvature ten
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11.3 The possible space-times with
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Table 11.2. Solutions with proper h
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11.4 Summary of solutions with homo
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p − 1 )(y∂ y + z∂z) Table 11.
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12Homogeneous space-times12.1 The p
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12.1 The possible metrics 173deriva
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12.3 Homogeneous non-null electroma
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12.4 Homogeneous perfect fluid solu
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12.4 Homogeneous perfect fluid solu
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12.6 Summary 181Table 12.1.Homogene
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13Hypersurface-homogeneous space-ti
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13.1 The possible metrics 185The ex
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13.1 The possible metrics 187Summar
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13.2 Formulations of the field equa
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13.3 Vacuum, Λ-term and Einstein-M
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13.4 Perfect fluid solutions homoge
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13.5 Summary of all metrics with G
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Table 13.4. Solutions given explici
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14.2 Robertson-Walker cosmologies 2
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214 14 Spatially-homogeneous perfec
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15Groups G 3 on non-null orbits V 2
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228 15 Groups G 3 on non-null orbit
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248 16 Spherically-symmetric perfec
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17Groups G 2 and G 1 on non-null or
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266 17 Groups G 2 and G 1 on non-nu
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276 18 Stationary gravitational fie
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19Stationary axisymmetric fields: b
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294 19 Stationary axisymmetric fiel
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20Stationary axisymmetric vacuumsol
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306 20 Stationary axisymmetric vacu
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320 21 Non-empty stationary axisymm
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342 22 Groups G 2 I on spacelike or
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23Inhomogeneous perfect fluid solut
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360 23 Inhomogeneous perfect fluid
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376 24 Groups on null orbits. Plane
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388 25 Collision of plane wavesIVII
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390 25 Collision of plane waveswhic
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392 25 Collision of plane wavesone
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394 25 Collision of plane wavesErez
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396 25 Collision of plane wavesfron
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398 25 Collision of plane waves2V u
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400 25 Collision of plane wavesThe
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402 25 Collision of plane waves(a,
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404 25 Collision of plane wavesAssu
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406 25 Collision of plane wavessolu
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408 26 The various classes of algeb
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27The line element for metrics with
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418 27 The line element for κ = σ
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420 27 The line element for κ = σ
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28Robinson-Trautman solutions28.1 R
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424 28 Robinson-Trautman solutionsT
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426 28 Robinson-Trautman solutionsr
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428 28 Robinson-Trautman solutionsW
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430 28 Robinson-Trautman solutionsf
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432 28 Robinson-Trautman solutionsE
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434 28 Robinson-Trautman solutionsh
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436 28 Robinson-Trautman solutionsw
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438 29 Twistingvacuum solutionsthen
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440 29 Twistingvacuum solutions29.1
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442 29 Twistingvacuum solutionsThe
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444 29 Twistingvacuum solutionsTabl
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446 29 Twistingvacuum solutions29.2
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450 29 Twistingvacuum solutionsFor
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454 29 Twistingvacuum solutionscove
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456 30 TwistingEinstein-Maxwell and
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31Non-diverging solutions (Kundt’
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472 31 Non-diverging solutions (Kun
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486 32 Kerr-Schild metricsThese rel
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488 32 Kerr-Schild metricsay ( u )
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490 32 Kerr-Schild metricsTheorem 3
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492 32 Kerr-Schild metricsTable 32.
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494 32 Kerr-Schild metricsHence k i
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496 32 Kerr-Schild metricsThe only
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498 32 Kerr-Schild metricssticking
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500 32 Kerr-Schild metricswhere bot
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502 32 Kerr-Schild metricsIf we now
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504 32 Kerr-Schild metrics(Martín
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33Algebraically special perfect flu
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508 33 Algebraically special perfec
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Part IVSpecial methods34Application
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520 34 Application of generation te
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554 35 Special vector and tensor fi
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572 36 Solutions with special subsp
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37Local isometric embedding offour-
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582 37 Embeddingof four-dimensional
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606 38 The interconnections between
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Table 38.10. Algebraically special
Page 647 and 648:
616 ReferencesAlencar, P.S.C. and L
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618 ReferencesBasu, A., Ganguly, S.
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620 ReferencesBirkhoff, G.D. (1923)
Page 653 and 654:
622 ReferencesBradley, M.and Karlhe
Page 655 and 656:
624 ReferencesCarminati, J.(1981).A
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626 ReferencesCharach, Ch.and Malin
Page 659 and 660:
628 ReferencesCollinson, C.D. (1964
Page 661 and 662:
630 ReferencesDas, K.C. and Chaudhu
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632 ReferencesDemiański, M.and New
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634 ReferencesEhlers, J.(1961).Beit
Page 667 and 668:
636 ReferencesFernandez-Jambrina, L
Page 669 and 670:
638 ReferencesGarcía D., A. and Br
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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
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658 ReferencesKyriakopoulos, E.(198
Page 691 and 692:
660 ReferencesLi, W.and Ernst, F.J.
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662 ReferencesLun, A.W.C., McIntosh
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664 ReferencesMarder, L.(1969).Grav
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666 ReferencesMehra, A.L. (1966). R
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668 ReferencesNeugebauer, G.and Mei
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670 ReferencesPant, D.N. (1994). Va
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672 ReferencesPiper, M.S.(1997).Com
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674 ReferencesRobertson, H.P. (1929
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676 ReferencesSchmidt, B.G. (1996).
Page 709 and 710:
678 ReferencesSingleton, D.B. (1990
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680 ReferencesStewart, B.W., Witten
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682 ReferencesTeixeira, A.F.F., Wol
Page 715 and 716:
684 ReferencesVaidya, P.C. and Pate
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686 References65th birthday, ed. M.
Page 719 and 720:
688 ReferencesXanthopoulos, B.C. (1
Page 721 and 722:
Indexacceleration, 70affine colline
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692 Indexdust solutionsFriedmann un
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694 IndexHarrison solutions, 272Har
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696 IndexNUT solution, 310, 449, 45
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698 IndexRiemann tensor, 25, 34cova
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700 Indextype D solutions (contd)Ro
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