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13.5 Summary of all metrics with G r on V 3 207with arbitrary constant a. Ifa ̸= 0 it can be taken to be 1. The case witha = 0, which was found by Wainwright (1985), is LRS, can be regardedas a stationary axisymmetric metric, and admits an H 5 ,4x∂ x +(2q +1)(y∂ y + z∂ z + t∂ t ) being a homothety generator; its γ = 1 limit is thecase Λ = 0 = b of (13.84) and the γ = 2 solution is one case of (13.79).Analogously to the use of (13.78), the perfect fluid equations for ametric with a G 3 II on T 3 and the ω 1 of Table 8.2 spacelike have beenreduced to a single second-order equation by Misra and Narain (1971);as a limit these contain the Einstein static solution. Similarly Van denBergh (1988c) reduced a diagonal metric with a G 3 V on T 3 (a specialcase of (23.22)) to a single second-order differential equation. The metric(14.46) has a G 3 VI or V on T 3 for some x and t. Another fluid solutionnot obeying the γ-law is, in comoving coordinates,ds 2 = f 2 [b(−dt 2 +dx 2 )+e b(t+x)−2x dy 2 +e −b(t−x)−2x dz 2 ],κ 0 µ =(−2ff ′′ + f ′2 − 2bff ′ +4ff ′ − b 2 f 2 +3bf 2 − 3f 2 )/bf 4 , (13.86)κ 0 p =2ff ′′ − f ′2 + bff ′ − 2ff ′ − b 2 f 2 + f 2 )/bf 4 ,f= c/(1 − e (2−b)x/2 ),where b and c are constants and ′ denotes d/dx (Sintes et al. 1998). Thishas a G 3 VIon T 3 and a conformal Killing vector ∂ t +∂ x +(1−b)y∂ y +z∂ z ;there is a three-parameter conformal group acting on null orbits.In a study of stationary axisymmetric fluids, González-Romero (1994)gave solutions for G 3 II, G 3 VI 0 and G 3 VII 0 on T 3 up to an ordinarydifferential equation and quadratures: these generalize or complement theknown rotating solutions (see §21.2). An explicit perfect fluid solutionwith an H 4 on V 4 and G 3 VII 0 on T 3 was found by Hermann (1983); itcan be written, using a tetrad aligned with the fluid velocity, asds 2 = r 2(1−s) [ −(2 + s)(sin bz dϕ + cos bz dt) 2 + s(cos bz dϕ − sin bz dt) 2]+dr 2 + r 2 dz 2 , b 2 = s(s + 2)(1 − s)/(1 + s), (13.87)κ 0 µ =(2− 2s 2 )/r 2 , κ 0 p = 2(1 − s) 2 /r 2 ,where 0
208 13 Hypersurface-homogeneous space-timesTable 13.2.Subgroups G 3 on V 3 occurring in metrics with multiply-transitivegroupsThe spaces (8.33) of constant curvature are omitted here.BianchiMaximal grouptype G 7 or G 6 G 5 or G 4 G 6 G 4 on S 3 G 4 on T 3 ,of G 3 on V 4 on V 4 on S 3 or T 3 , LRS non-LRSI (12.24) (12.14), (12.26), (14.2) (13.1), k = 0 (13.14)(12.8) (12.30)–(12.32),(12.34)–(12.36),(12.38)II (12.21), (12.34), (13.2), k = 0 (13.15)(12.36), (12.38)III (12.8) (12.21), (12.26), (13.1), k = −1 (13.14),(12.28), (12.29) (13.2), k = −1 (13.15)(12.36)IV §12.2 (12.31)V (14.2) (13.3) (13.15)VI §12.2 (12.30), (12.31) (13.14),(12.8) (12.34)–(12.36) (13.15)(12.38)VII §12.2 (12.14) (14.2) (13.1), k =0(12.8) (12.32) (13.3)VIII (12.8) (12.26), (12.28), (13.2), k = −1 (13.15)(12.29), (12.36)IX (12.24) (12.27) (14.2) (13.2), k =1Table 13.3. Solutions given in this book with a maximal G 4 on V 3MetricEnergy- (13.9) (13.3) (13.17)momentum (13.1) (13.2)Dust (13.80)–(13.82) (14.20) (14.22) ∄(14.15)Perfect fluid §§15.7, 16.1 (13.83), (13.79) ∄(13.35), (14.14) (14.20) (14.24), (14.44)(14.16)–(14.19) (14.21) with m = n =0(13.79), (33.34) (14.23), (14.25)Vacuum and (13.48) (includes, (13.46) (13.48), e 2 →−e 2Einstein–Maxwell e.g. (15.21)–(15.32), (includes (15.12),(with Λ ̸= 0) A of Table 18.2) B of Table 18.2)
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Exact Solutions of Einstein’s Fie
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CAMBRIDGE UNIVERSITY PRESSCambridge
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Contentsix8 Continuous groups of tr
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Contentsxi15 Groups G 3 on non-null
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Contentsxiii21.1.4 Complexification
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Contentsxv29.2 Some general classes
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Contentsxvii34.1.3 Complex invarian
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PrefaceWhen, in 1975, two of the au
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Prefacexxifor tolerating our incess
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xxivList of Tables13.2 Subgroups G
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NotationAll symbols are explained i
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NotationxxixCurvature 2-forms: Θ a
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2 1 Introductionother fields and ma
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4 1 Introductionin physical applica
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6 1 Introductionfluids, scalar, Dir
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8 1 Introductionsince it is in prin
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10 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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Page 240 and 241: Table 13.4. Solutions given explici
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258 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
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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)
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622 ReferencesBradley, M.and Karlhe
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624 ReferencesCarminati, J.(1981).A
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626 ReferencesCharach, Ch.and Malin
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628 ReferencesCollinson, C.D. (1964
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630 ReferencesDas, K.C. and Chaudhu
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632 ReferencesDemiański, M.and New
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634 ReferencesEhlers, J.(1961).Beit
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636 ReferencesFernandez-Jambrina, L
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638 ReferencesGarcía D., A. and Br
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640 ReferencesGowdy, R.H. (1975). C
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642 ReferencesHajj-Boutros, J.(1985
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644 ReferencesHarrison, B.K. (1978)
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646 ReferencesHerlt, E.(1972).Über
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648 ReferencesHoenselaers, C.(1992)
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650 ReferencesIvanov, B.Y. (1999).
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652 ReferencesKerns, R.M. and Wild,
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654 ReferencesKolassis, C.and Griff
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656 ReferencesKramer, D.and Neugeba
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658 ReferencesKyriakopoulos, E.(198
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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
Page 713 and 714:
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.
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688 ReferencesXanthopoulos, B.C. (1
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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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