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14.4 Cosmologies with a G 3 on S 3 219where ω 2 =3|Λ|,Λa 2 = 3. There is a special solution M 2 = 12Λ > 0, S 3 =a(e ωt − 1), F =(1− e −ωt ) 2/3 (Lorenz 1982a). With M =0̸= Λ, (14.26)gives (13.63). The case Λ = 0 was earlier given by Raychaudhuri (1958)and Heckmann and Schücking (1958). The LRS cases (ψ =0orπ in(13.30)) have been rediscovered several times.Choosing a time variable y ∝ (S 3 )˙, Jacobs (1968) was able to integrateG 3 Iγ-law perfect fluids with a general constant γ (1 ̸=γ ̸= 2) forF (y)and so give an exact solution. This general family can be written, with adifferent choice of new time variable τ, as (Wainwright 1984, Wainwrightand Ellis 1997)ds 2 = −G 2(γ−1) dτ 2 + τ 2p 1G 2q 1dx 2 + τ 2p 2G 2q 2dy 2 + τ 2p 3G 2q 3dz 2 ,(14.27a)G 2−γ = √ 3Σ + 9 4 mτ 2−γ , q α = 2 3 − p α, (14.27b)where p α , Σ and m, which contain two essential parameters, are as in(13.54), (13.29) and (14.1). A closed form in terms of S can be given forγ =4/3 (Jacobs 1968; see also Shikin (1968)), and solutions as powerseries in S for γ =1+n/(n + 1) and γ = 1+(2n +1)/(2n + 3) withinteger n (Jacobs 1968). The limits Σ = 0 and m = 0 are respectively theRobertson–Walker and vacuum solutions. Solutions for G 3 I with combinationsof fluids with γ = 1, 4/3, 5/3 and 2 can be expressed in terms ofelliptic functions (Jacobs 1968, Ellis and MacCallum 1969).The solutions with γ = 2, including Λ ̸= 0, are (Ellis and MacCallum1969)√Λ > 0, S 3 = (3 + M)/Λ sinh ωt, F = (tanh(ωt/2)) b , (14.28a)√Λ=0, S 3 = 3(3 + M)t, F = t b , (14.28b)√Λ < 0, S 3 = (3 + M)/|Λ| sin ωt, F = (tan(ωt/2)) b , (14.28c)where b ≡ 2/ √ 3(3 + M) and ω 2 =3|Λ|. The case (14.28b), which wasfound by Jacobs (1968) and Shikin (1968), can be expressed in the form(13.53), the only change being that ∑ 31 (p i ) 2 < 1, and it has a homothety ofthe form (13.52) (Koutras 1992b). Solution (14.28b) with b cos ψ = −1/3is of Petrov type D (Allnutt 1980), in addition to the LRS subcases.The G 3 I solutions with electromagnetic fields, given by Jacobs (1969),can be used to solve the G 3 II fluid case (Collins 1972). Collins (1971)found a special G 3 II solution, which can be written asds 2 = −G 2(γ−1) dτ 2 + τ 2p 1G 2p 1(dx + nzdy) 2 v + τ 2p 2G 2p 3dy 2 (14.29)+τ 2p 3G 2p 2dz 2 , G 2−γ = a +4Mτ 2−γ /(6 − γ)
220 14 Spatially-homogeneous perfectfluid cosmologies(Wainwright and Ellis 1997), where a is a constant appearing in the shear,M gives µ as in (14.1), the ψ of (13.54) is given by 8 cos ψ =2− 3γ, and(6 − γ)n 2 =(3γ − 2)(2 − γ)M: here 2 3
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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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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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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
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270 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).
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678 ReferencesSingleton, D.B. (1990
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680 ReferencesStewart, B.W., Witten
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682 ReferencesTeixeira, A.F.F., Wol
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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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