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31.8 Solutions includinga cosmological constant Λ 483(31.41), these solutions admit a group of motions G 4 on T 3 (cp. (13.48)).Unlike the vacuum case this class does not exhaust all type D solutions: itdoes not, for instance, contain an Einstein–Maxwell field given by KowalczyńskiandPlebański(1977) in the formds 2 =2x −2 [(dx/A) 2 + A 2 dy 2 +(dz/B) 2 − B 2 dt 2 ],A 2 = ax 2 + cx 3 − 2e 2 x 4 , B 2 = b − az 2 , a,b,c,e= const.(31.60)The Einstein–Maxwell type D solutions (double aligned) of Kundt’sclass have been determined by Plebański(1979). They are given byds 2 = x2 + l 2K(x) dx2 + K(x) [x 2 + l 2 dσ +]vdu − udv+2 (x2 + l 2 )1 − εuv (1 − εuv) 2 dudv,K(x) =2nx − (e 2 + g 2 ) − 2ε(x 2 − l 2 ), (31.61)Φ 11 =(e 2 + g 2 )/2(x 2 + l 2 ) 2 ,e,g,l,ε,n= const,and contain the solutions (24.21)–(24.22) – with Λ = 0 – with a group G 2on null orbits.As in the vacuum case, Theorem 31.1 can be used to generate newsolutions from known ones.31.8 Solutions including a cosmological constant ΛIf we want to include (add) an energy-momentum tensor −Λg ik , cp. (5.4),then we have to replace the field equations (31.16b) byR 12 =2κ 0 Φ 1 Φ 1 +Λ, R 34 =2κ 0 Φ 1 Φ 1 − Λ, (31.62)the rest of the field equations remaining unchanged. No general theoremsaying how to incorporate Λ into a vacuum or Einstein–Maxwell field isavailable, but solutions have been found in a number of subcases.Starting from type III and N vacuum solutions, one has ∆ ln P =Λ,i.e. P = 1+Λζζ/2 instead of P = 1, and H ,vv = − 1 2 P 2 W ,v W ,v − Λinstead of (31.29). The type N solutions falling into this class have beengiven by Ozsváth et al. (1985) (see also García D. and Plebański(1981)and Bičák and Podolsky (1999)), and the solutions with W ,v = 0 byLewandowski(1992).Metrics for which the multiple null eigenvector k is recurrent, k a;b =k a p b , have been studied by Leroy and McLenaghan (1973).Some special type II solutions with constant Φ 1 Φ 1 have been constructedby García D. and Alvarez C. (1984) and Khlebnikov (1986).
484 31 Non-diverging solutions (Kundt’s class)The hypersurface-homogeneous Einstein spaces of Kundt’s class havebeen found by MacCallum and Siklos (1992).Type D Einstein–Maxwell fields (double aligned) with Λ can be obtainedfrom (31.61) by simply adding the term −Λ(x 4 /3+2l 2 x 2 − l 4 )inthe expression for K(x) (Plebański1979).
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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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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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Page 469 and 470: 438 29 Twistingvacuum solutionsthen
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Page 487 and 488: 456 30 TwistingEinstein-Maxwell and
Page 501 and 502: 31Non-diverging solutions (Kundt’
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Page 517 and 518: 486 32 Kerr-Schild metricsThese rel
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Page 537 and 538: 33Algebraically special perfect flu
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Page 549 and 550: Part IVSpecial methods34Application
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534 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
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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