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34.5 The Belinski–Zakharov technique 545Finally, one requires det(g) =−W 2 . This implies for Pdet [P (λ = 0)] = 1. (34.114)In fact, any solution of the linear pair (34.104) is defined up to a scalingg → h(λ =0) −1 g. Thus, given any solution g of (34.105) the physicalmetric g ph (det g ph = −W 2 ) can be obtained byg ph = W (det g) 1/2 g, (34.115)in which case we have h = (det g) 1/2 W −1 .In order to construct a solution explicitly, we now introduce some assumptionson the pole structure of the matrix P in the complex λ-plane.Here, we shall assume that P and P −1 have singularities in λ and thatthese are simple poles, i.e. both P and P −1 are meromorphic. Of course,the poles will depend on W and V ; note that – whenever possible –we have suppressed the coordinate dependence of the various quantities.This is analogous to prescribing the zeros of the dressing matrix P in§34.4. Let us assume that P (λ) is not invertible at a number of points ν k(k =1,...,n) and that these are simple poles for P −1 . Then it can beshown that P has simple poles at µ k = W 2 νk −1 .From (34.108) it can be concluded that the poles of P are either real orcome in complex conjugate pairs. Thus the general forms of P and P −1aren∑]P =[(λ − µ k )R k +(λ − µ k ) −1 R k , (34.116a)P −1 =k=1n∑k=1[(λ − ν k ) −1 Q k +(λ − ν k )Q k], (34.116b)where the matrices R k and Q k are related through the condition PP −1 =1. By inserting (34.116) into (34.107) we find an equation for µ k thesolution of which is√µ k = w k ± (w k − V ) 2 + W 2 . (34.117)The R k cannot be chosen arbitrarily, rather they are given in terms ofvectors n k and m k byR k = n k m T k . (34.118)For a given seed metric g 0 one obtains, at least in principle, Φ 0 from(34.104). Then one defines matrices M k and with their help the vectorsm k byM k =Φ 0 (λ = µ k ), m k = M T k κ k , (34.119)
546 34 Application of generation techniques to general relativityTable 34.4.Applications of the Belinski–Zakharov methodThe Killing vector types are AS (axisymmetric stationary), CY (cylindricalsymmetry), CO (cosmological symmetry), BR (boost-rotation symmetry).SeedType ResultMinkowski AS Vac. Alekseev (1981), Gruszczak (1981)Azuma et al. (1993)Letelier and Oliveira (1998a)CY Vac. Tomimatsu (1989)BR Vac. Jantzen (1983)Euclidean AS Vac. Verdaguer (1982)Weyl AS Vac. Letelier (1982, 1985a, 1985c)Tomimatsu (1984)Carot and Verdaguer (1989)Kerr AS Vac. Tomimatsu (1980)v. Stockum AS Vac. Letelier (1986)Kasner CO Vac. Belinskii and Fargion (1980b)Carr and Verdaguer (1983)Ibañez and Verdaguer (1983, 1985, 1986)Das (1985), Griffiths and Miccichè (1999)CY Vac. Cespedes and Verdaguer (1987)Bianchi II CO Vac. Belinskii and Francaviglia (1982, 1984)vacuum Vac. Bradley et al. (1991)Einstein– CY Vac. Letelier (1985b)Rosen Fustero and Verdaguer (1986)Oliver and Verdaguer (1989)Non-diagonal AS Vac. Das and Chaudhuri(1991)Minkowski AS EM Alekseev (1980), Wang et al. (1983b)CY EM Dagotto et al. (1993)where κ k are arbitrary complex vectors. The n k are now constructedvia2n∑(n l = N k µ −1kΓ−1 kl , N k = g 0 m k , Γ kl = µ k µ l − W 2) m k g 0 m l .k=1Finally the new physical metric is given byg =2n∏k=1(µ k W −1 )2n∑k,l=1(g 0 − µ −1k µ−1 lΓ −1kl N kN T l(34.120)). (34.121)
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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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3Some topics in Riemannian geometry
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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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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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9Invariants and the characterizatio
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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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13.4 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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214 14 Spatially-homogeneous perfec
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15Groups G 3 on non-null orbits V 2
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17Groups G 2 and G 1 on non-null or
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276 18 Stationary gravitational fie
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19Stationary axisymmetric fields: b
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20Stationary axisymmetric vacuumsol
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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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Page 537 and 538: 33Algebraically special perfect flu
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Page 549 and 550: Part IVSpecial methods34Application
Page 551 and 552: 520 34 Application of generation te
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Page 585 and 586: 554 35 Special vector and tensor fi
Page 603 and 604: 572 36 Solutions with special subsp
Page 611 and 612: 37Local isometric embedding offour-
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596 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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