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References 683Tsamparlis, M.and Mason, D.P.(1983).On spacelike congruences in general relativity.JMP 24, 1577.See §6.2.Tseitlin, M.G. (1985). Solutions of two-dimensional Einstein equations parametrizedby arbitrary functions and generated by the O(2, 1) sigma model. Theor. Math.Phys. 64, 679.See §§10.11, 20.6.Tsoubelis, D.(1989).Plane gravitational waves colliding with shells of null dust.CQG6, L117.See §25.6.Tsoubelis, D.and Wang, A.(1991).On the gravitational interaction of plane symmetricclouds of null dust.JMP 32, 1017.See §25.6.Tsoubelis, D.and Wang, A.(1992).Head-on collision of gravitational plane waveswith noncollinear polarization: a new class of analytic models.JMP 33, 1054.See§25.4.Tsoubelis, D.and Wang, A.Z.(1989).Asymmetric collision of gravitational planewaves: a new class of exact solutions.GRG 21, 807.See §25.3.Tsoubelis, D.and Wang, A.Z.(1990).Impulsive shells of null dust colliding withgravitational plane waves.GRG 22, 1091.See §25.6.Tupper, B.O.J. (1976). A class of algebraically general solutions of the Einstein–Maxwell equations for non-null electromagnetic fields.II. GRG 7, 479.See §12.3.Tupper, B.O.J. (1981). The equivalence of electromagnetic fields and viscous fluids ingeneral relativity.JMP 22, 2666.See §5.2.Tupper, B.O.J. (1983). The equivalence of perfect fluid space-times and viscousmagnetohydrodynamic space-times in general relativity.GRG 15, 849.See §5.2.Tupper, B.O.J. (1996).Conformal symmetries of conformal-reducible spacetimes withnonzero Weyl tensor.CQG 13, 1679.See §35.4.Uggla, C.(1990).New exact perfect fluid solutions of Einstein’s equations.CQG 7,L171.See §14.4.Uggla, C.(1992).Inhomogeneous self-similar cosmological models.CQG 9, 2287. See§§23.1, 23.2, 23.3.Uggla, C., Bradley, M. and Marklund, M.(1995a).Classifying Einstein’s field equationswith applications to cosmology and astrophysics. CQG 12, 2525. See §§9.4,13.2.Uggla, C., Jantzen, R.T. and Rosquist, K. (1995b). Exact hypersurface-homogeneoussolutions in cosmology and astrophysics.PRD 51, 5522.See §§13.2, 35.3.Uggla, C.and Rosquist, K.(1990).New exact perfect fluid solutions of Einstein’sequations.II.CQG 7, L279.See §14.4.Uggla, C., Rosquist, K. and Jantzen, R.T. (1990). Geometrizing the dynamics ofBianchi cosmology.PRD 42, 404.See §13.2.Unti, T.W.J. and Torrence, R.J. (1966).Theorem on gravitational fields with geodesicrays.JMP 7, 535.See §26.4.Urbantke, H.(1972).Note on Kerr–Schild type vacuum gravitational fields.Acta Phys.Aust. 35, 396.See §32.2.Urbantke, H.(1975).Der metrische Ansatz von Trautman, Kerr und Schild und einigeseiner Anwendungen.Acta Phys. Aust. 41, 1.See §§26.1, 32.4.Vaidya, P.C. (1943). The external field of a radiating star in general relativity. Curr.Sci. 12, 183.See §15.4.Vaidya, P.C. (1968). Nonstatic analogues of Schwarzschild’s interior solution in generalrelativity.Phys. Rev. 174, 1615.See §16.2.Vaidya, P.C. (1973). Some algebraically special solutions of Einstein’s equations. II.Tensor 27, 276.See §32.4.Vaidya, P.C. (1974). A generalized Kerr–Schild solution of Einstein’s equations. Proc.Camb. Phil. Soc. 75, 383.See §32.4.Vaidya, P.C. (1977).The Kerr metric in cosmological background.Pramana 8, 512.See§21.2.
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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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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
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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
Page 657 and 658:
626 ReferencesCharach, Ch.and Malin
Page 659 and 660:
628 ReferencesCollinson, C.D. (1964
Page 661 and 662:
630 ReferencesDas, K.C. and Chaudhu
Page 663 and 664: 632 ReferencesDemiański, M.and New
Page 665 and 666: 634 ReferencesEhlers, J.(1961).Beit
Page 667 and 668: 636 ReferencesFernandez-Jambrina, L
Page 669 and 670: 638 ReferencesGarcía D., A. and Br
Page 671 and 672: 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
Page 689 and 690: 658 ReferencesKyriakopoulos, E.(198
Page 691 and 692: 660 ReferencesLi, W.and Ernst, F.J.
Page 693 and 694: 662 ReferencesLun, A.W.C., McIntosh
Page 695 and 696: 664 ReferencesMarder, L.(1969).Grav
Page 697 and 698: 666 ReferencesMehra, A.L. (1966). R
Page 699 and 700: 668 ReferencesNeugebauer, G.and Mei
Page 701 and 702: 670 ReferencesPant, D.N. (1994). Va
Page 703 and 704: 672 ReferencesPiper, M.S.(1997).Com
Page 705 and 706: 674 ReferencesRobertson, H.P. (1929
Page 707 and 708: 676 ReferencesSchmidt, B.G. (1996).
Page 709 and 710: 678 ReferencesSingleton, D.B. (1990
Page 711 and 712: 680 ReferencesStewart, B.W., Witten
Page 713: 682 ReferencesTeixeira, A.F.F., Wol
Page 717 and 718: 686 References65th birthday, ed. M.
Page 719 and 720: 688 ReferencesXanthopoulos, B.C. (1
Page 721 and 722: Indexacceleration, 70affine colline
Page 723 and 724: 692 Indexdust solutionsFriedmann un
Page 725 and 726: 694 IndexHarrison solutions, 272Har
Page 727 and 728: 696 IndexNUT solution, 310, 449, 45
Page 729 and 730: 698 IndexRiemann tensor, 25, 34cova
Page 731 and 732: 700 Indextype D solutions (contd)Ro
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