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2.11 Fibre bundles 29Jet bundles J (n) (X, U) are another type of fibre bundle of interest inexact solutions (see Chapter 10). They are used in describing systemsof differential equations. Here the base manifold X is the space of independentvariables and the fibre is the set U of values of the dependentvariables and their first n partial derivatives with respect to the X. A(partial) differential equation, or a system of such equations, specifies asubmanifold in a jet bundle.
3Some topics in Riemannian geometry3.1 IntroductionIn Chapter 2 we treated differential geometry without a metric. In orderto define covariant derivatives we imposed an extra structure on thedifferential manifold M, the connection. Adding a further structure, themetric g ab , and postulating g ab;c = 0, we arrive at Riemannian geometry.General relativity is based on the concept of space-time, which is a fourdimensionaldifferentiable (C ∞ , Hausdorff) manifold M endowed with aLorentzian metric g ab which can be transformed tog ab = η ab ≡ diag(1, 1, 1, −1) (3.1)at any point of M, i.e. space-time is a normal-hyperbolic Riemannianspace V 4 . In what follows, a knowledge of fundamental facts about Riemanniangeometry as given in most textbooks on general relativity is presumed;we give here only some notation and results used in the remainderof this book. For further details the reader is referred to standard textson Riemannian geometry, e.g. Eisenhart (1949) and Schouten (1954).3.2 The metric tensor and tetradsWe introduce as a new structure a symmetric tensor of type (0,2), calledthe metric tensor g, which endows each vector space T p with a scalarproduct (inner product)e a · e b = g ab . (3.2)The tensor g, sometimes called the line element ds 2 , is writteng =ds 2 = g ab ω a ω b . (3.3)30
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Exact Solutions of Einstein’s Fie
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CAMBRIDGE UNIVERSITY PRESSCambridge
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80 7 The Newman-Penrose and related
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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
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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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