exact solutions of einstein’s field...

Exact Solutions of Einstein’s Field Equations

A revised edition of the now classic text, Exact Solutions of Einstein’s Field Equationsgives a unique survey of the known solutions of Einstein’s field equations for vacuum,Einstein–Maxwell, pure radiation and perfect fluid sources. It starts by introducing thefoundations of differential geometry and Riemannian geometry and the methods usedto characterize, find or construct solutions. The solutions are then considered, orderedby their symmetry group, their algebraic structure (Petrov type) or other invariantproperties such as special subspaces or tensor fields and embedding properties.

This edition has been expanded and updated to include the new developments in thefield since the publication of the first edition. It contains five completely new chapters,covering topics such as generation methods and their application, colliding waves, clas-sification of metrics by invariants and inhomogeneous cosmologies. It is an importantsource and guide for graduates and researchers in relativity, theoretical physics, astro-physics and mathematics. Parts of the book can also be used for preparing lectures andas an introductory text on some mathematical aspects of general relativity.

han s s t ephan i gained his Diploma, Ph.D. and Habilitation at the Friedrich-Schiller-Universitat Jena. He became Professor of Theoretical Physics in 1992, beforeretiring in 2000. He has been lecturing in theoretical physics since 1964 and has pub-lished numerous papers and articles on relativity and optics. He is also the author offour books.

d i e tr i ch kramer is Professor of Theoretical Physics at the Friedrich-Schiller-Universitat Jena. He graduated from this university, where he also finished his Ph.D.(1966) and habilitation (1970). His current research concerns classical relativity. Themajority of his publications are devoted to exact solutions in general relativity.

malcolm maccal lum is Professor of Applied Mathematics at the School ofMathematical Sciences, Queen Mary, University of London, where he is also Vice-Principal for Science and Engineering. He graduated from Kings College, Cambridgeand went on to complete his M.A. and Ph.D. there. His research covers general rela-tivity and computer algebra, especially tensor manipulators and differential equations.He has published over 100 papers, review articles and books.

corne l i u s hoen s e laer s gained his Diploma at Technische UniversitatKarlsruhe, his D.Sc. at Hiroshima Daigaku and his Habilitation at Ludwig-MaximilianUniversitat Munchen. He is Reader in Relativity Theory at Loughborough University.He has specialized in exact solutions in general relativity and other non-linear par-tial differential equations, and published a large number of papers, review articles andbooks.

eduard herlt is wissenschaftlicher Mitarbeiter at the Theoretisch PhysikalischesInstitut der Friedrich-Schiller-Universitat Jena. Having studied physics as an under-graduate at Jena, he went on to complete his Ph.D. there as well as his Habilitation.He has had numerous publications including one previous book.

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Exact Solutions of Einstein’sField Equations

Second Edition

HANS STEPHANIFriedrich-Schiller-Universitat, Jena

DIETRICH KRAMERFriedrich-Schiller-Universitat, Jena

MALCOLM MACCALLUMQueen Mary, University of London

CORNELIUS HOENSELAERSLoughborough University

EDUARD HERLTFriedrich-Schiller-Universitat, Jena






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c© H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers,

E. Herlt 2003

This book is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without

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A catalogue record for this book is available from the British Library

Library of Congress Cataloguing in Publication data

Exact solutions of Einstein’s field equations. – 2nd ed. / H. Stephani . . . [et al.].

p. cm. – (Cambridge monographs on mathematical physics)Includes bibliographical references and index.

ISBN 0-521-46136-71. General relativity (Physics) 2. Gravitational waves.

3. Space and time. 4. Einstein field equations – Numerical solutions.I. Stephani, Hans. II. Series.

QC173.6 .E96 2003530.11–dc21 2002071495

ISBN 0 521 46136 7 hardback






Contents

Preface xix

List of tables xxiii

Notation xxvii

1 Introduction 11.1 What are exact solutions, and why study them? 11.2 The development of the subject 31.3 The contents and arrangement of this book 41.4 Using this book as a catalogue 7

Part I: General methods 9

2 Differential geometry without a metric 92.1 Introduction 92.2 Differentiable manifolds 102.3 Tangent vectors 122.4 One-forms 132.5 Tensors 152.6 Exterior products and p-forms 172.7 The exterior derivative 182.8 The Lie derivative 212.9 The covariant derivative 232.10 The curvature tensor 252.11 Fibre bundles 27

vii






viii Contents

3 Some topics in Riemannian geometry 303.1 Introduction 303.2 The metric tensor and tetrads 303.3 Calculation of curvature from the metric 343.4 Bivectors 353.5 Decomposition of the curvature tensor 373.6 Spinors 403.7 Conformal transformations 433.8 Discontinuities and junction conditions 45

4 The Petrov classification 484.1 The eigenvalue problem 484.2 The Petrov types 494.3 Principal null directions and determination of the

Petrov types 53

5 Classification of the Ricci tensor and theenergy-momentum tensor 57

5.1 The algebraic types of the Ricci tensor 575.2 The energy-momentum tensor 605.3 The energy conditions 635.4 The Rainich conditions 645.5 Perfect fluids 65

6 Vector fields 686.1 Vector fields and their invariant classification 68

6.1.1 Timelike unit vector fields 706.1.2 Geodesic null vector fields 70

6.2 Vector fields and the curvature tensor 726.2.1 Timelike unit vector fields 726.2.2 Null vector fields 74

7 The Newman–Penrose and relatedformalisms 75

7.1 The spin coefficients and their transformationlaws 75

7.2 The Ricci equations 787.3 The Bianchi identities 817.4 The GHP calculus 847.5 Geodesic null congruences 867.6 The Goldberg–Sachs theorem and its generalizations 87






Contents ix

8 Continuous groups of transformations; isometryand homothety groups 91

8.1 Lie groups and Lie algebras 918.2 Enumeration of distinct group structures 958.3 Transformation groups 978.4 Groups of motions 988.5 Spaces of constant curvature 1018.6 Orbits of isometry groups 104

8.6.1 Simply-transitive groups 1058.6.2 Multiply-transitive groups 106

8.7 Homothety groups 110

9 Invariants and the characterization of geometries 1129.1 Scalar invariants and covariants 1139.2 The Cartan equivalence method for space-times 1169.3 Calculating the Cartan scalars 120

9.3.1 Determination of the Petrov and Segre types 1209.3.2 The remaining steps 124

9.4 Extensions and applications of the Cartan method 1259.5 Limits of families of space-times 126

10 Generation techniques 12910.1 Introduction 12910.2 Lie symmetries of Einstein’s equations 129

10.2.1 Point transformations and their generators 12910.2.2 How to find the Lie point symmetries of a given

differential equation 13110.2.3 How to use Lie point symmetries: similarity

reduction 13210.3 Symmetries more general than Lie symmetries 134

10.3.1 Contact and Lie–Backlund symmetries 13410.3.2 Generalized and potential symmetries 134

10.4 Prolongation 13710.4.1 Integral manifolds of differential forms 13710.4.2 Isovectors, similarity solutions and conservation laws 14010.4.3 Prolongation structures 141

10.5 Solutions of the linearized equations 14510.6 Backlund transformations 14610.7 Riemann–Hilbert problems 14810.8 Harmonic maps 14810.9 Variational Backlund transformations 15110.10 Hirota’s method 152






x Contents

10.11 Generation methods including perfect fluids 15210.11.1 Methods using the existence of Killing vectors 15210.11.2 Conformal transformations 155

Part II: Solutions with groups of motions 157

11 Classification of solutions with isometries orhomotheties 157

11.1 The possible space-times with isometries 15711.2 Isotropy and the curvature tensor 15911.3 The possible space-times with proper

homothetic motions 16211.4 Summary of solutions with homotheties 167

12 Homogeneous space-times 17112.1 The possible metrics 17112.2 Homogeneous vacuum and null Einstein-Maxwell space-times 17412.3 Homogeneous non-null electromagnetic fields 17512.4 Homogeneous perfect fluid solutions 17712.5 Other homogeneous solutions 18012.6 Summary 181

13 Hypersurface-homogeneous space-times 18313.1 The possible metrics 183

13.1.1 Metrics with a G6 on V3 18313.1.2 Metrics with a G4 on V3 18313.1.3 Metrics with a G3 on V3 187

13.2 Formulations of the field equations 18813.3 Vacuum, Λ-term and Einstein–Maxwell solutions 194

13.3.1 Solutions with multiply-transitive groups 19413.3.2 Vacuum spaces with a G3 on V3 19613.3.3 Einstein spaces with a G3 on V3 19913.3.4 Einstein–Maxwell solutions with a G3 on V3 201

13.4 Perfect fluid solutions homogeneous on T3 20413.5 Summary of all metrics with Gr on V3 207

14 Spatially-homogeneous perfect fluid cosmologies 21014.1 Introduction 21014.2 Robertson–Walker cosmologies 21114.3 Cosmologies with a G4 on S3 21414.4 Cosmologies with a G3 on S3 218






Contents xi

15 Groups G3 on non-null orbits V2. Sphericaland plane symmetry 226

15.1 Metric, Killing vectors, and Ricci tensor 22615.2 Some implications of the existence of an isotropy

group I1 22815.3 Spherical and plane symmetry 22915.4 Vacuum, Einstein–Maxwell and pure radiation fields 230

15.4.1 Timelike orbits 23015.4.2 Spacelike orbits 23115.4.3 Generalized Birkhoff theorem 23215.4.4 Spherically- and plane-symmetric fields 233

15.5 Dust solutions 23515.6 Perfect fluid solutions with plane, spherical or

pseudospherical symmetry 23715.6.1 Some basic properties 23715.6.2 Static solutions 23815.6.3 Solutions without shear and expansion 23815.6.4 Expanding solutions without shear 23915.6.5 Solutions with nonvanishing shear 240

15.7 Plane-symmetric perfect fluid solutions 24315.7.1 Static solutions 24315.7.2 Non-static solutions 244

16 Spherically-symmetric perfect fluid solutions 24716.1 Static solutions 247

16.1.1 Field equations and first integrals 24716.1.2 Solutions 250

16.2 Non-static solutions 25116.2.1 The basic equations 25116.2.2 Expanding solutions without shear 25316.2.3 Solutions with non-vanishing shear 260

17 Groups G2 and G1 on non-null orbits 26417.1 Groups G2 on non-null orbits 264

17.1.1 Subdivisions of the groups G2 26417.1.2 Groups G2I on non-null orbits 26517.1.3 G2II on non-null orbits 267

17.2 Boost-rotation-symmetric space-times 26817.3 Group G1 on non-null orbits 271

18 Stationary gravitational fields 27518.1 The projection formalism 275






xii Contents

18.2 The Ricci tensor on Σ3 27718.3 Conformal transformation of Σ3 and the field equations 27818.4 Vacuum and Einstein–Maxwell equations for stationary

fields 27918.5 Geodesic eigenrays 28118.6 Static fields 283

18.6.1 Definitions 28318.6.2 Vacuum solutions 28418.6.3 Electrostatic and magnetostatic Einstein–Maxwell

fields 28418.6.4 Perfect fluid solutions 286

18.7 The conformastationary solutions 28718.7.1 Conformastationary vacuum solutions 28718.7.2 Conformastationary Einstein–Maxwell fields 288

18.8 Multipole moments 289

19 Stationary axisymmetric fields: basic conceptsand field equations 292

19.1 The Killing vectors 29219.2 Orthogonal surfaces 29319.3 The metric and the projection formalism 29619.4 The field equations for stationary axisymmetric Einstein–

Maxwell fields 29819.5 Various forms of the field equations for stationary axisym-

metric vacuum fields 29919.6 Field equations for rotating fluids 302

20 Stationary axisymmetric vacuum solutions 30420.1 Introduction 30420.2 Static axisymmetric vacuum solutions (Weyl’s

class) 30420.3 The class of solutions U = U(ω) (Papapetrou’s class) 30920.4 The class of solutions S = S(A) 31020.5 The Kerr solution and the Tomimatsu–Sato class 31120.6 Other solutions 31320.7 Solutions with factor structure 316

21 Non-empty stationary axisymmetric solutions 31921.1 Einstein–Maxwell fields 319

21.1.1 Electrostatic and magnetostatic solutions 31921.1.2 Type D solutions: A general metric and its limits 32221.1.3 The Kerr–Newman solution 325






Contents xiii

21.1.4 Complexification and the Newman–Janis ‘complextrick’ 328

21.1.5 Other solutions 32921.2 Perfect fluid solutions 330

21.2.1 Line element and general properties 33021.2.2 The general dust solution 33121.2.3 Rigidly rotating perfect fluid solutions 33321.2.4 Perfect fluid solutions with differential rotation 337

22 Groups G2I on spacelike orbits: cylindricalsymmetry 341

22.1 General remarks 34122.2 Stationary cylindrically-symmetric fields 34222.3 Vacuum fields 35022.4 Einstein–Maxwell and pure radiation fields 354

23 Inhomogeneous perfect fluid solutions withsymmetry 358

23.1 Solutions with a maximal H3 on S3 35923.2 Solutions with a maximal H3 on T3 36123.3 Solutions with a G2 on S2 362

23.3.1 Diagonal metrics 36323.3.2 Non-diagonal solutions with orthogonal transitivity 37223.3.3 Solutions without orthogonal transitivity 373

23.4 Solutions with a G1 or a H2 374

24 Groups on null orbits. Plane waves 37524.1 Introduction 37524.2 Groups G3 on N3 37624.3 Groups G2 on N2 37724.4 Null Killing vectors (G1 on N1) 379

24.4.1 Non-twisting null Killing vector 38024.4.2 Twisting null Killing vector 382

24.5 The plane-fronted gravitational waves with parallel rays(pp-waves) 383

25 Collision of plane waves 38725.1 General features of the collision problem 38725.2 The vacuum field equations 38925.3 Vacuum solutions with collinear polarization 39225.4 Vacuum solutions with non-collinear polarization 39425.5 Einstein–Maxwell fields 397






xiv Contents

25.6 Stiff perfect fluids and pure radiation 40325.6.1 Stiff perfect fluids 40325.6.2 Pure radiation (null dust) 405

Part III: Algebraically special solutions 407

26 The various classes of algebraically specialsolutions. Some algebraically general solutions 407

26.1 Solutions of Petrov type II, D, III or N 40726.2 Petrov type D solutions 41226.3 Conformally flat solutions 41326.4 Algebraically general vacuum solutions with geodesic

and non-twisting rays 413

27 The line element for metrics with κ = σ = 0 =R11 = R14 = R44, Θ + iω = 0 416

27.1 The line element in the case with twisting rays (ω = 0) 41627.1.1 The choice of the null tetrad 41627.1.2 The coordinate frame 41827.1.3 Admissible tetrad and coordinate transformations 420

27.2 The line element in the case with non-twisting rays (ω = 0) 420

28 Robinson–Trautman solutions 42228.1 Robinson–Trautman vacuum solutions 422

28.1.1 The field equations and their solutions 42228.1.2 Special cases and explicit solutions 424

28.2 Robinson–Trautman Einstein–Maxwell fields 42728.2.1 Line element and field equations 42728.2.2 Solutions of type III, N and O 42928.2.3 Solutions of type D 42928.2.4 Type II solutions 431

28.3 Robinson–Trautman pure radiation fields 43528.4 Robinson–Trautman solutions with a cosmological

constant Λ 436

29 Twisting vacuum solutions 43729.1 Twisting vacuum solutions – the field equations 437

29.1.1 The structure of the field equations 43729.1.2 The integration of the main equations 43829.1.3 The remaining field equations 44029.1.4 Coordinate freedom and transformation

properties 441






Contents xv

29.2 Some general classes of solutions 44229.2.1 Characterization of the known classes of solutions 44229.2.2 The case ∂ζI = ∂ζ(G

2 − ∂ζG) = 0 44529.2.3 The case ∂ζI = ∂ζ(G

2 − ∂ζG) = 0, L,u = 0 44629.2.4 The case I = 0 44729.2.5 The case I = 0 = L,u 44929.2.6 Solutions independent of ζ and ζ 450

29.3 Solutions of type N (Ψ2 = 0 = Ψ3) 45129.4 Solutions of type III (Ψ2 = 0,Ψ3 = 0) 45229.5 Solutions of type D (3Ψ2Ψ4 = 2Ψ2

3, Ψ2 = 0) 45229.6 Solutions of type II 454

30 Twisting Einstein–Maxwell and pure radiationfields 455

30.1 The structure of the Einstein–Maxwell field equations 45530.2 Determination of the radial dependence of the metric and the

Maxwell field 45630.3 The remaining field equations 45830.4 Charged vacuum metrics 45930.5 A class of radiative Einstein–Maxwell fields (Φ02 = 0) 46030.6 Remarks concerning solutions of the different Petrov types 46130.7 Pure radiation fields 463

30.7.1 The field equations 46330.7.2 Generating pure radiation fields from vacuum by

changing P 46430.7.3 Generating pure radiation fields from vacuum by

changing m 46630.7.4 Some special classes of pure radiation fields 467

31 Non-diverging solutions (Kundt’s class) 47031.1 Introduction 47031.2 The line element for metrics with Θ + iω = 0 47031.3 The Ricci tensor components 47231.4 The structure of the vacuum and Einstein–Maxwell

equation 47331.5 Vacuum solutions 476

31.5.1 Solutions of types III and N 47631.5.2 Solutions of types D and II 478

31.6 Einstein–Maxwell null fields and pure radiation fields 48031.7 Einstein–Maxwell non-null fields 48131.8 Solutions including a cosmological constant Λ 483






xvi Contents

32 Kerr–Schild metrics 48532.1 General properties of Kerr–Schild metrics 485

32.1.1 The origin of the Kerr–Schild–Trautman ansatz 48532.1.2 The Ricci tensor, Riemann tensor and Petrov type 48532.1.3 Field equations and the energy-momentum tensor 48732.1.4 A geometrical interpretation of the Kerr–Schild

ansatz 48732.1.5 The Newman–Penrose formalism for shearfree and

geodesic Kerr–Schild metrics 48932.2 Kerr–Schild vacuum fields 492

32.2.1 The case ρ = −(Θ + iω) = 0 49232.2.2 The case ρ = −(Θ + iω) = 0 493

32.3 Kerr–Schild Einstein–Maxwell fields 49332.3.1 The case ρ = −(Θ + iω) = 0 49332.3.2 The case ρ = −(Θ + iω) = 0 495

32.4 Kerr–Schild pure radiation fields 49732.4.1 The case ρ = 0, σ = 0 49732.4.2 The case σ = 0 49932.4.3 The case ρ = σ = 0 499

32.5 Generalizations of the Kerr–Schild ansatz 49932.5.1 General properties and results 49932.5.2 Non-flat vacuum to vacuum 50132.5.3 Vacuum to electrovac 50232.5.4 Perfect fluid to perfect fluid 503

33 Algebraically special perfect fluid solutions 50633.1 Generalized Robinson–Trautman solutions 50633.2 Solutions with a geodesic, shearfree, non-expanding multiple

null eigenvector 51033.3 Type D solutions 512

33.3.1 Solutions with κ = ν = 0 51333.3.2 Solutions with κ = 0, ν = 0 513

33.4 Type III and type N solutions 515

Part IV: Special methods 518

34 Application of generation techniques to generalrelativity 518

34.1 Methods using harmonic maps (potential spacesymmetries) 51834.1.1 Electrovacuum fields with one Killing vector 51834.1.2 The group SU(2,1) 521






Contents xvii

34.1.3 Complex invariance transformations 52534.1.4 Stationary axisymmetric vacuum fields 526

34.2 Prolongation structure for the Ernst equation 52934.3 The linearized equations, the Kinnersley–Chitre B group and

the Hoenselaers–Kinnersley–Xanthopoulos transformations 53234.3.1 The field equations 53234.3.2 Infinitesimal transformations and transformations

preserving Minkowski space 53434.3.3 The Hoenselaers–Kinnersley–Xanthopoulos transfor-

mation 53534.4 Backlund transformations 53834.5 The Belinski–Zakharov technique 54334.6 The Riemann–Hilbert problem 547

34.6.1 Some general remarks 54734.6.2 The Neugebauer–Meinel rotating disc solution 548

34.7 Other approaches 54934.8 Einstein–Maxwell fields 55034.9 The case of two space-like Killing vectors 550

35 Special vector and tensor fields 55335.1 Space-times that admit constant vector and tensor fields 553

35.1.1 Constant vector fields 55335.1.2 Constant tensor fields 554

35.2 Complex recurrent, conformally recurrent, recurrent andsymmetric spaces 55635.2.1 The definitions 55635.2.2 Space-times of Petrov type D 55735.2.3 Space-times of type N 55735.2.4 Space-times of type O 558

35.3 Killing tensors of order two and Killing–Yano tensors 55935.3.1 The basic definitions 55935.3.2 First integrals, separability and Killing or Killing–

Yano tensors 56035.3.3 Theorems on Killing and Killing–Yano tensors in four-

dimensional space-times 56135.4 Collineations and conformal motions 564

35.4.1 The basic definitions 56435.4.2 Proper curvature collineations 56535.4.3 General theorems on conformal motions 56535.4.4 Non-conformally flat solutions admitting proper

conformal motions 567






xviii Contents

36 Solutions with special subspaces 57136.1 The basic formulae 57136.2 Solutions with flat three-dimensional slices 573

36.2.1 Vacuum solutions 57336.2.2 Perfect fluid and dust solutions 573

36.3 Perfect fluid solutions with conformally flat slices 57736.4 Solutions with other intrinsic symmetries 579

37 Local isometric embedding of four-dimensionalRiemannian manifolds 580

37.1 The why of embedding 58037.2 The basic formulae governing embedding 58137.3 Some theorems on local isometric embedding 583

37.3.1 General theorems 58337.3.2 Vector and tensor fields and embedding class 58437.3.3 Groups of motions and embedding class 586

37.4 Exact solutions of embedding class one 58737.4.1 The Gauss and Codazzi equations and the possible

types of Ωab 58737.4.2 Conformally flat perfect fluid solutions of embedding

class one 58837.4.3 Type D perfect fluid solutions of embedding class one 59137.4.4 Pure radiation field solutions of embedding class one 594

37.5 Exact solutions of embedding class two 59637.5.1 The Gauss–Codazzi–Ricci equations 59637.5.2 Vacuum solutions of embedding class two 59837.5.3 Conformally flat solutions 599

37.6 Exact solutions of embedding class p > 2 603

Part V: Tables 605

38 The interconnections between the mainclassification schemes 605

38.1 Introduction 60538.2 The connection between Petrov types and groups of motions 60638.3 Tables 609

References 615

Index 690






Preface

When, in 1975, two of the authors (D.K. and H.S.) proposed to changetheir field of research back to the subject of exact solutions of Einstein’sfield equations, they of course felt it necessary to make a careful studyof the papers published in the meantime, so as to avoid duplication ofknown results. A fairly comprehensive review or book on the exact solu-tions would have been a great help, but no such book was available. Thisprompted them to ask ‘Why not use the preparatory work we have todo in any case to write such a book?’ After some discussion, they agreedto go ahead with this idea, and then they looked for coauthors. Theysucceeded in finding two.

The first was E.H., a member of the Jena relativity group, who had beenengaged before in exact solutions and was also inclined to return to them.

The second, M.M., became involved by responding to the existing au-thors’ appeal for information and then (during a visit by H.S. to London)agreeing to look over the English text. Eventually he agreed to write someparts of the book.

The quartet’s original optimism somewhat diminished when referencesto over 2000 papers had been collected and the magnitude of the taskbecame all too clear. How could we extract even the most importantinformation from this mound of literature? How could we avoid constantrewriting to incorporate new information, which would have made thejob akin to the proverbial painting of the Forth bridge? How could wedecide which topics to include and which to omit? How could we checkthe calculations, put the results together in a readable form and still finishin reasonable time?

We did not feel that we had solved any of these questions in a completelyconvincing manner. However, we did manage to produce an outcome,which was the first edition of this book, Kramer et al. (1980).

xix






xx Preface

In the years since then so many new exact solutions have been publishedthat the first edition can no longer be used as a reliable guide to thesubject. The authors therefore decided to prepare a new edition. Althoughthey knew from experience the amount of work to be expected, it tookthem longer than they thought and feared. We looked at over 4000 newpapers (the cut-off date for the systematic search for papers is the endof 1999). In particular so much research had been done in the field ofgeneration techniques and their applications that the original chapter hadto be almost completely replaced, and C.H. was asked to collaborate onthis, and agreed.

Compared with the first edition, the general arrangement of the ma-terial has not been changed. But we have added five new chapters, thusreflecting the developments of the last two decades (Chapters 9, 10, 23, 25and 36), and some of the old chapters have been substantially rewritten.Unfortunately, the sheer number of known exact solutions has forced usto give up the idea of presenting them all in some detail; instead, in manycases we only give the appropriate references.

As with the first edition, the labour of reading those papers conceiv-ably relevant to each chapter or section, and then drafting the relatedmanuscript, was divided. Roughly, D.K., M.M. and C.H. were responsi-ble for most of the introductory Part I, M.M., D.K. and H.S. dealt withgroups (Part II), H.S., D.K. and E.H. with algebraically special solutions(Part III) and H.S. and C.H. with Part IV (special methods) and Part V(tables). Each draft was then criticized by the other authors, so that itswriter could not be held wholly responsible for any errors or omissions.Since we hope to maintain up-to-date information, we shall be glad tohear from any reader who detects such errors or omissions; we shall bepleased to answer as best we can any requests for further information.M.M. wishes to record that any infelicities remaining in the English arosebecause the generally good standard of his colleagues’ English lulled himinto a false sense of security.

This book could not have been written, of course, without the efforts ofthe many scientists whose work is recorded here, and especially the manycontemporaries who sent preprints, references and advice or informed usof mistakes or omissions in the first edition of this book. More immedi-ately we have gratefully to acknowledge the help of the students in Jena,and in particular of S. Falkenberg, who installed our electronic files, ofA. Koutras, who wrote many of the old chapters in LaTeX and simulta-neously checked many of the solutions, and of the financial support of theMax-Planck-Group in Jena and the Friedrich-Schiller-Universitat Jena.Last but not least, we have to thank our wives, families and colleagues






Preface xxi

for tolerating our incessant brooding and discussions and our obsessionwith the book.

Hans StephaniJena

Dietrich KramerJena

Malcolm MacCallumLondon

Cornelius HoenselaersLoughborough

Eduard HerltJena






List of Tables

3.1 Examples of spinor equivalents, defined as in (3.70). 42

4.1 The Petrov types 504.2 Normal forms of the Weyl tensor, and Petrov types 514.3 The roots of the algebraic equation (4.18) and their mul-

tiplicities. 55

5.1 The algebraic types of the Ricci tensor 595.2 Invariance groups of the Ricci tensor types 60

8.1 Enumeration of the Bianchi types 968.2 Killing vectors and reciprocal group generators by Bianchi

type 107

9.1 Maximum number of derivatives required to characterizea metric locally 121

11.1 Metrics with isometries listed by orbit and group action,and where to find them 163

11.2 Solutions with proper homothety groups Hr, r > 4 16511.3 Solutions with proper homothety groups H4 on V4 16611.4 Solutions with proper homothety groups on V3 168

12.1 Homogeneous solutions 181

13.1 The number of essential parameters, by Bianchi type, ingeneral solutions for vacuum and for perfect fluids withgiven equation of state 189

xxiii






xxiv List of Tables

13.2 Subgroups G3 on V3 occurring in metrics with multiply-transitive groups 208

13.3 Solutions given in this book with a maximal G4 on V3 20813.4 Solutions given explicitly in this book with a maximal G3

on V3 209

15.1 The vacuum, Einstein–Maxwell and pure radiation solu-tions with G3 on S2 (Y,aY

,a > 0) 231

16.1 Key assumptions of some static spherically-symmetric per-fect fluid solutions in isotropic coordinates 251

16.2 Key assumptions of some static spherically-symmetric per-fect fluid solutions in canonical coordinates 252

16.3 Some subclasses of the class F = (ax2+2bx + c)−5/2 ofsolutions 257

18.1 The complex potentials E and Φ for some physical prob-lems 281

18.2 The degenerate static vacuum solutions 285

21.1 Stationary axisymmetric Einstein–Maxwell fields 325

24.1 Metrics ds2 = x−1/2(dx2+dy2)−2xdu [dv + M(x, y, u)du]with more than one symmetry 382

24.2 Symmetry classes of vacuum pp-waves 385

26.1 Subcases of the algebraically special (not conformally flat)solutions 408

28.1 The Petrov types of the Robinson–Trautman vacuum so-lutions 424

29.1 The possible types of two-variable twisting vacuum met-rics 443

29.2 Twisting algebraically special vacuum solutions 444

32.1 Kerr–Schild space-times 492

34.1 The subspaces of the potential space for stationaryEinstein–Maxwell fields, and the corresponding subgroupsof SU(2, 1) 523

34.2 Generation by potential space transformations 53034.3 Applications of the HKX method 53734.4 Applications of the Belinski–Zakharov method 546






List of Tables xxv

37.1 Upper limits for the embedding class p of various metricsadmitting groups 586

37.2 Embedding class one solutions 59537.3 Metrics known to be of embedding class two 603

38.1 The algebraically special, diverging vacuum solutions ofmaximum mobility 607

38.2 Robinson–Trautman vacuum solutions admitting two ormore Killing vectors 608

38.3 Petrov types versus groups on orbits V4 60938.4 Petrov types versus groups on non-null orbits V3 61038.5 Petrov types versus groups on non-null orbits V2 and V1 61038.6 Energy-momentum tensors versus groups on orbits V4

(with LξFab = 0 for the Maxwell field) 61138.7 Energy-momentum tensors versus groups on non-null or-

bits V3 61138.8 Energy-momentum tensors versus groups on non-null or-

bits V2 and V1 61238.9 Algebraically special vacuum, Einstein–Maxwell and pure

radiation fields (non-aligned or with κκ + σσ = 0) 61338.10 Algebraically special (non-vacuum) Einstein–Maxwell and

pure radiation fields, aligned and with κκ + σσ = 0. 614






Notation

All symbols are explained in the text. Here we list only some importantconventions which are frequently used throughout the book.

Complex conjugates and constants

Complex conjugation is denoted by a bar over the symbol. The abbrevi-ation const is used for ‘constant’.

Indices

Small Latin indices run, in an n-dimensional Riemannian space Vn, from1 to n, and in space-time V4 from 1 to 4. When a general basis ea or itsdual ωa is in use, indices from the first part of the alphabet (a, b, . . . , h)will normally be tetrad indices and i, j, . . . are reserved for a coordinatebasis ∂/∂xi or its dual dxi. For a vector v and a l-form σ we writev = vaea = vi∂/∂xi, σ = σaω

a = σidxi. Small Greek indices run from1 to 3, if not otherwise stated. Capital Latin indices are either spinorindices (A, B = 1, 2) or indices in group space (A, B = 1, . . . , r), or theylabel the coordinates in a Riemannian 2-space V2 (M, N = 1, 2).

Symmetrization and antisymmetrization of index pairs are indicated byround and square brackets respectively; thus

v(ab) ≡ 12(vab + vba), v[ab] ≡ 1

2(vab − vba).

The Kronecker delta, δab , has the value 1 if a = b and zero otherwise.

Metric and tetrads

Line element in terms of dual basis ωa: ds2 = gabωaωb.

Signature of space-time metric: (+ + +−).

xxvii






xxviii Notation

Commutation coefficients: Dcab; [ea, eb] = Dc

abec.(Complex) null tetrad: ea = (m, m, l, k), gab = 2m(amb) − 2k(alb),

ds2 = 2ω1ω2 − 2ω3ω4.Orthonormal basis: Ea.Projection tensor: hab ≡ gab + uaub, uau

a = −1.

Bivectors

Levi-Civita tensor in four dimensions: εabcd; εabcdmamblckd = i.

in two dimensions: εab = −εba, ε12 = 1Dual bivector: Xab ≡ 1

2εabcdXcd.

(Complex) self-dual bivector: X∗ab ≡ Xab + iXab.

Basis of self-dual bivectors: Uab ≡ 2m[alb], Vab ≡ 2k[amb],Wab ≡ 2m[amb] − 2k[alb].

Derivatives

Partial derivative: comma in front of index or coordinate, e.g.

f,i ≡ ∂f/∂xi ≡ ∂if, f,ζ ≡ ∂f/∂ζ.

Directional derivative: denoted by stroke or comma, f|a ≡ f,a ≡ ea(f);if followed by a numerical (tetrad) index, we prefer the stroke, e.g.f|4 = f,ik

i. Directional derivatives with respect to the null tetrad(m, m, l, k) are symbolized by δf = f|1, δf = f|2, ∆f = f|3, Df = f|4.

Covariant derivative: ∇; in component calculus, semicolon. (Sometimesother symbols are used to indicate that in V4 a metric different fromgab is used, e.g. hab||c = 0, γab:c = 0.)

Lie derivative of a tensor T with respect to a vector v: LvT .Exterior derivative: d.

When a dot is used to denote a derivative without definition, e.g. Q,it means differentiation with respect to the time coordinate in use; aprime used similarly, e.g. Q′, refers either to the unique essential spacecoordinate in the problem or to the single argument of a function.

Connection and curvature

Connection coefficients: Γabc, va;c = va,c + Γa

bcvb.

Connection 1-forms: Γ ab ≡ Γa

bcωc, dωa = −Γ a

b ∧ ωb.Riemann tensor: Rd

abc, 2va;[bc] = vdRdabc.






Notation xxix

Curvature 2-forms: Θab ≡ 1

2Rabcdω

c ∧ ωd = dΓ ab + Γ a

c ∧ Γ cb.

Ricci tensor, Einstein tensor, and scalar curvature:

Rab ≡ Rcacb, Gab ≡ Rab − 1

2Rgab, R ≡ Raa.

Weyl tensor in V4:

Cabcd ≡ Rabcd + 13Rga[cgd]b − ga[cRd]b + gb[cRd]a.

Null tetrad components of the Weyl tensor:

Ψ0 ≡ Cabcdkambkcmd, Ψ1 ≡ Cabcdk

albkcmd,

Ψ2 ≡ Cabcdkambmcld,

Ψ3 ≡ Cabcdkalbmcld, Ψ4 ≡ Cabcdm

albmcld.

Metric of a 2-space of constant curvature:

dσ2 = dx2 ± Σ2(x, ε)dy2,

Σ(x, ε) = sinx, x, sinhx resp. when ε = 1, 0 or −1.

Gaussian curvature: K.

Physical fields

Energy-momentum tensor: Tab, Tabuaub ≥ 0 if uau

a = −1.Electromagnetic field: Maxwell tensor Fab, Tab = F ∗c

a F∗bc/2.

Null tetrad components of Fab:

Φ0 ≡ Fabkamb, Φ1 ≡ 1

2Fab(kalb + mamb), Φ2 = Fabmalb.

Perfect fluid: pressure p, energy density µ, 4-velocity u,

Tab = (µ + p)uaub + pgab.

Cosmological constant: Λ.Gravitational constant: κ0.Einstein’s field equations: Rab − 1

2Rgab + Λgab = κ0Tab.

Symmetries

Group of motions (r-dim.), Gr; isotropy group (s-dim.), Is;homothety group (q-dim.), Hq.

Killing vectors: ξ, η, ζ, or ξA, A = 1, . . . , rKilling equation: (Lξg)ab = ξa;b + ξb;a = 0.Structure constants: CC

AB; [ξA, ξB] = CCABξC .

Orbits (m-dim.) of Gr or Hq: Sm (spacelike), Tm (timelike), Nm (null).






exact solutions of einstein’s field...

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