exact solutions of einstein''s field equations

Exact Solutions of Einstein’s Field Equations
A revised edition of the now classic text, Exact Solutions of Einstein’s Field Equations
gives 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 the
foundations of differential geometry and Riemannian geometry and the methods used
to characterize, find or construct solutions. The solutions are then considered, ordered
by their symmetry group, their algebraic structure (Petrov type) or other invariant
properties such as special subspaces or tensor fields and embedding properties.
This edition has been expanded and updated to include the new developments in the
field 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 important
source 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 and
as an introductory text on some mathematical aspects of general relativity.
hans stephani gained his Diploma, Ph.D. and Habilitation at the Friedrich-
Schiller-Universit¨at Jena. He became Professor of Theoretical Physics in 1992, before
retiring 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 of
four books.
dietrich kramer is Professor of Theoretical Physics at the Friedrich-Schiller-
Universit¨at Jena. He graduated from this university, where he also finished his Ph.D.
(1966) and habilitation (1970). His current research concerns classical relativity. The
majority of his publications are devoted to exact solutions in general relativity.
malcolm maccallum is Professor of Applied Mathematics at the School of
Mathematical Sciences, Queen Mary, University of London, where he is also Vice-
Principal for Science and Engineering. He graduated from Kings College, Cambridge
and 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.

cornelius hoenselaers gained his Diploma at Technische Universit¨at
Karlsruhe, his D.Sc. at Hiroshima Daigaku and his Habilitation at Ludwig-Maximilian
Universit¨at M¨unchen. 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 and
books.
eduard herlt is wissenschaftlicher Mitarbeiter at the Theoretisch Physikalisches
Institut der Friedrich-Schiller-Universit¨at 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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Issued as a paperback

Exact Solutions of Einstein’s
Field Equations
Second Edition
HANS STEPHANI
Friedrich-Schiller-Universit¨at, Jena
DIETRICH KRAMER
Friedrich-Schiller-Universit¨at, Jena
MALCOLM MACCALLUM
Queen Mary, University of London
CORNELIUS HOENSELAERS
Loughborough University
EDUARD HERLT
Friedrich-Schiller-Universit¨at, Jena
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
First published in print format
ISBN-13 978-0-521-46136-8 hardback
ISBN-13 978-0-511-06548-4 eBook (NetLibrary)
© H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers, E. Herlt 2003
2003

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Published in the United States by Cambridge University Press, New York
www.cambridge.org
Contents
Preface xix
List of tables xxiii
Notation xxvii
1 Introduction 1
1.1 What are exact solutions, and why study them? 1
1.2 The development of the subject 3
1.3 The contents and arrangement of this book 4
1.4 Using this book as a catalogue 7
Part I: General methods 9
2 Differential geometry without a metric 9
2.1 Introduction 9
2.2 Differentiable manifolds 10
2.3 Tangent vectors 12
2.4 One-forms 13

2.5 Tensors 15
2.6 Exterior products and p-forms 17
2.7 The exterior derivative 18
2.8 The Lie derivative 21
2.9 The covariant derivative 23
2.10 The curvature tensor 25
2.11 Fibre bundles 27
vii
viii Contents
3 Some topics in Riemannian geometry 30
3.1 Introduction 30
3.2 The metric tensor and tetrads 30
3.3 Calculation of curvature from the metric 34
3.4 Bivectors 35
3.5 Decomposition of the curvature tensor 37
3.6 Spinors 40
3.7 Conformal transformations 43
3.8 Discontinuities and junction conditions 45
4 The Petrov classification 48
4.1 The eigenvalue problem 48
4.2 The Petrov types 49
4.3 Principal null directions and determination of the
Petrov types 53
5 Classification of the Ricci tensor and the
energy-momentum tensor 57
5.1 The algebraic types of the Ricci tensor 57
5.2 The energy-momentum tensor 60
5.3 The energy conditions 63
5.4 The Rainich conditions 64
5.5 Perfect fluids 65

6 Vector fields 68
6.1 Vector fields and their invariant classification 68
6.1.1 Timelike unit vector fields 70
6.1.2 Geodesic null vector fields 70
6.2 Vector fields and the curvature tensor 72
6.2.1 Timelike unit vector fields 72
6.2.2 Null vector fields 74
7 The Newman–Penrose and related
formalisms 75
7.1 The spin coefficients and their transformation
laws 75
7.2 The Ricci equations 78
7.3 The Bianchi identities 81
7.4 The GHP calculus 84
7.5 Geodesic null congruences 86
7.6 The Goldberg–Sachs theorem and its generalizations 87
Contents ix
8 Continuous groups of transformations; isometry
and homothety groups 91
8.1 Lie groups and Lie algebras 91
8.2 Enumeration of distinct group structures 95
8.3 Transformation groups 97
8.4 Groups of motions 98
8.5 Spaces of constant curvature 101
8.6 Orbits of isometry groups 104
8.6.1 Simply-transitive groups 105
8.6.2 Multiply-transitive groups 106
8.7 Homothety groups 110
9 Invariants and the characterization of geometries 112
9.1 Scalar invariants and covariants 113

9.2 The Cartan equivalence method for space-times 116
9.3 Calculating the Cartan scalars 120
9.3.1 Determination of the Petrov and Segre types 120
9.3.2 The remaining steps 124
9.4 Extensions and applications of the Cartan method 125
9.5 Limits of families of space-times 126
10 Generation techniques 129
10.1 Introduction 129
10.2 Lie symmetries of Einstein’s equations 129
10.2.1 Point transformations and their generators 129
10.2.2 How to find the Lie point symmetries of a given
differential equation 131
10.2.3 How to use Lie point symmetries: similarity
reduction 132
10.3 Symmetries more general than Lie symmetries 134
10.3.1 Contact and Lie–B¨acklund symmetries 134
10.3.2 Generalized and potential symmetries 134
10.4 Prolongation 137
10.4.1 Integral manifolds of differential forms 137
10.4.2 Isovectors, similarity solutions and conservation laws 140
10.4.3 Prolongation structures 141
10.5 Solutions of the linearized equations 145
10.6 B¨acklund transformations 146
10.7 Riemann–Hilbert problems 148
10.8 Harmonic maps 148
10.9 Variational B¨acklund transformations 151
10.10 Hirota’s method 152
x Contents
10.11 Generation methods including perfect fluids 152
10.11.1 Methods using the existence of Killing vectors 152

10.11.2 Conformal transformations 155
Part II: Solutions with groups of motions 157
11 Classification of solutions with isometries or
homotheties 157
11.1 The possible space-times with isometries 157
11.2 Isotropy and the curvature tensor 159
11.3 The possible space-times with proper
homothetic motions 162
11.4 Summary of solutions with homotheties 167
12 Homogeneous space-times 171
12.1 The possible metrics 171
12.2 Homogeneous vacuum and null Einstein-Maxwell space-times 174
12.3 Homogeneous non-null electromagnetic fields 175
12.4 Homogeneous perfect fluid solutions 177
12.5 Other homogeneous solutions 180
12.6 Summary 181
13 Hypersurface-homogeneous space-times 183
13.1 The possible metrics 183
13.1.1 Metrics with a G
6
on V
3
183
13.1.2 Metrics with a G
4
on V
3
183
13.1.3 Metrics with a G
3

on V
3
187
13.2 Formulations of the field equations 188
13.3 Vacuum, Λ-term and Einstein–Maxwell solutions 194
13.3.1 Solutions with multiply-transitive groups 194
13.3.2 Vacuum spaces with a G
3
on V
3
196
13.3.3 Einstein spaces with a G
3
on V
3
199
13.3.4 Einstein–Maxwell solutions with a G
3
on V
3
201
13.4 Perfect fluid solutions homogeneous on T
3
204
13.5 Summary of all metrics with G
r
on V
3
207
14 Spatially-homogeneous perfect fluid cosmologies 210

14.1 Introduction 210
14.2 Robertson–Walker cosmologies 211
14.3 Cosmologies with a G
4
on S
3
214
14.4 Cosmologies with a G
3
on S
3
218
Contents xi
15 Groups G
3
on non-null orbits V
2
. Spherical
and plane symmetry 226
15.1 Metric, Killing vectors, and Ricci tensor 226
15.2 Some implications of the existence of an isotropy
group I
1
228
15.3 Spherical and plane symmetry 229
15.4 Vacuum, Einstein–Maxwell and pure radiation fields 230
15.4.1 Timelike orbits 230
15.4.2 Spacelike orbits 231
15.4.3 Generalized Birkhoff theorem 232
15.4.4 Spherically- and plane-symmetric fields 233

15.5 Dust solutions 235
15.6 Perfect fluid solutions with plane, spherical or
pseudospherical symmetry 237
15.6.1 Some basic properties 237
15.6.2 Static solutions 238
15.6.3 Solutions without shear and expansion 238
15.6.4 Expanding solutions without shear 239
15.6.5 Solutions with nonvanishing shear 240
15.7 Plane-symmetric perfect fluid solutions 243
15.7.1 Static solutions 243
15.7.2 Non-static solutions 244
16 Spherically-symmetric perfect fluid solutions 247
16.1 Static solutions 247
16.1.1 Field equations and first integrals 247
16.1.2 Solutions 250
16.2 Non-static solutions 251
16.2.1 The basic equations 251
16.2.2 Expanding solutions without shear 253
16.2.3 Solutions with non-vanishing shear 260
17 Groups G
2
and G
1
on non-null orbits 264
17.1 Groups G
2
on non-null orbits 264
17.1.1 Subdivisions of the groups G
2
264

17.1.2 Groups G
2
I on non-null orbits 265
17.1.3 G
2
II on non-null orbits 267
17.2 Boost-rotation-symmetric space-times 268
17.3 Group G
1
on non-null orbits 271
18 Stationary gravitational fields 275
18.1 The projection formalism 275
xii Contents
18.2 The Ricci tensor on Σ
3
277
18.3 Conformal transformation of Σ
3
and the field equations 278
18.4 Vacuum and Einstein–Maxwell equations for stationary
fields 279
18.5 Geodesic eigenrays 281
18.6 Static fields 283
18.6.1 Definitions 283
18.6.2 Vacuum solutions 284
18.6.3 Electrostatic and magnetostatic Einstein–Maxwell
fields 284
18.6.4 Perfect fluid solutions 286
18.7 The conformastationary solutions 287
18.7.1 Conformastationary vacuum solutions 287

18.7.2 Conformastationary Einstein–Maxwell fields 288
18.8 Multipole moments 289
19 Stationary axisymmetric fields: basic concepts
and field equations 292
19.1 The Killing vectors 292
19.2 Orthogonal surfaces 293
19.3 The metric and the projection formalism 296
19.4 The field equations for stationary axisymmetric Einstein–
Maxwell fields 298
19.5 Various forms of the field equations for stationary axisym-
metric vacuum fields 299
19.6 Field equations for rotating fluids 302
20 Stationary axisymmetric vacuum solutions 304
20.1 Introduction 304
20.2 Static axisymmetric vacuum solutions (Weyl’s
class) 304
20.3 The class of solutions U = U(ω) (Papapetrou’s class) 309
20.4 The class of solutions S = S(A) 310
20.5 The Kerr solution and the Tomimatsu–Sato class 311
20.6 Other solutions 313
20.7 Solutions with factor structure 316
21 Non-empty stationary axisymmetric solutions 319
21.1 Einstein–Maxwell fields 319
21.1.1 Electrostatic and magnetostatic solutions 319
21.1.2 Type D solutions: A general metric and its limits 322
21.1.3 The Kerr–Newman solution 325
Contents xiii
21.1.4 Complexification and the Newman–Janis ‘complex
trick’ 328
21.1.5 Other solutions 329

21.2 Perfect fluid solutions 330
21.2.1 Line element and general properties 330
21.2.2 The general dust solution 331
21.2.3 Rigidly rotating perfect fluid solutions 333
21.2.4 Perfect fluid solutions with differential rotation 337
22 Groups G
2
I on spacelike orbits: cylindrical
symmetry 341
22.1 General remarks 341
22.2 Stationary cylindrically-symmetric fields 342
22.3 Vacuum fields 350
22.4 Einstein–Maxwell and pure radiation fields 354
23 Inhomogeneous perfect fluid solutions with
symmetry 358
23.1 Solutions with a maximal H
3
on S
3
359
23.2 Solutions with a maximal H
3
on T
3
361
23.3 Solutions with a G
2
on S
2
362

23.3.1 Diagonal metrics 363
23.3.2 Non-diagonal solutions with orthogonal transitivity 372
23.3.3 Solutions without orthogonal transitivity 373
23.4 Solutions with a G
1
or a H
2
374
24 Groups on null orbits. Plane waves 375
24.1 Introduction 375
24.2 Groups G
3
on N
3
376
24.3 Groups G
2
on N
2
377
24.4 Null Killing vectors (G
1
on N
1
) 379
24.4.1 Non-twisting null Killing vector 380
24.4.2 Twisting null Killing vector 382
24.5 The plane-fronted gravitational waves with parallel rays
(pp-waves) 383
25 Collision of plane waves 387

25.1 General features of the collision problem 387
25.2 The vacuum field equations 389
25.3 Vacuum solutions with collinear polarization 392
25.4 Vacuum solutions with non-collinear polarization 394
25.5 Einstein–Maxwell fields 397
xiv Contents
25.6 Stiff perfect fluids and pure radiation 403
25.6.1 Stiff perfect fluids 403
25.6.2 Pure radiation (null dust) 405
Part III: Algebraically special solutions 407
26 The various classes of algebraically special
solutions. Some algebraically general solutions 407
26.1 Solutions of Petrov type II, D, III or N 407
26.2 Petrov type D solutions 412
26.3 Conformally flat solutions 413
26.4 Algebraically general vacuum solutions with geodesic
and non-twisting rays 413
27 The line element for metrics with κ = σ =0=
R
11
= R
14
= R
44
, Θ+iω =0 416
27.1 The line element in the case with twisting rays (ω =0) 416
27.1.1 The choice of the null tetrad 416
27.1.2 The coordinate frame 418
27.1.3 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 422
28.1 Robinson–Trautman vacuum solutions 422
28.1.1 The field equations and their solutions 422
28.1.2 Special cases and explicit solutions 424
28.2 Robinson–Trautman Einstein–Maxwell fields 427
28.2.1 Line element and field equations 427
28.2.2 Solutions of type III, N and O 429
28.2.3 Solutions of type D 429
28.2.4 Type II solutions 431
28.3 Robinson–Trautman pure radiation fields 435
28.4 Robinson–Trautman solutions with a cosmological
constant Λ 436
29 Twisting vacuum solutions 437
29.1 Twisting vacuum solutions – the field equations 437
29.1.1 The structure of the field equations 437
29.1.2 The integration of the main equations 438
29.1.3 The remaining field equations 440
29.1.4 Coordinate freedom and transformation
properties 441
Contents xv
29.2 Some general classes of solutions 442
29.2.1 Characterization of the known classes of solutions 442
29.2.2 The case ∂
ζ
I = ∂
ζ
(G
2
− ∂
ζ

G) = 0 445
29.2.3 The case ∂
ζ
I = ∂
ζ
(G
2
− ∂
ζ
G) =0,L
,u
= 0 446
29.2.4 The case I = 0 447
29.2.5 The case I =0=L
,u
449
29.2.6 Solutions independent of ζ and
ζ 450
29.3 Solutions of type N (Ψ
2
=0=Ψ
3
) 451
29.4 Solutions of type III (Ψ
2
=0, Ψ
3
= 0) 452
29.5 Solutions of type D (3Ψ
2

Ψ
4
=2Ψ
2
3
, Ψ
2
= 0) 452
29.6 Solutions of type II 454
30 Twisting Einstein–Maxwell and pure radiation
fields 455
30.1 The structure of the Einstein–Maxwell field equations 455
30.2 Determination of the radial dependence of the metric and the
Maxwell field 456
30.3 The remaining field equations 458
30.4 Charged vacuum metrics 459
30.5 A class of radiative Einstein–Maxwell fields (Φ
0
2
= 0) 460
30.6 Remarks concerning solutions of the different Petrov types 461
30.7 Pure radiation fields 463
30.7.1 The field equations 463
30.7.2 Generating pure radiation fields from vacuum by
changing P 464
30.7.3 Generating pure radiation fields from vacuum by
changing m 466
30.7.4 Some special classes of pure radiation fields 467
31 Non-diverging solutions (Kundt’s class) 470
31.1 Introduction 470

31.2 The line element for metrics with Θ + iω = 0 470
31.3 The Ricci tensor components 472
31.4 The structure of the vacuum and Einstein–Maxwell
equation 473
31.5 Vacuum solutions 476
31.5.1 Solutions of types III and N 476
31.5.2 Solutions of types D and II 478
31.6 Einstein–Maxwell null fields and pure radiation fields 480
31.7 Einstein–Maxwell non-null fields 481
31.8 Solutions including a cosmological constant Λ 483
xvi Contents
32 Kerr–Schild metrics 485
32.1 General properties of Kerr–Schild metrics 485
32.1.1 The origin of the Kerr–Schild–Trautman ansatz 485
32.1.2 The Ricci tensor, Riemann tensor and Petrov type 485
32.1.3 Field equations and the energy-momentum tensor 487
32.1.4 A geometrical interpretation of the Kerr–Schild
ansatz 487
32.1.5 The Newman–Penrose formalism for shearfree and
geodesic Kerr–Schild metrics 489
32.2 Kerr–Schild vacuum fields 492
32.2.1 The case ρ = −(Θ + iω) = 0 492
32.2.2 The case ρ = −(Θ + iω)=0 493
32.3 Kerr–Schild Einstein–Maxwell fields 493
32.3.1 The case ρ = −(Θ + iω) = 0 493
32.3.2 The case ρ = −(Θ + iω)=0 495
32.4 Kerr–Schild pure radiation fields 497
32.4.1 The case ρ =0,σ = 0 497
32.4.2 The case σ = 0 499
32.4.3 The case ρ = σ = 0 499

32.5 Generalizations of the Kerr–Schild ansatz 499
32.5.1 General properties and results 499
32.5.2 Non-flat vacuum to vacuum 501
32.5.3 Vacuum to electrovac 502
32.5.4 Perfect fluid to perfect fluid 503
33 Algebraically special perfect fluid solutions 506
33.1 Generalized Robinson–Trautman solutions 506
33.2 Solutions with a geodesic, shearfree, non-expanding multiple
null eigenvector 510
33.3 Type D solutions 512
33.3.1 Solutions with κ = ν = 0 513
33.3.2 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 general
relativity 518
34.1 Methods using harmonic maps (potential space
symmetries) 518
34.1.1 Electrovacuum fields with one Killing vector 518
34.1.2 The group SU(2,1) 521
Contents xvii
34.1.3 Complex invariance transformations 525
34.1.4 Stationary axisymmetric vacuum fields 526
34.2 Prolongation structure for the Ernst equation 529
34.3 The linearized equations, the Kinnersley–Chitre B group and
the Hoenselaers–Kinnersley–Xanthopoulos transformations 532
34.3.1 The field equations 532
34.3.2 Infinitesimal transformations and transformations
preserving Minkowski space 534
34.3.3 The Hoenselaers–Kinnersley–Xanthopoulos transfor-

mation 535
34.4 B¨acklund transformations 538
34.5 The Belinski–Zakharov technique 543
34.6 The Riemann–Hilbert problem 547
34.6.1 Some general remarks 547
34.6.2 The Neugebauer–Meinel rotating disc solution 548
34.7 Other approaches 549
34.8 Einstein–Maxwell fields 550
34.9 The case of two space-like Killing vectors 550
35 Special vector and tensor fields 553
35.1 Space-times that admit constant vector and tensor fields 553
35.1.1 Constant vector fields 553
35.1.2 Constant tensor fields 554
35.2 Complex recurrent, conformally recurrent, recurrent and
symmetric spaces 556
35.2.1 The definitions 556
35.2.2 Space-times of Petrov type D 557
35.2.3 Space-times of type N 557
35.2.4 Space-times of type O 558
35.3 Killing tensors of order two and Killing–Yano tensors 559
35.3.1 The basic definitions 559
35.3.2 First integrals, separability and Killing or Killing–
Yano tensors 560
35.3.3 Theorems on Killing and Killing–Yano tensors in four-
dimensional space-times 561
35.4 Collineations and conformal motions 564
35.4.1 The basic definitions 564
35.4.2 Proper curvature collineations 565
35.4.3 General theorems on conformal motions 565
35.4.4 Non-conformally flat solutions admitting proper

conformal motions 567
xviii Contents
36 Solutions with special subspaces 571
36.1 The basic formulae 571
36.2 Solutions with flat three-dimensional slices 573
36.2.1 Vacuum solutions 573
36.2.2 Perfect fluid and dust solutions 573
36.3 Perfect fluid solutions with conformally flat slices 577
36.4 Solutions with other intrinsic symmetries 579
37 Local isometric embedding of four-dimensional
Riemannian manifolds 580
37.1 The why of embedding 580
37.2 The basic formulae governing embedding 581
37.3 Some theorems on local isometric embedding 583
37.3.1 General theorems 583
37.3.2 Vector and tensor fields and embedding class 584
37.3.3 Groups of motions and embedding class 586
37.4 Exact solutions of embedding class one 587
37.4.1 The Gauss and Codazzi equations and the possible
types of Ω
ab
587
37.4.2 Conformally flat perfect fluid solutions of embedding
class one 588
37.4.3 Type D perfect fluid solutions of embedding class one 591
37.4.4 Pure radiation field solutions of embedding class one 594
37.5 Exact solutions of embedding class two 596
37.5.1 The Gauss–Codazzi–Ricci equations 596
37.5.2 Vacuum solutions of embedding class two 598
37.5.3 Conformally flat solutions 599

37.6 Exact solutions of embedding class p>2 603
Part V: Tables 605
38 The interconnections between the main
classification schemes 605
38.1 Introduction 605
38.2 The connection between Petrov types and groups of motions 606
38.3 Tables 609
References 615
Index 690
Preface
When, in 1975, two of the authors (D.K. and H.S.) proposed to change
their field of research back to the subject of exact solutions of Einstein’s
field equations, they of course felt it necessary to make a careful study
of the papers published in the meantime, so as to avoid duplication of
known results. A fairly comprehensive review or book on the exact solu-
tions would have been a great help, but no such book was available. This
prompted them to ask ‘Why not use the preparatory work we have to
do in any case to write such a book?’ After some discussion, they agreed
to go ahead with this idea, and then they looked for coauthors. They
succeeded in finding two.
The first was E.H., a member of the Jena relativity group, who had been
engaged 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 some
parts of the book.
The quartet’s original optimism somewhat diminished when references
to over 2000 papers had been collected and the magnitude of the task
became all too clear. How could we extract even the most important
information from this mound of literature? How could we avoid constant

rewriting to incorporate new information, which would have made the
job akin to the proverbial painting of the Forth bridge? How could we
decide which topics to include and which to omit? How could we check
the calculations, put the results together in a readable form and still finish
in reasonable time?
We did not feel that we had solved any of these questions in a completely
convincing 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 published
that the first edition can no longer be used as a reliable guide to the
subject. The authors therefore decided to prepare a new edition. Although
they knew from experience the amount of work to be expected, it took
them longer than they thought and feared. We looked at over 4000 new
papers (the cut-off date for the systematic search for papers is the end
of 1999). In particular so much research had been done in the field of
generation techniques and their applications that the original chapter had
to be almost completely replaced, and C.H. was asked to collaborate on
this, 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, thus
reflecting the developments of the last two decades (Chapters 9, 10, 23, 25
and 36), and some of the old chapters have been substantially rewritten.
Unfortunately, the sheer number of known exact solutions has forced us
to give up the idea of presenting them all in some detail; instead, in many
cases 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 related
manuscript, 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 with
groups (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 its
writer could not be held wholly responsible for any errors or omissions.
Since we hope to maintain up-to-date information, we shall be glad to
hear from any reader who detects such errors or omissions; we shall be
pleased to answer as best we can any requests for further information.
M.M. wishes to record that any infelicities remaining in the English arose
because the generally good standard of his colleagues’ English lulled him
into a false sense of security.
This book could not have been written, of course, without the efforts of
the many scientists whose work is recorded here, and especially the many
contemporaries who sent preprints, references and advice or informed us
of 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, of
A. Koutras, who wrote many of the old chapters in LaTeX and simulta-
neously checked many of the solutions, and of the financial support of the
Max-Planck-Group in Jena and the Friedrich-Schiller-Universit¨at 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 obsession
with the book.
Hans Stephani
Jena
Dietrich Kramer
Jena
Malcolm MacCallum
London

Cornelius Hoenselaers
Loughborough
Eduard Herlt
Jena

List of Tables
3.1 Examples of spinor equivalents, defined as in (3.70). 42
4.1 The Petrov types 50
4.2 Normal forms of the Weyl tensor, and Petrov types 51
4.3 The roots of the algebraic equation (4.18) and their mul-
tiplicities. 55
5.1 The algebraic types of the Ricci tensor 59
5.2 Invariance groups of the Ricci tensor types 60
8.1 Enumeration of the Bianchi types 96
8.2 Killing vectors and reciprocal group generators by Bianchi
type 107
9.1 Maximum number of derivatives required to characterize
a 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 H
r
, r>4 165
11.3 Solutions with proper homothety groups H
4
on V
4
166
11.4 Solutions with proper homothety groups on V
3

168
12.1 Homogeneous solutions 181
13.1 The number of essential parameters, by Bianchi type, in
general solutions for vacuum and for perfect fluids with
given equation of state 189
xxiii
xxiv List of Tables
13.2 Subgroups G
3
on V
3
occurring in metrics with multiply-
transitive groups 208
13.3 Solutions given in this book with a maximal G
4
on V
3
208
13.4 Solutions given explicitly in this book with a maximal G
3
on V
3
209
15.1 The vacuum, Einstein–Maxwell and pure radiation solu-
tions with G
3
on S
2
(Y
,a

Y
,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 =(ax
2
+2bx + c)
−5/2
of
solutions 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 ds
2
= x
−1/2
(dx
2
+dy
2
)−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 stationary
Einstein–Maxwell fields, and the corresponding subgroups
of SU(2, 1) 523
34.2 Generation by potential space transformations 530
34.3 Applications of the HKX method 537
34.4 Applications of the Belinski–Zakharov method 546