Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface -- List of Tables -- Notation -- Complex conjugates and constants -- Indices -- Metric and tetrads -- Bivectors -- Derivatives -- Connection and curvature -- Physical fields -- Symmetries -- 1 Introduction -- 1.1 What are exact solutions, and why study them? -- 1.2 The development of the subject -- 1.3 The contents and arrangement of this book -- 1.4 Using this book as a catalogue -- Part I General methods -- 2 Differential geometry without a metric -- 2.1 Introduction -- 2.2 Differentiable manifolds -- 2.3 Tangent vectors -- 2.4 One-forms -- 2.5 Tensors -- 2.6 Exterior products and p-forms -- 2.7 The exterior derivative -- 2.8 The Lie derivative -- 2.9 The covariant derivative -- 2.10 The curvature tensor -- 2.11 Fibre bundles -- 3 Some topics in Riemannian geometry -- 3.1 Introduction -- 3.2 The metric tensor and tetrads -- 3.3 Calculation of curvature from the metric -- 3.4 Bivectors -- 3.5 Decomposition of the curvature tensor -- 3.6 Spinors -- 3.7 Conformal transformations -- 3.8 Discontinuities and junction conditions -- 4 The Petrov classification -- 4.1 The eigenvalue problem -- 4.2 The Petrov types -- 4.3 Principal null directions and determination of the Petrov types -- 5 Classification of the Ricci tensor and the energy-momentum tensor -- 5.1 The algebraic types of the Ricci tensor -- 5.2 The energy-momentum tensor -- 5.3 The energy conditions -- 5.4 The Rainich conditions -- 5.5 Perfect fluids -- 6 Vector fields -- 6.1 Vector fields and their invariant classification -- 6.2 Vector fields and the curvature tensor -- 6.2.1 Timelike unit vector fields -- 6.2.2 Null vector fields -- 7 The Newman-Penrose and related formalisms -- 7.1 The spin coefficients and their transformation laws -- 7.2 The Ricci equations -- 7.3 The Bianchi identities.

7.4 The GHP calculus -- 7.5 Geodesic null congruences -- 7.6 The Goldberg-Sachs theorem and its generalizations -- 8 Continuous groups of transformations -- isometry and homothety groups -- 8.1 Lie groups and Lie algebras -- 8.2 Enumeration of distinct group structures -- 8.3 Transformation groups -- 8.4 Groups of motions -- 8.5 Spaces of constant curvature -- 8.6 Orbits of isometry groups -- 8.6.1 Simply-transitive groups -- 8.6.2 Multiply-transitive groups -- 8.7 Homothety groups -- 9 Invariants and the characterization of geometries -- 9.1 Scalar invariants and covariants -- 9.2 The Cartan equivalence method for space-times -- 9.3 Calculating the Cartan scalars -- 9.3.1 Determination of the Petrov and Segre types -- 9.3.2 The remaining steps -- 9.4 Extensions and applications of the Cartan method -- 9.5 Limits of families of space-times -- 10 Generation techniques -- 10.1 Introduction -- 10.2 Lie symmetries of Einstein's equations -- 10.2.1 Point transformations and their generators -- 10.2.2 How to find the Lie point symmetries of a given differential equation -- 10.2.3 How to use Lie point symmetries: similarity reduction -- 10.3 Symmetries more general than Lie symmetries -- 10.3.1 Contact and Lie-Bäcklund symmetries -- 10.3.2 Generalize and potential symmetries -- 10.4 Prolongation -- 10.4.1 Integral manifolds of differential forms -- 10.4.2 Isovectors, similarity solutions and conservation laws -- 10.4.3 Prolongation structures -- 10.5 Solutions of the linearized equations -- 10.6 Bäcklund transformations -- 10.7 Riemann-Hilbert problems -- 10.8 Harmonic maps -- 10.9 Variational Bäcklund transformations -- 10.10 Hirota's method -- 10.11 Generation methods including perfect fluids -- 10.11.1 Methods using the existence of Killing vectors -- 10.11.2 Conformal transformations -- Part II Solutions with groups of motions.

11 Classification of solutions with isometries or homotheties -- 11.1 The possible space-times with isometries -- 11.2 Isotropy and the curvature tensor -- 11.3 The possible space-times with proper homothetic motions -- 11.4 Summary of solutions with homotheties -- 12 Homogeneous space-times -- 12.1 The possible metrics -- 12.2 Homogeneous vacuum and null Einstein-Maxwell space-times -- 12.3 Homogeneous non-null electromagnetic fields -- 12.4 Homogeneous perfect fluid solutions -- 12.5 Other homogeneous solutions -- 12.6 Summary -- 13 Hypersurface-homogeneous space-times -- 13.1 The possible metrics -- 13.1.1 Metrics with a G6 on V3 -- 13.1.2 Metrics with a G4 on V3 -- Spatial rotation isotropy -- Boost isotropy -- Null rotation isotropy -- 13.1.3 Metrics with a G3 on V3 -- 13.2 Formulations of the field equations -- 13.3 Vacuum, A-term and Einstein-Maxwell solutions -- 13.3.1 Solutions with multiply-transitive groups -- 13.3.2 Vacuum spaces with a G3 on V3 -- 13.3.3 Einstein spaces with a G3 on V3 -- 13.3.4 Einstein-Maxwell solutions with a G3 on V3 -- 13.4 Perfect fluid solutions homogeneous on T3 -- 13.5 Summary of all metrics with Gr on V3 -- 14 Spatially-homogeneous perfect fluid cosmologies -- 14.1 Introduction -- 14.2 Robertson-Walker cosmologies -- 14.3 Cosmologies with a G4 on S3 -- 14.4 Cosmologies with a G3 on S3 -- 15 Groups G3 on non-null orbits V 2- Spherical and plane symmetry -- 15.1 Metric, Killing vectors, and Ricci tensor -- 15.2 Some implications of the existence of an isotropy group I1 -- 15.3 Spherical and plane symmetry -- 15.4 Vacuum, Einstein-Maxwell and pure radiation fields -- 15.4.1 Timelike orbits -- 15.4.2 Spacelike orbits -- 15.4.3 Generalize Birkhoff theorem -- 15.4.4 Spherically- and plane-symmetric fields -- 15.5 Dust solutions -- 15.6 Perfect fluid solutions with plane, spherical or pseudospherical symmetry.

15.6.1 Some basic properties -- 15.6.2 Static solutions -- 15.6.3 Solutions without shear and expansion -- 15.6.4 Expanding solutions without shear -- 15.6.5 Solutions with nonvanishing shear -- Solutions with shear but without acceleration -- 15.7 Plane-symmetric perfect fluid solutions -- 15.7.1 Static solutions -- 15.7.2 Non-static solutions -- 16 Spherically-symmetric perfect fluid solutions -- 16.1 Static solutions -- 16.1.1 Field equations and first integrals -- 16.1.2 Solutions -- 16.2 Non-static solutions -- 16.2.1 The basic equations -- 16.2.2 Expandingsolutions without shear -- Some basic properties -- Known classes of solutions of… -- Examples of solvable classes F(x) -- The Kustaanheimo-Qvist class of solutions -- Solutions with a homogeneous distribution of matter Mu = Mu(t) -- Solutions with an equation of state p = p (Mu) -- 16.2.3 Solutions with non-vanishing shear -- Solutions with shear but without expansion -- Solutions with shear, acceleration and expansion -- 17 Groups G2 and G1 on non-null orbits -- 17.1 Groups G2 on non-null orbits -- 17.1.1 Subdivisions of the groups G2 -- 17.1.2 Groups G2I on non-null orbits -- 17.1.3 G2II on non-null orbits -- 17.2 Boost-rotation-symmetric space-times -- 17.3 Group G1 on non-null orbits -- 18 Stationary gravitational fields -- 18.1 The projection formalism -- 18.2 The Ricci tensor on… -- 18.3 Conformal transformation of… -- 18.4 Vacuum and Einstein-Maxwell equations for stationary fields -- 18.5 Geodesic eigenrays -- 18.6 Static fields -- 18.6.1 Definitions -- 18.6.2 Vacuum solutions -- 18.6.3 Electrostatic and magnetostatic Einstein-Maxwell fields -- 18.6.4 Perfect fluid solutions -- 18.7 The conformastationary solutions -- 18.7.1 Conformastationary vacuum solutions -- 18.7.2 Conformastationary Einstein-Maxwell fields -- 18.8 Multipole moments.

19 Stationary axisymmetric fields: basic concepts and field equations -- 19.1 The Killing vectors -- 19.2 Orthogonal surfaces -- 19.3 The metric and the projection formalism -- 19.4 The field equations for stationary axisymmetric Einstein-Maxwell fields -- 19.5 Various forms of the field equations for stationary axisymmetric vacuum fields -- 19.6 Field equations for rotating fluids -- 20 Stationary axisymmetric vacuum solutions -- 20.1 Introduction -- 20.2 Static axisymmetric vacuum solutions (Weyl's class) -- 20.3 The class of solutions U = U (Omega) (Papapetrou's class) -- 20.4 The class of solutions S = S(A ) -- 20.5 The Kerr solution and the Tomimatsu-Sato class -- 20.6 Other solutions -- 20.7 Solutions with factor structure -- 21 Non-empty stationary axisymmetric solutions -- 21.1 Einstein-Maxwell fields -- 21.1.1 Electrostatic and magnetostatic solutions -- 21.1.2 Type D solutions: A general metric and its limits -- 21.1.3 The Kerr-Newman solution -- 21.1.4 Complexification and the Newman-Janis 'complex trick' -- 21.1.5 Other solutions -- 21.2 Perfect fluid solutions -- 21.2.1 Line element and general properties -- 21.2.2 The general dust solution -- 21.2.3 Rigidly rotating perfect fluid solutions -- 21.2.4 Perfect fluid solutions with differential rotation -- 22 Groups G2I on spacelike orbits: cylindrical symmetry -- 22.1 General remarks -- 22.2 Stationary cylindrically-symmetric fields -- Vacuum solutions -- …term solutions -- Einstein-Maxwell fields -- Dust solutions -- Static perfect fluid solutions -- Stationary perfect fluids -- 22.3 Vacuum fields -- 22.4 Einstein-Maxwell and pure radiation fields -- 23 Inhomogeneous perfect fluid solutions with symmetry -- 23.1 Solutions with a maximal H3 on S3 -- 23.2 Solutions with a maximal H3 on T3 -- 23.3 Solutions with a G2 on S2 -- 23.3.1 Diagonal metrics.

23.3.2 Non-diagonal solutions with orthogonal transitivity.