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EbookNice Team
Status:
Available4.6
31 reviewsISBN 10: 0521461368
ISBN 13: 978-0521461368
Author: Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt
A completely revised and updated edition of this classic text, covering important new methods and many recently discovered solutions. This edition contains new chapters on generation methods and their application, classification of metrics by invariants, and treatments of homothetic motions and methods from dynamical systems theory. It also includes colliding waves, inhomogeneous cosmological solutions, and spacetimes containing special subspaces.
Notation
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 Rainy conditions
5.5 Perfect fluids
6 Vector fields
6.1 Vector fields and their invariant classification
6.1.1 Timelike unit vector fields
6.1.2 Geodesic null vector fields
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 Generalized 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
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 13
13.5 Summary of all metrics with G, 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 V2. Spherical and plane symmetry
15.1 Metric, Killing vectors, and Ricci tensor
15.2 Some implications of the existence of an isotropy group 11
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 Generalized 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
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 Expanding solutions without shear
16.2.3 Solutions with non-vanishing shear
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 G21 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 23
18.3 Conformal transformation of E3 and the field equations
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-Maxwel 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 axisym-metric 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 (w) (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
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
23.3.3 Solutions without orthogonal transitivity
23.4 Solutions with a G1 or a H2
24 Groups on null orbits. Plane waves
24.1 Introduction
24.2 Groups G3 on N3
24.3 Groups G2 on N2
24.4 Null Killing vectors (G₁ on N₁)
24.4.1 Non-twisting null Killing vector
24.4.2 Twisting null Killing vector
24.5 The plane-fronted gravitational waves with parallel rays (pp-waves)
25 Collision of plane waves
25.1 General features of the collision problem
25.2 The vacuum field equations
25.3 Vacuum solutions with collinear polarization
25.4 Vacuum solutions with non-collinear polarization
25.5 Einstein-Maxwell fields
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Tags: Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt, Exact solutionsEinstein s field
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