Cover
Annotation
Title
Copyright
Contents
The scope of this text
Preface to the second edition
Acknowledgements
1 How the theory of relativity came into being (a brief historical sketch)
1.1 Special versus general relativity
1.2 Space and inertia in Newtonian physics
1.3 Newton’s theory and the orbits of planets
1.4 The basic assumptions of general relativity
Part I Elements of differential geometry
2 A short sketch of 2-dimensional differential geometry
2.1 Constructing parallel straight lines in a flat space
2.2 Generalisation of the notion of parallelism to curved surfaces
3 Tensors, tensor densities
3.1 What are tensors good for?
3.2 Differentiable manifolds
3.3 Scalars
3.4 Contravariant vectors
3.5 Covariant vectors
3.6 Tensors of second rank
3.7 Tensor densities
3.8 Tensor densities of arbitrary rank
3.9 Algebraic properties of tensor densities
3.10 Mappings between manifolds
3.11 The Levi-Civita symbol
3.12 Multidimensional Kronecker deltas
3.13 Examples of applications of the Levi-Civita symbol and of the multidimensional Kronecker delta
3.14 Exercises
4 Covariant derivatives
4.1 Differentiation of tensors
4.2 Axioms of the covariant derivative
4.3 A field of bases on a manifold and scalar components of tensors
4.4 The affine connection
4.5 The explicit formula for the covariant derivative of tensor density fields
4.6 Exercises
5 Parallel transport and geodesic lines
5.1 Parallel transport
5.2 Geodesic lines
5.3 Exercises
6 The curvature of a manifold; flat manifolds
6.1 The commutator of second covariant derivatives
6.2 The commutator of directional covariant derivatives
6.3 The relation between curvature and parallel transport
6.4 Covariantly constant fields of vector bases
6.5 A torsion-free flat manifold
6.6 Parallel transport in a flat manifold
6.7 Geodesic deviation
6.8 Algebraic and differential identities obeyed by the curvature tensor
6.9 Exercises
7 Riemannian geometry
7.1 The metric tensor
7.2 Riemann spaces
7.3 The signature of a metric, degenerate metrics
7.4 Christoffel symbols
7.5 The curvature of a Riemann space
7.6 Flat Riemann spaces
7.7 Subspaces of a Riemann space
7.8 Flat Riemann spaces that are globally non-Euclidean
7.9 The Riemann curvature versus the normal curvature of a surface
7.10 The geodesic line as the line of extremal distance
7.11 Mappings between Riemann spaces
7.12 Conformally related Riemann spaces
7.13 Conformal curvature
7.14 Timelike, null and spacelike intervals in a 4-dimensional spacetime
7.15 Embeddings of Riemann spaces in Riemann spaces of higher dimension
7.16 The Petrov classification
7.17 Exercises
8 Symmetries of Riemann spaces, invariance of tensors
8.1 Symmetry transformations
8.2 The Killing equations
8.3 The connection between generators and the invariance transformations
8.4 Finding the Killing vector fields
8.5 A Killing vector field along a geodesic is a geodesic deviation field
8.6 Invariance of other tensor fields
8.7 The Lie derivative
8.8 The algebra of Killing vector fields
8.9 Surface-forming vector fields
8.10 Spherically symmetric 4-dimensional Riemann spaces
8.11 * Conformal Killing fields and their finite basis
8.12 * The maximal dimension of an invariance group
8.13 Exercises
9 Methods to calculate the curvature quickly: differential forms and algebraic computer programs
9.1 The basis of differential forms
9.2 The connection forms
9.3 The Riemann tensor
9.4 Using computers to calculate the curvature
9.5 Exercises
10 The spatially homogeneous Bianchi-type spacetimes
10.1 The Bianchi classification of 3-dimensional Lie algebras
10.2 The dimension of the group versus the dimension of the orbit
10.3 Action of a group on a manifold
10.4 Groups acting transitively, homogeneous spaces
10.5 Invariant vector fields
10.6 The metrics of the Bianchi-type spacetimes
10.7 The isotropic Bianchi-type (Robertson–Walker) spacetimes
10.8 Exercises
11 * The Petrov classification by the spinor method
11.1 What is a spinor?
11.2 Translating spinors to tensors and vice versa
11.3 The spinor image of the Weyl tensor
11.4 The Petrov classification in the spinor representation
11.5 The Weyl spinor represented as a 3 × 3 complex matrix
11.6 The equivalence of the Penrose classes to the Petrov classes
11.7 The Petrov classification by the Debever method
11.8 Exercises
Part II The theory of gravitation
12 The Einstein equations and the sources of a gravitational field
12.1 Why Riemannian geometry?
12.2 Local inertial frames
12.3 Trajectories of free motion in Einstein’s theory
12.4 Special relativity versus gravitation theory
12.5 The Newtonian limit of general relativity
12.6 Sources of the gravitational field
12.7 The Einstein equations
12.8 Hilbert’s derivation of the Einstein equations
12.9 The Palatini variational principle
12.10 The asymptotically Cartesian coordinates and the asymptotically flat spacetime
12.11 The Newtonian limit of Einstein’s equations
12.12 Examples of sources in the Einstein equations: perfect fluid and dust
12.13 Equations of motion of a perfect fluid
12.14 The cosmological constant
12.15 An example of an exact solution of Einstein’s equations: a Bianchi type I spacetime with dust source
12.16 * Other gravitation theories
12.16.1 The Brans–Dicke theory
12.16.2 The Bergmann–Wagoner theory
12.16.3 The Einstein–Cartan theory
12.16.4 The bi-metric Rosen theory
12.17 Matching solutions of Einstein’s equations
12.18 The weak-field approximation to general relativity
12.19 Exercises
13 The Maxwell and Einstein–Maxwell equations and the Kaluza–Klein theory
13.1 The Lorentz-covariant description of electromagnetic field
13.2 The covariant form of the Maxwell equations
13.3 The energy-momentum tensor of electromagnetic field
13.4 The Einstein–Maxwell equations
13.5 * The variational principle for the Maxwell and Einstein–Maxwell equations
13.6 * The Kaluza–Klein theory
13.7 Exercises
14 Spherically symmetric gravitational fields of isolated objects
14.1 The curvature coordinates
14.2 Symmetry inheritance
14.3 Spherically symmetric electromagnetic field in vacuum
14.4 The Schwarzschild and Reissner–Nordstr¨om solutions
14.5 Orbits of planets in the gravitational field of the Sun
14.6 Deflection of light rays in the Schwarzschild field
14.7 Measuring the deflection of light rays
14.8 Gravitational lenses
14.9 The spurious singularity of the Schwarzschild solution at r = 2m
14.10 * Embedding the Schwarzschild spacetime in a flat Riemannian space
14.11 Interpretation of the spurious singularity at r = 2m; black holes
14.12 The Schwarzschild metric in other coordinate systems
14.13 The equation of hydrostatic equilibrium
14.14 The ‘interior Schwarzschild solution’
14.15 * The maximal analytic extension of the Reissner–Nordstr¨om metric
14.16 Motion of particles in the Reissner–Nordstr¨om spacetime with e2 < m2
14.17 Exercises
15 Relativistic hydrodynamics and thermodynamics
15.1 Motion of a continuous medium in Newtonian hydrodynamics
15.2 Motion of a continuous medium in relativistic hydrodynamics
15.3 The equations of evolution of θ, σαβ, ωαβ and u˙α; the Raychaudhuri equation
15.4 Singularities and singularity theorems
15.5 Relativistic thermodynamics
15.6 Exercises
16 Relativistic cosmology I: general geometry
16.1 A continuous medium as a model of the Universe
16.2 The geometric optics approximation
16.3 The redshift
16.4 The optical tensors
16.5 The apparent horizon
16.6 * The double-null tetrad
16.7 * The equations of propagation of the optical scalars
16.8 * The Goldberg–Sachs theorem
16.9 * The area distance
16.10 * The reciprocity theorem
16.11 Other observable quantities
16.12 The Fermi–Walker transport
16.13 Position drift of light sources
16.14 Exercises
17 Relativistic cosmology II: the Robertson–Walker geometry
17.1 The Robertson–Walker metrics as models of the Universe
17.2 Optical observations in an R–W universe
17.2.1 The redshift
17.2.2 The redshift–distance relation
17.2.3 Number counts
17.3 The Friedmann equation
17.4 The Friedmann solutions with Λ = 0
17.5 The redshift–distance relation in the Λ = 0 Friedmann models
17.6 The Newtonian cosmology
17.7 The Friedmann solutions with the cosmological constant
17.8 The ΛCDM model
17.9 The redshift–distance relation in the Λ ̸= 0 Friedmann models
17.10 The redshift drift: a test for accelerating expansion
17.11 Horizons in the Robertson–Walker models
17.12 The inflationary models and the ‘problems’ they solved
17.13 The value of the cosmological constant
17.14 The ‘history of the Universe’
17.15 Invariant definitions of the Robertson–Walker models
17.16 Different representations of the R–W metrics
17.17 Exercises
18 Relativistic cosmology III: the Lemaˆıtre–Tolman geometry
18.1 The comoving-synchronous coordinates
18.2 The spherically symmetric inhomogeneous models
18.3 The Lemaˆıtre–Tolman model
18.4 Conditions of regularity at the centre
18.5 Formation of voids in the Universe
18.6 Formation of other structures in the Universe
18.6.1 Density to density evolution
18.6.2 Velocity to density evolution
18.6.3 Velocity to velocity evolution
18.7 The influence of cosmic expansion on planetary orbits
18.8 * The apparent horizons for a central observer in L–T models
18.9 * Black holes in the evolving Universe
18.10 * Shell crossings and necks/wormholes
18.10.1 E < 0
18.10.2 E = 0
18.10.3 E > 0
18.10.4 Final comment
18.11 The redshift along radial rays
18.12 The blueshift
18.13 * Apparent horizons for noncentral observers
18.14 The influence of inhomogeneities in matter distribution on the cosmic microwave background radiation
18.15 Matching the L–T models to the Schwarzschild and Friedmann solutions
18.16 * The shell focussing singularity
18.17 * Extending an L–T spacetime through a shell crossing singularity
18.18 * Singularities and cosmic censorship
18.19 Solving the ‘horizon problem’ without inflation
18.20 * The evolution of R(t,M) versus the evolution of ρ(t,M)
18.21 * Increasing and decreasing density perturbations
18.22 Mimicking accelerating expansion of the Universe by inhomogeneities in matter distribution
18.23 Drift of light rays
18.24 * L&T curio shop
18.24.1 Lagging cores of the Big Bang
18.24.2 Strange or nonintuitive properties of the L–T model
18.24.3 Chances to fit an L–T model to observations
18.24.4 An ‘in one ear and out the other’ Universe
18.24.5 A ‘string of beads’ Universe
18.24.6 Uncertainties in inferring the spatial distribution of matter
18.24.7 Is the distribution of matter in our Universe fractal?
18.24.8 General results related to the L–T models
18.25 Exercises
19 Relativistic cosmology IV: simple generalisations of L–T and related geometries
19.1 The plane- and hyperbolically symmetric spacetimes
19.2 G3/S2-symmetric dust solutions with R,r ̸= 0
19.3 Plane symmetric dust solutions with R,r ̸= 0
19.4 G3/S2-symmetric dust in electromagnetic field, the case R,r ̸= 0
19.4.1 Integrals of the field equations
19.4.2 Matching the charged dust metric to the Reissner–Nordstr¨om metric
19.4.3 Prevention of the Big Crunch singularity by electric charge
19.4.4 * Charged dust in curvature and mass-curvature coordinates
19.4.5 Regularity conditions at the centre
19.4.6 * Shell crossings in charged dust
19.5 The Datt–Ruban solution
19.6 Exercises
20 Relativistic cosmology V: the Szekeres geometries
20.1 The Szekeres–Szafron family of metrics
20.1.1 The β,z = 0 subfamily
20.1.2 The β,z ̸= 0 subfamily
20.1.3 Interpretation of the Szekeres–Szafron coordinates
20.1.4 Common properties of the two subfamilies
20.1.5 * The invariant definitions of the Szekeres–Szafron metrics
20.2 The Szekeres solutions and their properties
20.2.1 The β,z = 0 subfamily
20.2.2 The β,z ̸= 0 subfamily
20.2.3 * The β,z = 0 family as a limit of the β,z ̸= 0 family
20.3 Properties of the quasi-spherical Szekeres solutions with β,z ̸= 0 = Λ
20.3.1 Basic physical restrictions
20.3.2 The significance of ℰ
20.3.3 Conditions of regularity at the origin
20.3.4 Shell crossings
20.3.5 Regular maxima and minima
20.3.6 The mass dipole
20.3.7 * The absolute apparent horizon
20.3.8 * The apparent horizon and its relation to the AAH
20.3.9 * Which is the true horizon – the AH or the AAH?
20.4 * The Goode–Wainwright representation of the Szekeres solutions
20.5 Selected interesting subcases of the Szekeres–Szafron family
20.5.1 The Szafron–Wainwright model
20.5.2 The toroidal universe of Senin
20.6 Selected further reading on the Szekeres models
20.7 Exercises
21 The Kerr metric
21.1 The Kerr–Schild metrics
21.2 The derivation of the Kerr metric by the original method
21.3 Basic properties
21.4 * Derivation of the Kerr metric by Carter’s method – from the separability of the Klein–Gordon equation
21.5 The event horizons and the stationary limit hypersurfaces
21.6 The Hamiltonian and the Poisson bracket
21.7 General geodesics
21.8 Geodesics in the equatorial plane
21.9 *The maximal analytic extension of the Kerr metric
21.10 * The Penrose process
21.11 Stationary–axisymmetric spacetimes and locally nonrotating observers
21.12 * Ellipsoidal spacetimes
21.13 A Newtonian analogue of the Kerr solution
21.14 A source of the Kerr field?
21.15 Exercises
22 Relativity enters technology: the Global Positioning System
22.1 Purpose and setup
22.2 The principle of position determination
22.3 The reference frames and the Sagnac effect
22.4 Earth’s gravitation and the SI time units
22.5 Selected corrections of the orbits of the GPS satellites
22.5.1 Corrections for gravity and velocity
22.5.2 The eccentricity correction
22.6 The 9 largest relativistic effects in the GPS
22.7 Exercises
23 Subjects omitted from this book
24 Comments to selected exercises and calculations
24.1 Exercise 1 to Chapter 14
24.2 Exercise 14 to Chapter 14
24.3 Verifying Eqs. (19.35) with (19.31) and (19.32) with (19.28)
24.4 Verifying the Einstein equations (20.2), (20.9) and (20.11)
24.5 Equation (20.179) defines η at the AAH uniquely
24.6 The four curves in Fig. 20.4 meet at one point
24.7 The discarded case in Eqs. (20.2)–(20.11)
24.8 Hints for verifying Eq. (21.28)
References
Index