An Introduction to General Relativity and Cosmology

Cover
About the authors
Half title
Imprint
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–Nordström 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–Nordström metric
	14.16 Motion of particles in the Reissner–Nordström 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 u̇α ; 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 6= 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 Lemaı̂tre–Tolman geometry
	18.1 The comoving-synchronous coordinates
	18.2 The spherically symmetric inhomogeneous models
	18.3 The Lemaı̂tre–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 6= 0
	19.3 Plane symmetric dust solutions with R,r 6= 0
	19.4 G3 /S2 -symmetric dust in electromagnetic field, the case R,r 6= 0
	19.4.1 Integrals of the field equations
	19.4.2 Matching the charged dust metric to the Reissner–Nordström 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 6= 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 6= 0 subfamily
		20.2.3 * The β,z = 0 family as a limit of the β,z 6= 0 family
	20.3 Properties of the quasi-spherical Szekeres solutions with β,z 6= 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