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• Half-title
• Title
• Copyright
• Contents
• Preface
• Notation
• I. Special Relativity
• 1 Introduction: Inertial systems and the Galilei invariance of Classical Mechanics
• 1.1 Inertial systems
• 1.2 Invariance under translations
• 1.3 Invariance under rotations
• 1.4 Invariance under Galilei transformations
• 1.5 Some remarks on the homogeneity of time
• Exercises
• 2 Light propagation in moving coordinate systems and Lorentz transformations
• 2.1 The Michelson experiment
• 2.2 The Lorentz transformations
• 2.3 Some properties of Lorentz transformations
• Exercises
• 3 Our world as a Minkowski space
• 3.1 The concept of Minkowski space
• 3.2 Four-vectors and light cones
• 3.3 Measuring length and time in Minkowski space
• 3.4 Two thought experiments
• 3.4.1 A rod moving through a tube
• 3.4.2 The twin paradox
• 3.5 Causality, and velocities larger than that of light
• Exercises
• 4 Mechanics of Special Relativity
• 4.1 Kinematics
• 4.2 Equations of motion
• 4.3 Hyperbolic motion
• 4.4 Systems of particles
• Exercises
• 5 Optics of plane waves
• 5.1 Invariance of phase and null vectors
• 5.2 The Doppler effect – shift in the frequency of a wave
• 5.3 Aberration – change in the direction of a light ray
• 5.4 The visual shape of moving bodies
• 5.5 Reflection at a moving mirror
• 5.6 Dragging of light within a fluid
• Exercises
• 6 Four-dimensional vectors and tensors
• 6.1 Some definitions
• 6.2 Tensor algebra
• 6.3 Symmetries of tensors
• 6.4 Algebraic properties of second rank tensors
• 6.5 Tensor analysis
• Exercises
• 7 Electrodynamics in vacuo
• 7.1 The Maxwell equations in three-dimensional notation
• 7.2 Current four-vector, four-potential, and the retarded potentials
• 7.3 Field tensor and the Maxwell equations
• 7.4 Poynting’s theorem, Lorentz force, and the energy-momentum tensor
• 7.5 The variational principle for the Maxwell equations
• Exercises
• 8 Transformation properties of electromagnetic fields: examples
• 8.1 Current and four-potential
• 8.2 Field tensor and energy-momentum tensor
• Exercises
• 9 Null vectors and the algebraic properties of electromagnetic field tensors
• 9.1 Null tetrads and Lorentz transformations
• 9.2 Self-dual bivectors and the electromagnetic field tensor
• 9.3 The algebraic classification of electromagnetic fields
• 9.4 The physical interpretation of electromagnetic null fields
• Exercises
• 10 Charged point particles and their field
• 10.1 The equations of motion of charged test particles
• 10.2 The variational principle for charged particles
• 10.3 Canonical equations
• 10.4 The field of a charged particle in arbitrary motion
• 10.5 The equations of motion of charged particles – the self-force
• Exercises
• Further reading for Chapter 10
• 11 Pole-dipole particles and their field
• 11.1 The current density
• 11.2 The dipole term and its field
• 11.3 The force exerted on moving dipoles
• Exercises
• 12 Electrodynamics in media
• 12.1 Field equations and constitutive relations
• 12.2 Remarks on the matching conditions at moving surfaces
• 12.3 The energy-momentum tensor
• Exercises
• 13 Perfect fluids and other physical theories
• 13.1 Perfect fluids
• 13.2 Other physical theories – an outlook
• II. Riemannian geometry
• 14 Introduction: the force-free motion of particles in Newtonian mechanics
• 14.1 Coordinate systems
• 14.2 Equations of motion
• 14.3 The geodesic equation
• 14.4 Geodesic deviation
• Exercise
• 15 Why Riemannian geometry?
• 16 Riemannian space
• 16.1 The metric
• 16.2 Geodesics and Christoffel symbols
• 16.3 Coordinate transformations
• 16.4 Special coordinate systems
• 16.5 The physical meaning and interpretation of coordinate systems
• Exercises
• Further reading for Chapter 16
• 17 Tensor algebra
• 17.1 Scalars and vectors
• 17.2 Tensors and other geometrical objects
• 17.3 Algebraic operations with tensors
• 17.4 Tetrad and spinor components of tensors
• Exercises
• Further reading for Section 17.4
• 18 The covariant derivative and parallel transport
• 18.1 Partial and covariant derivatives
• 18.2 The covariant differential and local parallelism
• 18.3 Parallel displacement along a curve and the parallel propagator
• 18.4 Fermi–Walker transport
• 18.5 The Lie derivative
• Exercises
• Further reading for Chapter 18
• 19 The curvature tensor
• 19.1 Intrinsic geometry and curvature
• 19.2 The curvature tensor and global parallelism of vectors
• 19.3 The curvature tensor and second derivatives of the metric tensor
• 19.4 Properties of the curvature tensor
• 19.5 Spaces of constant curvature
• Exercises
• Further reading for Chapter 19
• 20 Differential operators, integrals and integral laws
• 20.1 The problem
• 20.2 Some important differential operators
• 20.3 Volume, surface and line integrals
• 20.4 Integral laws
• 20.5 Integral conservation laws
• Further reading for Chapter 20
• 21 Fundamental laws of physics in Riemannian spaces
• 21.1 How does one find the fundamental physical laws?
• 21.2 Particle mechanics
• 21.3 Electrodynamics in vacuo
• 21.4 Geometrical optics
• 21.5 Thermodynamics
• 21.6 Perfect fluids and dust
• 21.7 Other fundamental physical laws
• Exercises
• Further reading for Chapter 21
• III. Foundations of Einstein’s theory of gravitation
• 22 The fundamental equations of Einstein’s theory of gravitation
• 22.1 The Einstein field equations
• 22.2 The Newtonian limit
• 22.3 The equations of motion of test particles
• Further reading for Section 22.3
• 22.4 A variational principle for Einstein’s theory
• 23 The Schwarzschild solution
• 23.1 The field equations
• 23.2 The solution of the vacuum field equations
• 23.3 General discussion of the Schwarzschild solution
• 23.4 The motion of the planets and perihelion precession
• 23.5 The propagation of light in the Schwarzschild field
• 23.6 Further aspects of the Schwarzschild solution
• 23.7 The Reissner–Nordström solution
• Exercises
• 24 Experiments to verify the Schwarzschild metric
• 24.1 Some general remarks
• 24.2 Perihelion precession and planetary orbits
• 24.3 Light deflection by the Sun
• 24.4 Redshifts
• 24.5 Measurements of the travel time of radar signals (time delay)
• 24.6 Geodesic precession of a top
• Further reading for Chapter 24
• 25 Gravitational lenses
• 25.1 The spherically symmetric gravitational lens
• 25.2 Galaxies as gravitational lenses
• Exercise
• 26 The interior Schwarzschild solution
• 26.1 The field equations
• 26.2 The solution of the field equations
• 26.3 Matching conditions and connection to the exterior Schwarzschild solution
• 26.4 A discussion of the interior Schwarzschild solution
• Exercises
• IV. Linearized theory of gravitation, far fields and gravitational waves
• 27 The linearized Einstein theory of gravity
• 27.1 Justification for a linearized theory and its realm of validity
• 27.2 The fundamental equations of the linearized theory
• 27.3 A discussion of the fundamental equations and a comparison with special-relativistic electrodyn
• 27.4 The far field due to a time-dependent source
• 27.5 Discussion of the properties of the far field (linearized theory)
• 27.6 Some remarks on approximation schemes
• Further reading for Section 27.6
• Exercise
• 28 Far fields due to arbitrary matter distributions and balance equations for momentum and angular m
• 28.1 What are far fields?
• 28.2 The energy-momentum pseudotensor for the gravitational field
• 28.3 The balance equations for momentum and angular momentum
• 28.4 Is there an energy law for the gravitational field?
• Further reading for Chapter 28
• 29 Gravitational waves
• 29.1 Are there gravitational waves?
• 29.2 Plane gravitational waves in the linearized theory
• 29.3 Plane waves as exact solutions of Einstein’s equations
• 29.4 The experimental evidence for gravitational waves
• Exercises
• 30 The Cauchy problem for the Einstein field equations
• 30.1 The problem
• 30.2 Three-dimensional hypersurfaces and reduction formulae for the curvature tensor
• 30.3 The Cauchy problem for the vacuum field equations
• 30.4 The characteristic initial value problem
• 30.5 Matching conditions at the boundary surface of two metrics
• V. Invariant characterization of exact solutions
• 31 Preferred vector fields and their properties
• 31.1 Special simple vector fields
• 31.2 Timelike vector fields
• 31.3 Null vector fields
• Exercises
• 32 The Petrov classification
• 32.1 What is the Petrov classification?
• 32.2 The algebraic classification of gravitational fields
• 32.3 The physical interpretation of degenerate vacuum gravitational fields
• Exercises
• 33 Killing vectors and groups of motion
• 33.1 The problem
• 33.2 Killing vectors
• 33.3 Killing vectors of some simple spaces
• 33.4 Relations between the curvature tensor and Killing vectors
• 33.5 Groups of motion
• 33.6 Killing vectors and conservation laws
• Exercises
• 34 A survey of some selected classes of exact solutions
• 34.1 Degenerate vacuum solutions
• 34.2 Vacuum solutions with special symmetry properties
• 34.3 Perfect fluid solutions with special symmetry properties
• Exercises
• VI. Gravitational collapse and black holes
• 35 The Schwarzschild singularity
• 35.1 How does one examine the singular points of a metric?
• 35.2 Radial geodesics near r = 2M
• 35.3 The Schwarzschild solution in other coordinate systems
• 35.4 The Schwarzschild solution as a black hole
• Exercises
• 36 Gravitational collapse – the possible life history of a spherically symmetric star
• 36.1 The evolutionary phases of a spherically symmetric star
• 36.2 The critical mass of a star
• 36.3 Gravitational collapse of spherically symmetric dust
• Further reading for Chapter 36
• 37 Rotating black holes
• 37.1 The Kerr solution
• 37.2 Gravitational collapse – the possible life history of a rotating star
• 37.3 Some properties of black holes
• 37.4 Are there black holes?
• Further reading for Chapter 37
• 38 Black holes are not black – Relativity Theory and Quantum Theory Theory and Quantum Theory
• 38.1 The problem
• 38.2 Unifled quantum field theory and quantization of the gravitational field
• 38.3 Semiclassical gravity
• 38.4 Quantization in a given classical gravitational field
• 38.5 Black holes are not black – the thermodynamics of black holes
• Further reading for Chapter 38
• 39 The conformal structure of infinity
• 39.1 The problem and methods to answer it
• 39.2 Infinity of the three-dimensional Euclidean space (E3)
• 39.3 The conformal structure of Minkowski space
• 39.4 Asymptotically flat gravitational fields
• 39.5 Examples of Penrose diagrams
• Exercises
• VII. Cosmology
• 40 Robertson–Walker metrics and their properties
• 40.1 The cosmological principle and Robertson–Walker metrics
• 40.2 The motion of particles and photons
• 40.3 Distance definitions and horizons
• 40.4 Some remarks on physics in closed universes
• Exercises
• 41 The dynamics of Robertson–Walker metrics and the Friedmann universes
• 41.1 The Einstein field equations for Robertson–Walker metrics
• 41.2 The most important Friedmann universes
• 41.3 Consequences of the field equations for models with arbitrary equation of state having positive
• Exercises
• 42 Our universe as a Friedmann model
• 42.1 Redshift and mass density
• 42.2 The earliest epochs of our universe and the cosmic background radiation
• 42.3 A Schwarzschild cavity in the Friedmann universe
• 43 General cosmological models
• 43.1 What is a cosmological model?
• 43.2 Solutions of Bianchi type I with dust
• 43.3 The Gödel universe
• 43.4 Singularity theorems
• Exercises
• Further reading for Chapter 43
• Bibliography
• Alternative textbooks on relativity and useful review volumes
• Monographs and research articles
• Index