Preface
About the author
List of symbols
1 Concepts in special relativity
1.1 Galilean relativity
1.2 Inertial frames
1.3 Special relativity
1.4 Velocity addition, length contraction, and time dilation
1.5 Questions
References
2 Tensors in relativity
2.1 Motivation
2.2 General tensors and their basic properties
2.3 Lorentz tensors
2.4 Example: 4-momentum and force
2.5 Example: Doppler effect
2.6 Questions
Reference
3 The equivalence principle and local inertial frames
3.1 Inertial versus gravitational mass
3.2 Einstein's equivalence principle
3.3 Local inertial frames
3.4 Questions
References
4 The motion of freely falling particles in general relativity
4.1 Local inertial frames and the geodesic equation
4.2 The metric tensor
4.3 Gravity as geometry
4.4 The Newtonian limit
4.5 Questions
5 The Schwarzschild metric and black holes
5.1 Spherical symmetry and the Schwarzschild metric
5.2 Geodesics in spherically symmetric spacetimes
5.3 Particle geodesics in a Schwarzschild spacetime
5.4 Deflection of light by the Sun
5.5 Falling into a black hole
5.6 Questions
References
6 Tensors and geometry
6.1 Covariant derivatives
6.2 Basic properties of covariant derivatives
6.3 Riemann and Ricci tensors
6.4 Symmetries and Bianchi identities
6.5 Questions
Reference
7 Einstein's field equations
7.1 Overview
7.2 Energy-momentum tensor and conservation laws
7.2.1 Conservation of electric charge
7.2.2 Conservation of energy-momentum
7.2.3 The energy-momentum tensor
7.3 The field equations for general relativity
7.4 The cosmological constant
7.5 Questions
References
8 Solving the field equations: vacuum solutions
8.1 The vacuum field equations
8.2 The Schwarzschild-de Sitter solution
8.2.1 Vacuum field equations for static spherically symmetric
metrics
8.2.2 Deriving the Schwarzschild-de Sitter metric
8.3 Gravitational waves
8.3.1 Weak-field approximation
8.3.2 Harmonic gauge
8.3.3 Plane waves and polarisation
8.3.4 Detection of gravitational waves
8.4 Questions
References
9 Solving the field equations: cosmological solutions
9.1 The cosmological principle
9.2 The Friedmann-Robertson-Walker metric
9.2.1 Checking homogeneity and isotropy
9.2.2 Galaxies, distances, and the cosmological redshift
9.3 Friedmann-Robertson-Walker universes
9.3.1 A perfect (fluid) world
9.3.2 Local conservation of energy and momentum
9.3.3 Cosmic microwave background
9.3.4 Our accelerating Universe
9.4 Questions
References
Appendices
A Derivation of Lorentz transformations
B Derivation of Einstein's field equations
C Remarks on selected questions
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