A First Course in General Relativity

Cover
Half-title
Title
Copyright
Contents
Dedication
Preface to the second edition
Preface to the first edition
1 Special relativity
1.1 Fundamental principles of special relativity (SR) theory
1.2 Definition of an inertial observer in SR
1.3 New units
1.4 Spacetime diagrams
1.5 Construction of the coordinates used by another observer
1.6 Invariance of the interval
1.7 Invariant hyperbolae
1.8 Particularly important results
Time dilation
Lorentz contraction
Conventions
Failure of relativity?
1.9 The Lorentz transformation
1.10 The velocity-composition law
1.11 Paradoxes and physical intuition
1.12 Further reading
1.13 Appendix: The twin ‘paradox’ dissected
The problem
Brief solution
Fuller discussion
1.14 Exercises
2 Vector analysis in special relativity
2.1 Definition of a vector
2.2 Vector algebra
Basis vectors
Transformation of basis vectors
Inverse transformations
2.3 The four-velocity
2.4 The four-momentum
Conservation of four-momentum
Center of momentum (CM) frame
2.5 Scalar product
Magnitude of a vector
Scalar product of two vectors
2.6 Applications
Four-velocity and acceleration as derivatives
Energy and momentum
2.7 Photons
No four-velocity
Four-momentum
Zero rest-mass particles
2.8 Further reading
2.9 Exercises
3 Tensor analysis in special relativity
3.1 The metric tensor
3.2 Definition of tensors
Aside on the usage of the term ‘function’
Components of a tensor
3.3 The (0 1) tensors: one-forms
General properties
Basis one-forms
Picture of a one-form
Gradient of a function is a one-form
Notation for derivatives
Normal one-forms
3.4 The (0 2) tensors
Components
Symmetries
3.5 Metric as a mapping of vectors into one-forms
The inverse: going from…
Why distinguish one-forms from vectors?
Magnitudes and scalar products of one-forms
Normal vectors and unit normal one-forms
3.6 Finally: (M N) tensors
Vector as a function of one-forms
(M 0) tensors
(M N) tensors
Circular reasoning?
3.7 Index ‘raising’ and ‘lowering’
Mixed components of metric
Metric and nonmetric vector algebras
3.8 Differentiation of tensors
3.9 Further reading
3.10 Exercises
4 Perfect fluids in special relativity
4.1 Fluids
4.2 Dust: the number–flux vector…
The number density n
The flux across a surface
The number–flux four-vector…
4.3 One-forms and surfaces
Number density as a timelike flux
A one-form defines a surface
The flux across the surface
Representation of a frame by a one-form
4.4 Dust again: the stress–energy tensor
Energy density
Stress–energy tensor
4.5 General fluids
Definition of macroscopic quantities
First law of thermodynamics
The general stress–energy tensor
The spatial components of T, T ij
Symmetry of Talphabeta in MCRF
Conservation of energy–momentum
Conservation of particles
4.6 Perfect fluids
No heat conduction
No viscosity
Form of T
Aside on the meaning of pressure
The conservation laws
4.7 Importance for general relativity
4.8 Gauss’ law
4.9 Further reading
4.10 Exercises
5 Preface to curvature
5.1 On the relation of gravitation to curvature
The gravitational redshift experiment
Nonexistence of a Lorentz frame at rest on Earth
The principle of equivalence
The redshift experiment again
Local inertial frames
Tidal forces
The role of curvature
5.2 Tensor algebra in polar coordinates
Vectors and one-forms
Curves and vectors
Polar coordinate basis one-forms and vectors
Metric tensor
5.3 Tensor calculus in polar coordinates
Derivatives of basis vectors
Derivatives of general vectors
The Christoffel symbols
The covariant derivative
Divergence and Laplacian
Derivatives of one-forms and tensors of higher types
5.4 Christoffel symbols and the metric
Calculating the Christoffel symbols from the metric
The tensorial nature of…
5.5 Noncoordinate bases
Polar coordinate basis
Polar unit basis
General remarks on noncoordinate bases
Noncoordinate bases in this book
5.6 Looking ahead
5.7 Further reading
5.8 Exercises
6 Curved manifolds
6.1 Differentiable manifolds and tensors
Manifolds
Differential structure
6.2 Riemannian manifolds
The metric and local flatness
Lengths and volumes
Proof of the local-flatness theorem
6.3 Covariant differentiation
Divergence formula
6.4 Parallel-transport, geodesics, and curvature
Parallel-transport
Geodesics
6.5 The curvature tensor
Geodesic deviation
6.6 Bianchi identities: Ricci and Einstein tensors
The Ricci tensor
The Einstein tensor
6.7 Curvature in perspective
6.8 Further reading
6.9 Exercises
7 Physics in a curved spacetime
7.1 The transition from differential geometry to gravity
7.2 Physics in slightly curved spacetimes
7.3 Curved intuition
7.4 Conserved quantities
7.5 Further reading
7.6 Exercises
8 The Einstein field equations
8.1 Purpose and justification of the field equations
Geometrized units
8.2 Einstein’s equations
8.3 Einstein’s equations for weak gravitational fields
Nearly Lorentz coordinate systems
Background Lorentz transformations
Gauge transformations
Riemann tensor
Weak-field Einstein equations
8.4 Newtonian gravitational fields
Newtonian limit
The far field of stationary relativistic sources
Definition of the mass of a relativistic body
8.5 Further reading
8.6 Exercises
9 Gravitational radiation
9.1 The propagation of gravitational waves
The transverse-traceless gauge
The effect of waves on free particles
Tidal accelerations: gravitational wave forces
Measuring the stretching of space
Polarization of gravitational waves
An exact plane wave
Geometrical optics: waves in a curved spacetime
9.2 The detection of gravitational waves
General considerations
A resonant detector
Bar detectors in operation
Measuring distances with light
Beam detectors
Interferometer observations
9.3 The generation of gravitational waves
Simple estimates
Slow motion wave generation
Order-of-magnitude estimates
Exact solution of the wave equation
9.4 The energy carried away by gravitational waves
Preview
The energy flux of a gravitational wave
Energy lost by a radiating system
An example. The Hulse–Taylor binary pulsar
9.5 Astrophysical sources of gravitational waves
Overview
Binary systems
Spinning neutron stars
Gravitational collapse
Gravitational waves from the Big Bang
9.6 Further reading
9.7 Exercises
10 Spherical solutions for stars
10.1 Coordinates for spherically symmetric spacetimes
Flat space in spherical coordinates
Two-spheres in a curved spacetime
Meshing the two-spheres into a three-space for t = const
Spherically symmetric spacetime
10.2 Static spherically symmetric spacetimes
The metric
Physical interpretation of metric terms
The Einstein tensor
10.3 Static perfect fluid Einstein equations
Stress–energy tensor
Equation of state
Equations of motion
Einstein equations
10.4 The exterior geometry
Schwarzschild metric
Generality of the metric
10.5 The interior structure of the star
General rules for integrating the equations
The structure of Newtonian stars
10.6 Exact interior solutions
The Schwarzschild constant-density interior solution
Buchdahl’s interior solution
10.7 Realistic stars and gravitational collapse
Buchdahl’s theorem
Formation of stellar-mass black holes
Quantum mechanical pressure
White dwarfs
Neutron stars
10.8 Further reading
10.9 Exercises
11 Schwarzschild geometry and black holes
11.1 Trajectories in the Schwarzschild spacetime
Black holes in Newtonian gravity
Conserved quantities
Types of orbits
Perihelion shift
Binary pulsars
Post-Newtonian gravity
Gravitational deflection of light
Gravitational lensing
11.2 Nature of the surface r = 2M
Coordinate singularities
Infalling particles
Inside r = 2M
Coordinate systems
Kruskal–Szekeres coordinates
11.3 General black holes
Formation of black holes in general
General properties of black holes
Kerr black hole
Dragging of inertial frames
Ergoregion
The Kerr horizon
Equatorial photon motion in the Kerr metric
The Penrose process
11.4 Real black holes in astronomy
Black holes of stellar mass
Supermassive black holes
Intermediate-mass black holes
Dynamical black holes
11.5 Quantum mechanical emission of radiation by black holes: the Hawking process
11.6 Further reading
11.7 Exercises
12 Cosmology
12.1 What is cosmology?
The universe in the large
The cosmological arena
12.2 Cosmological kinematics: observing the expanding universe
Homogeneity and isotropy of the universe
Models of the universe: the cosmological principle
Cosmological metrics
Three types of universe
Cosmological redshift as a distance measure
Cosmography: measures of distance in the universe
The universe is accelerating!
12.3 Cosmological dynamics: understanding the expanding universe
Dynamics of Robertson–Walker universes: Big Bang and dark energy
Critical density and the parameters of our universe
12.4 Physical cosmology: the evolution of the universe we observe
Decoupling: forming the cosmic microwave background radiation
Dark matter and galaxy formation: the universe after decoupling
The early universe: fundamental physics meets cosmology
Beyond general relativity
12.5 Further reading
12.6 Exercises
Appendix A Summary of linear algebra
Vector space
Matrices
References
Index