Table of contents for Relativity, gravitation, and cosmology : a basic introduction / Ta-Pei Cheng.

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Contents
Part I RELATIVITY--Metric Description of Spacetime
1 Introduction and overview 3
1.1 Relativity as a coordinate symmetry 5
1.1.1 From Newtonian relativity to aether 5
1.1.2 Einsteinian relativity 6
1.1.3 Coordinate symmetry transformations 7
1.1.4 New kinematics and dynamics 7
1.2 GR as a gravitational field theory 8
1.2.1 Einstein's motivations for the general theory 8
1.2.2 Geometry as gravity 10
1.2.3 Mathematical language of relativity 11
1.2.4 GR is the framework for cosmology 12
Review questions 12
2 Special relativity and the flat spacetime 14
2.1 Coordinate symmetries 14
2.1.1 Rotational symmetry 14
2.1.2 Newtonian physics and Galilean symmetry 16
2.1.3 Electrodynamics and Lorentz symmetry 17
2.1.4 Velocity addition rule amended 18
2.2 The new kinematics of space and time 19
2.2.1 Relativity of spatial congruity 20
2.2.2 Relativity of simultaneity--the new kinematics 20
2.2.3 The invariant space-time interval 22
2.3 Geometric formulation of SR 24
2.3.1 General coordinates and the metric tensor 24
2.3.2 Derivation of Lorentz transformation 28
2.3.3 The spacetime diagram 30
2.3.4 Time dilation and length contraction 32
Review questions 34
Problems 35
3 The principle of equivalence 38
3.1 Newtonian gravitation potential--a review 38
3.2 EP introduced 39
3.2.1 Inertial mass vs. gravitational mass 40
3.2.2 EP and its significance 41
3.3 Implications of the strong EP 43
3.3.1 Gravitational redshift and time dilation 43
3.3.2 Light ray deflection calculated 48
3.3.3 Energy considerations of a gravitating light pulse 51
3.3.4 Einstein's inference of a curved spacetime 52
Review questions 53
Problems 53
4 Metric description of a curved space 55
4.1 Gaussian coordinates 56
4.2 Metric tensor 57
4.2.1 Geodesic as the shortest curve 59
4.2.2 Local Euclidean coordinates 61
4.3 Curvature 63
4.3.1 Gaussian curvature 63
4.3.2 Spaces with constant curvature 64
4.3.3 Curvature measures deviation from Euclidean relations 66
Review questions 68
Problems 69
5 GR as a geometric theory of gravity--I 71
5.1 Geometry as gravity 71
5.1.1 EP physics and a warped spacetime 73
5.1.2 Curved spacetime as gravitational field 74
5.2 Geodesic equation as GR equation of motion 75
5.2.1 The Newtonian limit 76
5.2.2 Gravitational redshift revisited 78
5.3 The curvature of spacetime 79
5.3.1 Tidal force as the curvature of spacetime 80
5.3.2 The GR field equation described 83
Review questions 85
Problems 85
6 Spacetime outside a spherical star 87
6.1 Description of Schwarzschild spacetime 87
6.1.1 Spherically symmetric metric tensor 88
6.1.2 Schwarzschild geometry 90
6.2 Gravitational lensing 92
6.2.1 Light ray deflection revisited 93
6.2.2 The lens equation 93
6.3 Precession of Mercury's perihelion 97
6.4 Black holes 102
6.4.1 Singularities of the Schwarzschild metric 102
6.4.2 Time measurements in the Schwarzschild spacetime 102
6.4.3 Lightcones of the Schwarzschild black hole 105
6.4.4 Orbit of an object around a black hole 108
6.4.5 Physical reality of black holes 108
Review questions 111
Problems 112
Part II COSMOLOGY
7 The homogeneous and isotropic universe 115
7.1 The cosmos observed 116
7.1.1 Matter distribution on the cosmic distance scale 116
7.1.2 Cosmological redshift: Hubble's law 116
7.1.3 Age of the universe 120
7.1.4 Dark matter and mass density of the universe 121
7.2 The cosmological principle 125
7.3 The Robertson-Walker metric 127
7.3.1 Proper distance in the RW geometry 129
7.3.2 Redshift and luminosity distance 130
Review questions 133
Problems 134
8 The expanding universe and thermal relics 136
8.1 Friedmann equations 137
8.1.1 The quasi-Newtonian interpretation 139
8.2 Time evolution of model universes 142
8.3 Big bang cosmology 145
8.3.1 Scale-dependence of radiation's temperature 145
8.3.2 Different thermal equilibrium stages 147
8.4 Primordial nucleosynthesis 149
8.5 Photon decoupling and the CMB 153
8.5.1 Universe became transparent to photons 153
8.5.2 The discovery of CMB radiation 154
8.5.3 Photons, neutrinos, and the radiation-matter equality time 155
8.5.4 CMB temperature fluctuation 159
Review questions 163
Problems 164
9 In.ation and the accelerating universe 165
9.1 The cosmological constant 166
9.1.1 Vacuum energy as source of gravitational repulsion 167
9.1.2 The static universe 168
9.2 The inflationary epoch 170
9.2.1 Initial conditions for the standard big bang model 171
9.2.2 The inflation scenario 173
9.2.3 Inflation and the conditions it left behind 175
9.3 CMB anisotropy and evidence for k = 0 178
9.3.1 Three regions of the angular power spectrum 179
9.3.2 The primary peak and spatial geometry of the universe 181
9.4 The accelerating universe in the present epoch 183
9.4.1 Distant supernovae and the 1998 discovery 184
9.4.2 Transition from deceleration to acceleration 187
9.5 The concordant picture 190
Review questions 193
Problems 193
Part IIIRELATIVITY--Full Tensor Formulation
10 Tensors in special relativity 197
10.1 General coordinate systems 197
10.2 Four-vectors in Minkowski spacetime 200
10.3 Manifestly covariant formalism for E&M 204
10.3.1 The electromagnetic field tensor 205
10.3.2 Electric charge conservation 208
10.4 Energy-momentum tensors 208
Review questions 212
Problems 213
11 Tensors in general relativity 215
11.1 Derivatives in a curved space 215
11.1.1 General coordinate transformations 216
11.1.2 Covariant differentiation 218
11.1.3 Christoffel symbols and metric tensor 220
11.2 Parallel transport 222
11.2.1 Component changes under parallel transport 223
11.2.2 The geodesic as the straightest possible curve 224
11.3 Riemannian curvature tensor 225
11.3.1 The curvature tensor in an n-dimensional space 226
11.3.2 Symmetries and contractions of the curvature tensor 228
Review questions 231
Problems 231
12 GR as a geometric theory of gravity--II 233
12.1 The principle of general covariance 233
12.1.1 Geodesic equation from SR equation of motion 235
12.2 Einstein field equation 236
12.2.1 Finding the relativistic gravitational field equation 236
12.2.2 Newtonian limit of the Einstein equation 237
12.3 The Schwarzschild exterior solution 239
12.4 The Einstein equation for cosmology 244
12.4.1 Solution for a homogeneous and isotropic 3D space 244
12.4.2 Friedmann equations 246
12.4.3 Einstein equation with a cosmological constant term 247
Review questions 248
Problems 248
13 Linearized theory and gravitational waves 250
13.1 The linearized Einstein theory 251
13.1.1 The coordinate change called gauge transformation 252
13.1.2 The wave equation in the Lorentz gauge 253
13.2 Plane waves and the polarization tensor 254
13.3 Gravitational wave detection 255
13.3.1 Effect of gravitational waves on test particles 255
13.3.2 Gravitational wave interferometers 257
13.4 Evidence for gravitational wave 259
13.4.1 Energy flux in linearized gravitational waves 260
13.4.2 Emission of gravitational radiation 262
13.4.3 Binary pulsar PSR 1913+16 264
Review questions 268
Problems 269
A Supplementary notes 271
A.1 The twin paradox (Section 2.3.4) 271
A.2 A glimpse of advanced topics in black hole physics (Section 6.4) 275
A.3 False vacuum and hidden symmetry (Section 9.2.2) 279
A.4 The problem of as quantum vacuum energy (Section 9.4) 280
B Answer keys to review questions 283
C Solutions of selected problems 293
References 330
Bibliography 333
Index 335

Library of Congress Subject Headings for this publication:

General relativity (Physics) -- Textbooks.
Space and time.
Gravity.
Cosmology.