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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.