Relativity, Gravitation, and Cosmology

Transcription

1 Relativity, Gravitation, and Cosmology A basic introduction TA-PEI CHENG University of Missouri St. Louis OXFORD UNIVERSITY PRESS

2 Contents Parti RELATIVITY Metric Description of Spacetime 1 Introduction and overview Relativity as a coordinate symmetry From Newtonian relativity to aether Einsteinian relativity Coordinate symmetry transformations New kinematics and dynamics GR as a gravitational field theory Einstein's motivations for the general theory Geometry as gravity Mathematical language of relativity GR is the framework for cosmology 12 Review questions 12 2 Special relativity and the flat spacetime Coordinate symmetries Rotational symmetry Newtonian physics and Galilean symmetry Electrodynamics and Lorentz symmetry Velocity addition rule amended The new kinematics of space and time Relativity of spatial congruity Relativity of simultaneity the new kinematics The invariant space-time interval Geometric formulation of SR General coordinates and the metric tensor Derivation of Lorentz transformation The spacetime diagram Time-dilation and length contraction 32 Review questions 35 Problems 35 3 The principle of equivalence Newtonian gravitation potential a review EP introduced Inertial mass vs. gravitational mass EP and its significance 41

3 x Contents 3.3 Implications of the strong EP Gravitational redshift and time dilation Light ray deflection calculated Energy considerations of a gravitating light pulse Einstein's inference of a curved spacetime 52 Review questions 53 Problems 53 Metric description of a curved space Gaussian coordinates Metric tensor Geodesic as the shortest curve y Local Euclidean coordinates Curvature Gaussian curvature Spaces with constant curvature Curvature measures deviation from Euclidean relations 66 Review questions 68 Problems 69 GR as a geometric theory of gravity Geometry as gravity EP physics and a warped spacetime Curved spacetime as gravitational field Geodesic equation as GR equation of motion The Newtonian limit Gravitational redshift revisited The curvature of spacetime Tidal force as the curvature of spacetime The GR field equation described 83 Review questions 85 Problems 85 Spacetime outside a spherical star Description of Schwarzschild spacetime Spherically symmetric metric tensor Schwarzschild geometry Gravitational lensing Light ray deflection revisited The lens equation Precession of Mercury's perihelion Black holes Singularities of the Schwarzschild metric Time measurements in the Schwarzschild spacetime Lightcones of the Schwarzschild black hole 105

4 Contents xi Orbit of an object around a black hole Physical reality of black holes 108 Review questions 111 Problems 112 Part II COSMOLOGY 7 The homogeneous and isotropic universe The cosmos observed Matter distribution on the cosmic distance scale Cosmological redshift: Hubble's law Age of the universe Dark matter and mass density of the universe The cosmological principle The Robertson-Walker metric Proper distance in the RW geometry Redshift and luminosity distance 130 Review questions 133 Problems The expanding universe and thermal relics Friedmann equations The quasi-newtonian interpretation Time evolution of model universes Big bang cosmology Scale-dependence of radiation's temperature Different thermal equilibrium stages Primordial nucleosynthesis Photon decoupling and the CMB Universe became transparent to photons The discovery of CMB radiation Photons, neutrinos, and the radiation-matter equality time CMB temperature fluctuation 159 Review questions 162 Problems Inflation and the accelerating universe The cosmological constant Vacuum energy as source of gravitational repulsion The static universe The inflationary epoch Initial conditions for the standard big bang model 171

5 xii Contents The inflation scenario Inflation and the conditions it left behind CMB anisotropy and evidence for k = Three regions of the angular power spectrum The primary peak and spatial geometry of the universe The accelerating universe in the present epoch Distant supernovae and the 1998 discovery Transition from deceleration to acceleration The concordant picture 189 Review questions 193 Problems 193 Part III RELATIVITY Full Tensor Formulation 10 Tensors in special relativity General coordinate systems Four-vectors in Minkowski spacetime Manifestly covariant formalism for E&M The electromagnetic field tensor Electric charge conservation Energy-momentum tensors 208 Review questions 213 Problems Tensors in general relativity Derivatives in a curved space General coordinate transformations Covariant differentiation Christoffel symbols and metric tensor Parallel transport Component changes under parallel transport The geodesic as the straightest possible curve Riemannian curvature tensor The curvature tensor in an n-dimensional space Symmetries and contractions of the curvature tensor 228 Review questions 230 Problems GR as a geometric theory of gravity - II The principle of general covariance Geodesic equation from SR equation of motion 235

6 Contents xiii 12.2 Einstein field equation Finding the relativistic gravitational field equation Newtonian limit of the Einstein equation The Schwarzschild exterior solution The Einstein equation for cosmology Solution-for a homogeneous and isotropic 3D space Friedmann equations Einstein equation with a cosmological constant term 247 Review questions 248 Problems Linearized theory and gravitational waves The linearized Einstein theory The coordinate change called gauge transformation The wave equation in the Lorentz gauge Plane waves and the polarization tensor Gravitational wave detection Effect of gravitational waves on test particles Gravitational wave interferometers Evidence for gravitational wave Energy flux in linearized gravitational waves Emission of gravitational radiation Binary pulsar PSR Review questions 268 Problems 269 A Supplementary notes 271 A.I 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 A 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