Special and General Relativity with Applications to White Dwarfs, Neutron Stars and Black Holes

Norman K. Glendenning

Special and General Relativity With Applications to White Dwarfs, Neutron Stars and Black Holes

First Edition

42) Springer Contents

Preface vii 1 Introduction 1 1.1 Compact Stars 2 1.2 Compact Stars and Relativistic Physics 5 1.3 Compact Stars and Dense-Matter Physics 6

2 Special Relativity 9 2.1 Lorentz Invariance 11 2.1.1 Lorentz transformations 11 2.1.2 Time Dilation 14 2.1.3 Covariant vectors 14 2.1.4 Energy and Momentum 16 2.1.5 Energy-momentum tensor of a perfect fluid 17 2.1.6 Light cone 18

3 General Relativity 19 3.1 Scalars, Vectors, and Tensors in Curvilinear Coordinates 20 3.1.1 Photon in a gravitational field 28 3.1.2 Tidal gravity 29 3.1.3 Curvature of spacetime 30 3.1.4 Energy conservation and curvature 30 3.2 Gravity 32 3.2.1 Einstein's Discovery 32 3.2.2 Particle Motion in an Arbitrary Gravitational Field 32 3.2.3 Mathematical definition of local Lorentz frames . 35 3.2.4 Geodesics 36 3.2.5 Comparison with Newton's gravity 38 3.3 Covariance 39 3.3.1 Principle of general covariance 39 3.3.2 Covariant differentiation 40 3.3.3 Geodesic equation from covariance principle 41 3.3.4 Covariant divergente and conserved quantities . 42 3.4 Riemann Curvature Tensor 45 x Contents

3.4.1 Second covariant derivative of scalars and vectors 45 3.4.2 Symmetries of the Riemann tensor 46 3.4.3 Test for flatness 47 3.4.4 Second covariant derivative of tensors 47 3.4.5 Bianchi identities 48 3.4.6 Einstein tensor 48 3.5 Einstein's Field Equations 50 3.6 Relativistic Stars 52 3.6.1 Metric in static isotropic spacetime 53 3.6.2 The Schwarzschild solution 54 3.6.3 Riemann tensor outside a Schwarzschild star 55 3.6.4 Energy-Momentum tensor of matter 56 3.6.5 The Oppenheimer—Volkoff equations 57 3.6.6 Gravitational collapse and limiting mass 62 3.7 Action Principle in Gravity 63 3.7.1 Derivations 65 3.8 Problems for Chapter 3 68

4 Compact Stars: From Dwarfs to Black Holes 70 4.1 Birth and Death of Stars 70 4.2 Aim of this Chapter 78 4.3 Gravitational Units and Neutron Star Size 79 4.3.1 Units 79 4.3.2 Size and number of baryons in a star 82 4.3.3 Gravitational energy of a neutron star 84 4.4 Partial Decoupling of Matter from Gravity 85 4.5 Equations of Relativistic Stellar Structure 87 4.5.1 Interpretation 87 4.5.2 Boundary conditions and stellar sequences 90 4.6 Electrical Neutrality of Stars 92 4.7 "Constancy" of the Chemical Potential 93 4.8 Gravitational Redshift 95 4.8.1 Integrity of an atom in strong flelds 95 4.8.2 Redshift in a general static field 96 4.8.3 Comparison of emitted and received light 100 4.8.4 Measurements of M/R from redshift 100 4.9 White Dwarfs and Neutron Stars 101 4.9.1 Overview 101 4.9.2 Fermi-Gas equation of state for nucleons and electrons 103 4.9.3 High and low—density limits 109 4.9.4 Polytropes and Newtonian white dwarfs 112 4.9.5 Nonrelativistic electron region 116 4.9.6 Ultrarelativistic electron region: asymptotic white dwarf mass 116 Contents xi 4.9.7 Nature of limiting mass of dwarfs and neutron stars 120 4.9.8 Degenerate ideal gas neutron star 121 4.10 Improvements in White Dwarf Models 123 4.10.1 Nature of matter at dwarf and neutron star densities 123 4.10.2 Low–density equation of state 126 4.10.3 Carbon and oxygen white dwarfs 127 4.11 Temperature and Neutron Star Surface 130 4.12 Stellar Sequences from White Dwarfs to Neutron Stars . 133 4.13 Density Distribution in Neutron Stars 136 4.14 Baryon Number of a Star 137 4.15 Binding Energy of a Neutron Star 138 4.16 Star of Uniform Density 140 4.17 Scaling Solution of the OV Equations 142 4.18 Bound an Maximum Mass of Neutron Stars 144 4.19 Stability 148 4.19.1 Necessary condition for stability 149 4.19.2 Normal modes of vibration: Sufficient condition for stability 151 4.20 Beyond the Maximum-Mass Neutron Star 152 4.21 Hyperons and Quarks in Neutron Stars 155 4.22 First Order Phase Transitions in Stars 156 4.22.1 Degrees of freedom and driving forces 157 4.22.2 Isospin symmetry energy as a driving forte 159 4.22.3 Geometrical phases 162 4.22.4 Color-flavor locked quark-matter phase (CFL) 162 4.23 Signal of Quark Deconfinement in Neutron Stars 165 4.24 Neutron Star Twins 171 4.24.1 Particle populations in twins 173 4.24.2 Test for stability 174 4.24.3 Formation and detection 175 4.25 Black Holes 176 4.25.1 Interior and exterior regions 176 4.25.2 No statics within 179 4.25.3 Black hole densities 182 4.25.4 Black Hole Evaporation 182 4.25.5 Kerr Metric for Rotating Black Hole 183 4.26 Problems for Chapter 4 184

5 Cosmology 187 5.1 Foreword 187 5.2 Units and Data 188 5.3 World Lines and Weyl's Hypothesis 188 5.4 Metric for a uniform isotropic universe 189 5.5 Friedmann—Lemaitre Equations 190 5.6 Temperature Variation with Expansion 192 xii Contents

5.7 Expansion in the Three Ages 192 5.8 Redshift 194 5.9 Hubble constant and Universe age 194 5.10 Evolution of the Early Universe 195 5.11 Temperature and Density of the Early Universe 196 5.12 Derivation of the Planck Scale 197 5.13 Time-scale of Neutrino Interactions 198 5.14 Neutrino Reaction Time-scale Becomes Longer than the Age of the Universe 198 5.15 Ionization of Hydrogen 199 5.16 Present Photon and Baryon Densities 199 5.17 Expansion Since Equality of Radiation and Mass 200 5.18 Helium Abundance 201 5.19 Helium Abundance is Primeval 201 5.20 Redshift and Scale Factor Relationship 201 5.21 Collapse Time of a Dust Cloud 202 5.22 Jeans Mass 203 5.23 Jeans Mass in the Radiation Era 203 5.24 Jeans Mass in the Matter Era 204 5.25 Early Matter Dominated Universe 205 5.26 Curvature 206 5.27 Acceleration 207

References 209

Index 217