Special and General Relativity: With Applications to White Dwarfs, Neutron Stars and Black Holes (Astronomy and Astrophysics Library)

235 Pages·2007·8.26 MB·English

by  Norman K. Glendenning

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ASTRONOMY AND
ASTROPHYSICS LIBRARY
LIBRARY
SeriesEditors: G.Borner,Garching,Germany
A.Burkert,Miinchen,Germany
W.B.Burton,Charlottesville,VA,USAand
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M.A.Dopita,Canberra,Australia
A.Eckart,Koln,Germany
T.Encrenaz,Meudon,France
E.K.Grebel, Basel,Switzerland
B.Leibundgut,Garching,Germany
J.Lequeux,Paris,France
A.Maeder,Sauverny,Switzerland
V.Trimble,CollegePark,MD,andIrvine,CA,USA
Norman K. Glendenning
Special and General
Relativity
With Applications to White Dwarfs,
Neutron Stars and Black Holes
First Edition
fd
Springer
Norman K.Glendenning
LaurenceBerkeley National Laboratory
NuclearScience Division & Institute for
Nuclear& Particle Astrophysics
OneCyclotronRd,MS70R319
Berkeley,CA94720
U.S.A.
Library ofCongress Control Number: 2007921529
ISBN 10:0-387-47106-5 eISBN 10:0-387-47109-X
ISBN 13:978-0-387-47106-8 eISBN 13:978-0-387-47109-9
Printed onacid-free paper.
©2007SpringerScience+Business Media, LLC
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I dedicate this book to my youngest child, Nathan
Preface
There are magnificent tomeson General Relativity, the greatest ofall being
Gravitationby C. Misner, K. Thorne, and J. Wheeler, and Gravitation and
Cosmology by S. Weinberg, both of which include some of the interesting
historical context. Nevertheless, a smaller book that the covers the math-
ematics of curved spacetimes through to the development of the General
Relativity-briefly but completely-and provides as wellsuccinct chapters
on White Dwarfs, Neutron Stars, Black Holes, and Cosmology, has been
suggested by some who have read my earlier book on Compact Stars. I
would like to mention as wella little known book by the great Paul Dirac.
It will probably hold the distinction of being the briefest of all expositions
ofGeneralRelativity alone. There are no applications to relativisticobjects
aside from black holes.
Norman K. Glendenning
Lawrence Radiation Laboratory
One Cyclotron Road
University of California
Berkeley, California 94720
January 1, 2007
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 divergence 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-Volkoffequations . 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 fields 95
4.8.2 Redshift in a general static field 96
4.8.3 Comparison ofemitted 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 electrons103
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 Ultrarelativisticelectronregion: asymptoticwhitedwarf
mass. . . . . . . . . . . . . . . . . . . . . . . . . .. 116
Contents xi
4.9.7 Nature oflimiting mass ofdwarfs and neutron stars 120
4.9.8 Degenerate ideal gas neutron star. . . . . . . . . . . 121
4.10 Improvements in White DwarfModels . 123
4.10.1 Nature of matter at dwarf and neutron star densities 123
4.10.2 Low-density equation ofstate . 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 ofa Neutron Star . 138
4.16 Star of Uniform Density . 140
4.17 Scaling Solution ofthe OV Equations 142
4.18 Bound on 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 offreedom and driving forces 157
4.22.2 Isospin symmetry energy as a driving force 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-Lemaltre 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 ofthe Early Universe 195
5.11 Temperature and Density ofthe Early Universe 196
5.12 Derivation of the Planck Scale . 197
5.13 Time-scale of Neutrino Interactions . 198
5.14 NeutrinoReaction Time-scale Becomes Longer thanthe Age
ofthe 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