Compact objects in astrophysics : white dwarfs, neutron stars, and black holes

Compact objects are an important class of astronomical objects in current research. Supermassive black holes play an important role in the understanding of the formation of galaxies in the early Universe. Old white dwarfs are nowadays used to calibrate the age of the Universe. Mergers of neutron stars and black holes are the sources of intense gravitational waves which will be measured in the next ten years by gravitational wave detectors.

Camenzind's Compact Objects in Astrophysics gives a comprehensive introduction and up-to-date overview about the physical processes behind these objects, covering the field from the beginning to most recent results, including all relevant observations.

After a presentation of the taxonomy of compact objects, the basic principles of general relativity are given. The author then discusses in detail the physics and observations of white dwarfs and neutron stars (including the most recent equations of state for neutron star matter), the gravitational field of rapidly rotating compact objects, rotating black holes (including ray tracing and black hole magnetospheres), gravitational waves, and the new understanding of accretion processes by means of the magnetorotational instability of accretion disks.
This modern treatise of compact object astrophysics uses the 3+1 split approach to Einstein's equations, and to relativistic hydrodynamics and magnetohydrodynamics. In each chapter problems and solutions help deepen the understanding of the subject. Both advanced students and researchers will appreciate this book as an advanced textbook and reference on this fascinating field of astrophysics.

1 Compact Objects in Astrophysics	p. 1
1.1 Why is Newtonian Gravity Obsolete?	p. 1
1.2 Einstein was Skeptical about the Existence of Black Holes	p. 3
1.3 Subrahmanyan Chandrasekhar and Compact Objects	p. 4
1.4 Classes of Compact Objects	p. 5
1.4.1 White Dwarfs and Neutron Stars	p. 8
1.4.2 Compact X-Ray Sources	p. 9
1.4.3 Radio Pulsars	p. 11
1.5 Supermassive Black Holes in Galactic Centers	p. 16
1.6 Gamma-Ray Bursters	p. 19
Problems	p. 25
2 Gravity of Compact Objects	p. 27
2.1 Geometric Concepts and General Relativity	p. 27
2.2 The Basic Principles of General Relativity	p. 29
2.2.1 Einstein's Equivalence Principle and Metricity	p. 29
2.2.2 Metric Theories of Gravity	p. 33
2.3 Basic Calculus on Manifolds	p. 37
2.3.1 Tensors and Forms on Manifolds	p. 37
2.3.2 The Metric Field and Pseudo-Riemannian Manifolds	p. 42
2.3.3 The Calculus of Forms on Lorentzian Manifolds	p. 44
2.4 Affine Connection and Covariant Derivative	p. 47
2.4.1 Affine Connection	p. 47
2.4.2 Covariant Derivative of Vector Fields	p. 47
2.4.3 Covariant Derivative for Tensor Fields	p. 48
2.4.4 Parallel Transport and Metric Connection	p. 50
2.4.5 Metric Connection	p. 52
2.4.6 Divergence of Vector Fields	p. 55
2.5 Curvature of Pseudo-Riemannian Manifolds	p. 56
2.5.1 Mathematical Definition of Torsion and Curvature	p. 57
2.5.2 Bianchi Identities for Metric Connection	p. 58
2.5.3 Ricci, Weyl and Einstein Tensor	p. 60
2.5.4 Cartan's Structure Equations	p. 61
2.6 Gravity is a Lorentzian Connection on Spacetime	p. 65
2.6.1 The Four Key Principles of General Relativity	p. 65
2.6.2 The Hilbert Action and Einstein's Field Equations	p. 68
2.6.3 On the Cosmological Constant	p. 69
2.6.4 Limits of General Relativity	p. 71
2.7 Gravitational Waves	p. 73
2.7.1 The Geodesic Deviation - Relativistic Tidal Forces	p. 73
2.7.2 Gravity Wave Experiments	p. 74
2.7.3 The Nature of Gravitational Waves	p. 76
2.7.4 Degrees of Freedom	p. 79
2.7.5 Gravitational Wave Solutions	p. 83
2.7.6 The Quadrupole Formula	p. 87
2.8 3+1 Split of Einstein's Equations	p. 91
2.8.1 Induced Spatial Metric and Extrinsic Curvature	p. 92
2.8.2 Hypersurface Embedding	p. 93
2.8.3 Split of Affine Connection and Curvature	p. 95
2.8.4 Split of Einstein's Equations	p. 98
2.8.5 Black Hole Simulations and Gravitational Waves	p. 100
Problems	p. 101
3 Matter Models for Compact Objects	p. 105
3.1 General Relativistic Hydrodynamics	p. 105
3.1.1 Relativistic Plasma Equations	p. 106
3.1.2 On Numerics of Hydrodynamics	p. 110
3.2 The Boltzmann Equation in GR	p. 113
3.2.1 The Geodesics Spray on the Cotangent Bundle	p. 113
3.2.2 Particle Number Current and Energy-Momentum Tensor	p. 116
3.2.3 The Relativistic Boltzmann Equation	p. 117
3.2.4 Liouville Operator in 3+1 Split	p. 118
3.2.5 Transformation into the Local Rest Frame	p. 119
Problems	p. 120
4 Relativistic Stellar Structure	p. 123
4.1 Spacetime of Relativistic Stars	p. 123
4.2 Derivation of the TOV Equations	p. 125
4.2.1 The Curvature of Static Spacetimes	p. 125
4.2.2 Matter in the Interior	p. 127
4.2.3 The Exterior Schwarzschild Solution	p. 130
4.2.4 Stable Branches for Degenerate Stars	p. 131
4.2.5 Metric for Relativistic Stars	p. 131
4.3 A Variational Principle for the Stellar Structure	p. 132
Problems	p. 134
5 White Dwarfs	p. 137
5.1 Observations of Isolated White Dwarfs	p. 138
5.1.1 Sirius B	p. 138
5.1.2 Field White Dwarfs and Classification	p. 139
5.1.3 White Dwarfs in Globular Clusters	p. 143
5.1.4 Magnetic White Dwarfs	p. 143
5.1.5 Ultracool White Dwarfs as Cosmochronometers	p. 145
5.2 What is Inside a White Dwarf?	p. 151
5.3 Equation of State below the Neutron Drip Density	p. 153
5.4 Structure of White Dwarfs and the Chandrasekhar Mass	p. 159
5.4.1 Polytropic Approximation	p. 160
5.4.2 Beyond the Chandrasekhar Treatment	p. 162
5.4.3 Comparison with Observations	p. 162
5.5 The Relativistic Instability of White Dwarf Stars	p. 167
5.5.1 Necessary Condition for Stability	p. 168
5.5.2 The Total Energy in the Post-Newtonian Limit	p. 169
5.5.3 GR White Dwarf Instability	p. 171
5.6 Cooling White Dwarfs	p. 174
5.6.1 Structure of the Surface Layers	p. 175
5.6.2 Cooling Curves and Crystallization	p. 177
5.6.3 Testing WD Crystallization Theory	p. 179
5.7 White Dwarfs in Binary Systems	p. 180
Problems	p. 185
6 Neutron Stars	p. 187
6.1 The Structure of a Neutron Star	p. 188
6.2 Equations of State beyond Neutron Drip	p. 189
6.2.1 From Neutron Drip to Saturation	p. 190
6.2.2 Nuclear EoS for Dense Neutron Matter	p. 199
6.2.3 Relativistic Mean Field Theory above Saturation	p. 206
6.2.4 Analytical Fits to EoS	p. 216
6.3 Neutron Star Models	p. 219
6.3.1 Hadronic Models	p. 219
6.3.2 Quark Matter Cores	p. 224
6.3.3 Grand Canonical Potential for Quark Matter	p. 231
6.3.4 Strange Quark Stars	p. 241
6.3.5 The Structure of Massive Neutron Stars	p. 242
6.4 Neutron Stars in Close Binary Systems	p. 244
6.4.1 Post-Newtonian Potentials for Many-Body Systems	p. 244
6.4.2 Periastron Shift in Two-Body Systems	p. 248
6.4.3 The Shapiro Time Delay in a Binary System	p. 250
6.4.4 Decay of Binary Orbits due to Gravitational Radiation	p. 251
6.5 Masses of Neutron Stars from Radio Pulsar Timing	p. 255
6.5.1 What is Pulsar Timing?	p. 255
6.5.2 The Timing Formula	p. 259
6.5.3 Timing of the Binary System PSR B1913+16	p. 263
6.5.4 Masses of Companion Stars	p. 264
6.5.5 The Double Pulsar System PSR 0737-3039A+B	p. 265
6.6 Neutron Stars in our Galaxy	p. 269
6.6.1 100 Million Neutron Stars in the Galaxy	p. 269
6.6.2 Thermal Emission from Isolated Neutron Stars	p. 272
6.6.3 Rotation-Powered Pulsars	p. 284
6.6.4 Accretion-Powered Neutron Stars and the Mass-Radius Relation	p. 294
Problems	p. 303
7 Rapidly Rotating Neutron Stars	p. 307
7.1 Spacetime of Stationary and Axisymmetric Rotating Bodies	p. 308
7.1.1 Physical Interpretation of the Metric	p. 309
7.1.2 Geodetic and Lense-Thirring Precession	p. 312
7.1.3 On General 3+1 Split of Spacetime	p. 315
7.2 Einstein's Field Equations for Rotating Objects	p. 317
7.2.1 Ricci Tensors of Time-Slices	p. 318
7.2.2 Extrinsic Curvature and 4D Ricci Tensors	p. 319
7.2.3 3+1 Split of Einstein's Equations	p. 320
7.3 Stellar Structure Equations in Isotropic Gauge	p. 321
7.3.1 The Isotropic Gauge	p. 321
7.3.2 Structure Equations for Rotating Stars	p. 322
7.3.3 Mechanical Equilibrium and Effective Potential	p. 324
7.3.4 Stellar Parameters	p. 326
7.4 The Slow-Rotation Approximation	p. 332
7.5 Numerical Integration of the Stellar Structure Equations	p. 335
7.5.1 Comparison of Numerical Codes	p. 337
7.5.2 Properties of Rotating Equilibrium Stellar Structures	p. 338
7.6 Towards Analytical Vacuum Solutions for Rotating Neutron Stars	p. 342
7.6.1 Weyl-Papapetrou Form	p. 342
7.6.2 Ernst Equations	p. 343
7.6.3 Manko's Solution	p. 345
7.7 On Oscillation and Formation of Rotating Neutron Stars	p. 350
Problems	p. 353
8 Black Holes	p. 355
8.1 The Schwarzschild Black Hole	p. 355
8.1.1 Tortoise Coordinates and Null Cones	p. 356
8.1.2 Roads towards Black Hole Formation	p. 358
8.1.3 The Kruskal Extension	p. 359
8.1.4 Penrose Diagram - the Conformal Structure of Infinity	p. 363
8.2 Geodetic Motions in Schwarzschild Spacetime	p. 369
8.2.1 A Lagrangian	p. 369
8.2.2 The Effective Potential for Equatorial Motion	p. 371
8.2.3 Orbital Equation and Bound Orbits in Schwarzschild Spacetime	p. 373
8.3 The Kerr Black Hole	p. 378
8.3.1 Kerr Black Hole in Boyer-Lindquist Coordinates	p. 379
8.3.2 A Short Derivation of the Kerr Solution	p. 379
8.3.3 The Weyl-Papapetrou Form of the Kerr Metric	p. 384
8.3.4 Uniqueness of the Kerr Solution	p. 385
8.3.5 Global Properties of the Ken Metric	p. 386
8.3.6 On the Conformal Structure of the Kerr Solution	p. 393
8.3.7 Ernst's Equations for the Kerr Geometry	p. 394
8.3.8 The Kerr-Schild Metric and Two-Black-Hole States	p. 395
8.4 Rotational Energy and the Four Laws of Black Hole Evolution	p. 399
8.4.1 Surface Gravity and Angular Velocity of the Horizon	p. 400
8.4.2 First Law of Black Hole Dynamics	p. 402
8.4.3 Rotational Energy of Astrophysical Black Holes	p. 405
8.4.4 On the Second and Third Laws of Black Hole Dynamics	p. 406
8.5 Time Evolution of Black Holes	p. 408
8.5.1 Quasistationary Evolution of Accreting Black Holes	p. 408
8.5.2 Merging of Black Holes	p. 411
8.6 Geodesics in the Kerr Geometry	p. 412
8.6.1 Direct Integration of Geodesics Equations	p. 414
8.6.2 Geodesies in the Equatorial Plane	p. 416
8.6.3 Geodesics Including Lateral Motion	p. 424
8.6.4 Null Geodesics and Ray-Tracing in Kerr Geometry	p. 431
8.7 Dark Energy Stars	p. 442
8.7.1 Why Dark energy Stars?	p. 442
8.7.2 Structure of Gravastars	p. 443
8.7.3 The Necessity of an Anisotropic Crust	p. 445
Problems	p. 446
9 Astrophysical Black Holes	p. 449
9.1 Classes of Astrophysical Black Holes	p. 450
9.2 Measuring Black Hole Masses	p. 451
9.2.1 BHs in X-Ray Binaries	p. 451
9.2.2 Intermediate-Mass Black Holes	p. 456
9.2.3 Supermassive Black Holes in Nearby Galaxies	p. 456
9.2.4 Black Holes in Quasars	p. 468
9.3 Estimating Black Hole Spin	p. 470
9.3.1 Black Hole Spin and Radio Galaxies	p. 471
9.3.2 Spectral Fitting of Accretion Disks	p. 471
9.3.3 Relativistic Iron Lines	p. 472
9.3.4 Quasiperiodic Oscillations	p. 472
9.4 Black Holes and Galaxy Formation	p. 472
9.5 Black Hole Magnetospheres	p. 473
9.5.1 The 3+1 Formalism for Maxwell's Equations	p. 474
9.5.2 Plasma Equations in the 3+1 Split	p. 478
9.5.3 Time Evolution of Magnetic and Current Flux in Turbulent Disks	p. 480
9.5.4 Stationary Magnetospheres on Kerr Black Holes	p. 486
9.5.5 Relaxation of Black Hole Magnetospheres and the Blandford-Znajek Process	p. 499
9.6 Magnetic Spin-Down of Rotating Black Holes	p. 509
Problems	p. 511
10 Physics of Accretion Flows around Compact Objects	p. 513
10.1 Angular Momentum Transport	p. 514
10.2 Magnetohydrodynamics for Accretion Disks	p. 517
10.2.1 Equations of Magnetohydrodynamics	p. 517
10.2.2 Time and Space Discretization	p. 523
10.2.3 MRI Driven Turbulence in Disks	p. 525
10.2.4 Two-Temperature Plasmas and Radiation Pressure in Accretion Disks	p. 533
10.3 States of Turbulent Accretion Disks	p. 537
10.3.1 Turbulent Angular Momentum Transport in Accretion Disks	p. 538
10.3.2 Truncated Accretion and Standard Disk Models in ID	p. 540
10.3.3 Standard Thin Disk Solutions (SSD)	p. 545
10.3.4 Advection-Dominated Flows (ADAF)	p. 551
10.3.5 Super-Eddington Accretion	p. 552
10.3.6 Unified Models of Disk Accretion	p. 553
10.3.7 Fundamental Time-Scales for Accreting Black Holes	p. 555
10.4 Relativistic MHD - Turbulent Accretion onto Black Holes	p. 558
10.4.1 From SRMHD to GRMHD	p. 558
10.4.2 The Equations for GRMHD	p. 559
10.4.3 Nonradiative Accretion onto Rotating Black Holes	p. 563
10.5 Jets and the Ergosphere	p. 565
10.5.1 Jets as Outflows from the Ergospheric Region	p. 566
10.5.2 From the Ergosphere to the Cluster Gas	p. 572
Problems	p. 575
11 Epilogue and Future Prospects	p. 579
Astrophysical Constants and Symbols	p. 587
SLy4 Equation of State for Neutron Star Matter	p. 591
3+1 Split of Spacetime Curvature	p. 595
C.1 Gauss Decomposition	p. 595
C.2 Codazzi-Mainardi Equations	p. 596
3+1 Split of Rotating Neutron Star Geometry	p. 599
D.1 The 3+1 Split of the Connection	p. 599
D.2 The Curvature of Time Slices	p. 601
Equations of GRMHD	p. 605
E.1 Electromagnetic Fields	p. 605
E.2 Conservative Formulation of GRMHD	p. 607
E.3 Numerical Schemes	p. 609
Solutions	p. 613
Glossary	p. 641
References	p. 657
Index	p. 675

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