The Galactic Black Hole: Lectures on General Relativity and Astrophysics (Series in High Energy Physics, Cosmology and Gravitation)

The supermassive black hole in the center of our Milky Way is the nearest such object and relatively easy to observe and study. Not surprisingly therefore, it is the best studied supermassive black hole. Many astrophysical and even general relativistic effects can be investigated in great detail. The Galactic Black Hole: Lectures on General Relativity and Astrophysics provides a systematic introduction to the physics/astrophysics and mathematics of black holes at a level suitable for graduate students, postdocs, and researchers in physics, astrophysics, astronomy, and applied mathematics. The focus is mainly on the supermassive black hole in the center of our Milky Way but the results can be easily generalized taking it as an example. Leading international experts provide first-hand accounts of the observational and theoretical aspects of this black hole. Topics range from the properties of the Schwarzschild metric and the collapse of a black hole, to quantum gravity, and from the structure of the Galaxy to accretion of matter and the emission properties of the Galactic Center black hole. THE GALACTIC BLACK HOLE: LECTURES ON GENERAL RELATIVITY AND ASTROPHYSICS......Page 1 Back Cover......Page 2 Title Page......Page 5 Copyright Page......Page 6 Contents......Page 7 Preface......Page 13 Part 1: General introduction......Page 15 1.1 Newton’s gravitational theory in quasi-field-theoretical form......Page 17 1.2 Special relativity and Newton’s theory: a clash......Page 22 1.3 Accelerated frames of reference, equivalence principle and Einstein’s field equation......Page 25 1.4 The exterior Schwarzschild solution......Page 30 1.5 Flat Minkowski spacetime, null coordinates, and the Penrose diagram......Page 31 1.6 Schwarzschild spacetime and the Penrose–Kruskal diagram......Page 33 1.7 The interior Schwarzschild solution and the TOV equation......Page 39 1.8 Computer algebra......Page 43 References......Page 47 2.1.1 Deducing the large-scale structure of the Galaxy......Page 49 2.1.2 Unveiling Galactic structure: history......Page 50 2.1.3 ‘External’ views......Page 53 2.2.1 The Galactic rotation curve......Page 56 2.2.2 The disk: spiral arms and their tracers......Page 58 2.2.3 The bulge: photometric 3D models, bulge/disk models and mass......Page 61 2.2.4 The nuclear bulge or bar and the Central Molecular Zone......Page 65 2.2.5 Gas flows and infall: Feeding the nuclear region......Page 68 2.3 Galaxy evolution......Page 71 2.3.1 Hierarchical, bottom-up structure formation......Page 72 2.3.2 Evolutionary mechanisms: mergers and ‘internal’ processes......Page 74 2.4.1 Black hole mass and bulge mass/luminosity......Page 77 2.4.2 Black hole mass and bulge velocity dispersion......Page 79 References......Page 82 3.1 Introduction......Page 86 3.2.1 Scenario and model......Page 91 3.2.2.1 The problem......Page 92 3.2.2.2 Exterior solution......Page 93 3.2.2.3 Interior solution......Page 94 3.2.2.4 Matching of geometries......Page 96 3.2.3 Physical interpretation......Page 98 3.3 Rotating matter and black hole formation......Page 102 References......Page 107 4.1 Introduction......Page 109 4.2 The nuclear stellar bulge......Page 110 4.3 The Central Molecular Zone......Page 114 4.5 The Galactic Center magnetosphere......Page 116 4.6 The circumnuclear disk and Sagittarius A West......Page 121 4.7 Star formation......Page 125 4.8 A provocative supernova remnant: Sgr A East......Page 128 4.9 The vicinity of Sgr A*......Page 131 References......Page 132 Part 2: General relativity and black holes......Page 137 5.1 Introduction......Page 139 5.2.1.1 Particle motion......Page 140 5.2.2.1 Killing vectors......Page 141 5.2.2.3 Symmetries of Schwarzschild spacetime......Page 142 5.2.3.2 Effective potential......Page 143 5.2.4.1 Bound and unbound trajectories......Page 144 5.2.4.2 Circular motion......Page 147 5.2.5 Equations of motion in ‘tilted’ spherical coordinates......Page 148 5.2.6.2 Types of trajectory......Page 149 5.2.7 Gravitational capture......Page 151 5.3.1.2 Killing vectors......Page 152 5.3.1.3 Killing tensor......Page 153 5.3.2.1 Integrals of motion......Page 154 5.3.2.4 Effective potential......Page 155 5.3.2.5 Motion in the θ-direction......Page 156 5.3.3.1 Circular orbits......Page 157 5.3.3.2 Last stable circular orbits......Page 159 5.3.4 Motion off the equatorial plane......Page 161 5.3.5.2 Gravitational capture of ultrarelativistic particles......Page 162 5.4.1.1 Field equation......Page 163 5.4.1.3 Radial equation and effective potential......Page 164 5.4.1.5 Interpretation of basic solutions......Page 166 5.4.2.1 Retarded Green’s function......Page 167 5.4.2.3 Analytical properties......Page 168 5.4.2.5 Late-time behavior......Page 170 5.4.3.1 Electromagnetic waves and gravitational perturbations in the Kerr geometry......Page 171 5.4.3.3 Field restoration from solutions of the decoupled equations......Page 172 5.4.3.4 Separation of variables, spin-weighted spheroidal harmonics......Page 173 5.4.3.5 The radial equation......Page 174 5.4.4.1 Wave evolution and quasinormal modes in the Kerr spacetime......Page 175 5.4.4.2 Gravitational radiation from a particle plunging into the black hole......Page 176 5.5.1 Introduction......Page 177 5.5.2.1 Electrodynamics in the uniformly accelerated frame......Page 178 5.5.2.3 Electric charge in the homogeneous gravitational field......Page 180 5.5.3.1 Maxwell’s equations in (3 + 1)-form......Page 182 5.5.3.2 Boundary conditions at the event horizon......Page 183 5.5.4 Electric field of a pointlike charge near a black hole......Page 184 5.5.5.2 A black hole in a homogeneous magnetic field......Page 186 5.5.6.1 Potential difference......Page 187 5.5.6.2 Black hole magnetosphere and efficiency of the power generating process......Page 188 5.5.6.3 Black hole as a unipolar inductor......Page 189 References......Page 190 6.1 Introduction and motivation......Page 192 6.2 A first step beyond Newtonian gravity......Page 193 6.3 Constrained evolutionary structure of Einstein’s equations......Page 197 6.4 The 3 + 1 split and the Cauchy initial-value problem......Page 200 6.5.1 Horizons......Page 202 6.5.2 Poincaré charges......Page 203 6.5.3 Maximal and time-symmetric data......Page 204 6.5.5 Explicit time-symmetric data......Page 205 6.5.5.1 One black hole......Page 206 6.5.5.2 Two black holes......Page 207 6.5.5.4 Energy bounds from Hawking’s area law......Page 209 6.5.5.5 Other topologies......Page 211 6.5.6 Non-time-symmetric data......Page 215 6.6 Problems and recent developments......Page 216 6.7 Appendix: equation (6.2) satisfies the energy principle......Page 217 References......Page 218 7.1 Introduction......Page 221 7.2 The laws of black hole mechanics......Page 222 7.3 Hawking radiation......Page 226 7.4 Interpretation of entropy......Page 232 7.5 Primordial black holes......Page 235 References......Page 239 Part 3: Our galactic center......Page 241 8.1 Introduction and summary......Page 243 8.2 A brief history of imaging the Galactic Center in the near-infrared......Page 245 8.3 Speckle interferometry......Page 246 8.4.1 Imaging and proper motions......Page 247 8.4.3 Enclosed mass......Page 249 8.4.4 Orbital curvatures......Page 251 8.4.5 Is there an infrared counterpart of Sgr A ∗?......Page 254 8.4.6 LBT and the Galactic Center......Page 256 References......Page 258 9.1 Introduction......Page 260 9.2 Stellar dynamics near a black hole......Page 262 9.2.1 Physical scales......Page 263 9.2.1.2 Length scales......Page 264 9.2.2 A relaxed stellar system around a MBH......Page 265 9.3 The stellar collider in the Galactic Center......Page 267 9.3.1 The case for a dense stellar cusp in the Galactic Center......Page 268 9.3.2 Tidal spin-up......Page 271 9.3.3 Tidal scattering......Page 273 9.4 The gravitational telescope in the Galactic Center......Page 275 9.4.1 Gravitational lensing by a point mass......Page 277 9.4.2 Pinpointing the MBH with lensed images......Page 278 9.4.3 The detection of gravitational lensing......Page 281 9.4.4 Magnification bias......Page 284 9.4.5 Beyond the point mass lens approximation......Page 285 References......Page 288 10.1 Introduction......Page 290 10.2.1 Adiabatic spherical accretion......Page 291 10.2.1.1 A comment on the relativistic solution......Page 296 10.2.2 Supersonic non-adiabatic spherical accretion......Page 298 10.2.3 Radiation from spherical accretion......Page 302 10.2.4 Calculation of the spectrum due to spherical accretion......Page 304 10.3 Non-spherical accretion models......Page 306 10.3.1 Keplerian flow with magnetic dynamo......Page 307 10.3.1.1 Calculation of the spectrum......Page 310 10.3.2 Sub-Eddington two-temperature accretion (ADAFs)......Page 313 10.3.2.1 Comments on ADAFs......Page 319 10.5 Summary......Page 321 References......Page 322 11.1 Introduction......Page 324 11.2 Radio properties of Sgr A*......Page 325 11.2.1 Variability of Sgr A*......Page 326 11.2.2 Size of Sgr A*—VLBI observations......Page 328 11.2.3 Position of Sgr A*......Page 331 11.2.4 Radio spectrum of Sgr A*......Page 333 11.2.5 Polarization of Sgr A*......Page 334 11.3 Radio and X-ray emission from a black hole jet......Page 335 11.3.1 The flat radio spectrum......Page 336 11.3.2 The X-ray spectrum......Page 343 11.3.3 Numerical results......Page 344 11.3.4 The circular polarization......Page 345 11.4 Imaging the event horizon—an outlook......Page 350 References......Page 354 Appendix A. List of authors......Page 357 Appendix B. Units and constants......Page 360 Index......Page 363 Pt. 1. General introduction. The Schwarzschild black hole: a general relativistic introduction / Christian Heinicke and Friedrich W. Helhl The Milky Way: structure, constituents and evolution / Susanne Hüttemeister The collapse to a black hole / Gernot Neugebauer The environment of the Galaxy's central black hole / Mark R. Morris pt. 2. General relativity and black holes. Particles and fields near black holes / Valeri Frolov Close encounters of black holes / Domenico Giulini Quantum aspects of black holes / Claus Kiefer pt. 3. Our galactic center. The mass of the galactic Center black hole / Andreas Eckert Stars and singularities: stellar phenomena near a massive black hole / Tal Alexander Black hole accretion models for the galactic center / Robert F. Coker Radio and x-ray emission from the galactic black hole / Heino Falcke "These lectures provide a systematic introduction to the physics/astrophysics and mathematics of black holes at a level suitable for graduate students, postdocs and researchers in physics, astrophysics, astronomy, and applied mathematics. The focus is mainly on the supermassive black hole in the center of our Milky Way but the results can be easily generalized taking it as an example. Leading international experts provide first hand accounts of the observational and theoretical aspects of this black hole. Topics range from the properties of the Schwarzschild metric and the collapse to a black hole, to quantum gravity; and from the structure of the Galaxy to accretion of matter and the emission properties of the Galactic Center black hole."--BOOK JACKET