THE GALACTIC Lectures on and

Edited by

Heino Falcke Max Planck Institute for , Bonn,

and

Friedrich W Hehl Institute for Theoretical Physics, University of , Germany

SUB Gfittingen 7 215 826 302

2003 A 1697

IoP INSTITUTE OF PHYSICS PUBLISHING BRISTOL AND PHILADELPHIA Contents

Preface xi

PART 1 General introduction 1 1 The Schwarzschild black hole: a general relativistic introduction Christian Heinicke and Friedrich WHehl 3 1.1 Newton's gravitational theory in quasi-field-theoretical form 3 1.2 Special relativity and Newton's theory: a clash 8 1.3 Accelerated frames of reference, equivalence principle and Einstein's field equation 11 1.4 The exterior Schwarzschild solution 16 1.5 Flat Minkowski spacetime, null coordinates, and the Penrose diagram 17 1.6 Schwarzschild spacetime and the Penrose-Kruskal diagram 19 1.7 The interior Schwarzschild solution and the TOV equation 25 1.8 Computer algebra 29 References 33 2 The Milky Way: structure, constituents and evolution Susanne Hiittemeister 35 2.1 The overall structure of the Milky Way 35 2.1.1 Deducing the large-scale structure of the Galaxy 35 2.1.2 Unveiling Galactic structure: history 36 2.1.3 'External' views 39 2.2 The constituents . 42 2.2.1 The Galactic rotation curve 42 2.2.2 The disk: spiral arms and their tracers 44 2.2.3 The bulge: photometric 3D models, bulge/disk models and mass 47 2.2.4 The nuclear bulge or bar and the Central Molecular Zone 51 2.2.5 Gas flows and infall: Feeding the nuclear region 54 2.3 Galaxy evolution 57 vi Contents

2.3.1 Hierarchical,, bottom-up structure formation 58 2.3.2 Evolutionary mechanisms: mergers and 'internal' processes 60 2.4 The relation between black holes and bulges 63 2.4.1 Black hole mass and bulge mass/luminosity 63 2.4.2 Black hole mass and bulge velocity dispersion 65 References 68 3 The collapse to a black hole Gernot Neugebauer 72 3.1 Introduction % 72 3.2 Oppenheimer-Snyder collapse 77 3.2.1 Scenario and model 77 3.2.2 Solution of the field equations 78 3.2.3 Physical interpretation 84 3.3 Rotating matter and black hole formation 88 References 93 4 The environment of the Galaxy's central black hole Mark R Morris 95 4.1 Introduction 95 4.2 The nuclear stellar bulge 96 4.3 The Central Molecular Zone 100 4.4 Hot gas 102 4.5 The Galactic Center magnetosphere 102 4.6 The circumnuclear disk and Sagittarius A West 107 4.7 Star formation 111 4.8 A provocative supernova remnant: Sgr A East 114 4.9 The vicinity of Sgr A* 117 4.10 Perspective 118 References 118

PART 2 General relativity and black holes 123 5 Particles and fields near black holes ValeriFrolov 125 5.1 Introduction 125 5.2 Particle motion near a non-rotating black hole 126 5.2.1 Equations of motion 126 5.2.2 Symmetries and integrals of motion 127 5.2.3 Equations of motion of a free test particle 129 5.2.4 Types of trajectory 130 5.2.5 Equations of motion in'tilted'spherical coordinates 134 5.2.6 Motion of ultrarelativistic particles 135 5.2.7 Gravitational capture 137 Contents vii

5.3 Particle motion near a rotating black hole 138 5.3.1 Gravitational field of a rotating black hole 138 5.3.2 Equations of motion of a free test particle 140 5.3.3. Motion in the equatorial plane 143 5.3.4 Motion off the equatorial plane 147 5.3.5 Gravitational capture 148 5.4 Propagation of fields in the black hole spacetime 149 5.4.1 Scalar massless field in the Schwarzschild metric 149 5.4.2 Evolution of the scalar massless field around a non- rotating black hole 153 5.4.3 Wave fields in the Kerr metric 157 5.4.4 Effects connected with black hole rotation 161 5.5 Black hole electrodynamics 163 5.5.1 Introduction 163 5.5.2 Electrodynamics in a homogeneous gravitational field 164 5.5.3 Membrane interpretation 168 5.5.4 Electric field of a pointlike charge near a black hole 170 5.5.5 Black hole in a magnetic field 172 5.5.6 Mechanism of the power generation 173 References 176 6 Close encounters of black holes Domenico Giulini 178 6.1 Introduction and motivation 178 6.2 A first step beyond Newtonian gravity 179 63 Constrained evolutionary structure of Einstein's equations 183 6.4 The 3 + 1 split and the Cauchy initial-value problem 186 6.5 Black hole data - 188 6.5.1 Horizons 188 6.5.2 Poincare charges 189 6.5.3 Maximal and time-symmetric data 190 6.5.4 Solution strategy for maximal data 191 6.5.5 Explicit time-symmetric data 191 6.5.6 Non-time-symmetric data 201 6.6 Problems and recent developments 202 6.7 Appendix: equation (6.2) satisfies the energy principle 203 References 204 7 Quantum aspects of black holes Claus Kiefer 207 7.1 Introduction 207 7.2 The laws of black hole mechanics 208 7.3 Hawking radiation 212 7.4 Interpretation of entropy 218 viii Contents

7.5 Primordial black holes 221 References 225

PART 3 Our galactic center 227 8 The mass of the Galactic Center black hole Andreas Eckart 229 8.1 Introduction and summary 229 8.2 A brief history of imaging the Galactic Center in the near-infrared 231 8.3 Speckle interferometry 232 8.4 The center of the Milky Way 233 8.4.1 Imaging and proper motions 233 8.4.2 Spectroscopy 235 8.4.3 Enclosed mass 235 8.4.4 Orbital curvatures 237 8.4.5 Is there an infrared counterpart of Sgr A*? 240 8.4.6 LBT and the Galactic Center 242 Note added in proof 244 References 244 9 Stars and singularities: stellar phenomena near a massive black hole Tal Alexander 246 9.1 Introduction 246 9.2 Stellar dynamics near a black hole 248 9.2.1 Physical scales 249 9.2.2 A relaxed stellar system around a MBH 251 - 9.3 The stellar collider in the Galactic Center 253 9.3.1 The case for a dense stellar cusp in the Galactic Center 254 9.3.2 Tidal spin-up 257 9.3.3 Tidal scattering 259 9.4 The gravitational telescope in the Galactic Center 261 9.4.1 Gravitational lensing by a point mass 263 9.4.2 Pinpointing the MBH with lensed images 264 9.4.3 The detection of gravitational lensing 267 9.4.4 Magnification bias 270 9.4.5 Beyond the point mass lens approximation 271 9.5 Summary 274 References 274 10 Black hole accretion models for the Galactic Center Robert F Coker 276 10.1 Introduction 276 10.2 Accreting gas with zero angular momentum 277 10.2.1 Adiabatic spherical accretion 277 Contents ix

10.2.2 Supersonic non-adiabatic spherical accretion 284 10.2.3 Radiation from spherical accretion 288 10.2.4 Calculation of the spectrum due to spherical accretion 290 10.3 Non-spherical accretion models 292 10.3.1 Keplerian flow with magnetic dynamo 293 10.3.2 Sub-Eddington two-temperature accretion (ADAFs) 299 10.4 Comment on X-ray emission from Sgr A* 307 10.5 Summary 307 Acknowledgements 308 References 308 11 Radio and X-ray emission from the Galactic Black Hole Heino Falcke 310 11.1 Introduction 310 11.2 Radio properties of Sgr A* 311 11.2.1 Variability of Sgr A* 312 11.2.2 Size of Sgr A*—VLBI observations 314 11.2.3 Position of Sgr A* 317 11.2.4 Radio spectrum of Sgr A* 319 11.2.5 Polarization of Sgr A* 320 11.3 Radio and X-ray emission from a black hole jet 321 11.3.1 The flat radio spectrum 322 11.3.2 The X-ray spectrum 329 11.3.3 Numerical results 330 11.3.4 The circular polarization 331 11.3.5 Comparison with other supermassive black holes 336 11.4 Imaging the —an outlook 336 References 340 A List of authors 343 B Units and constants 346 Index 349