Notes on Relativity and Cosmology - Physics Department, UCSB
Notes on Relativity and Cosmology - Physics Department, UCSB Notes on Relativity and Cosmology - Physics Department, UCSB
6 CONTENTS 8 General Relativity and Curved Spacetime 193 8.1 A return to geometry . . . . . . . . . . . . . . . . . . . . . . . . . 194 8.1.1 Straight Lines in Curved Space . . . . . . . . . . . . . . . 196 8.1.2 Curved Surfaces are Locally Flat . . . . . . . . . . . . . . 198 8.1.3 From curved space to curved spacetime . . . . . . . . . . 199 8.2 More on Curved Space . . . . . . . . . . . . . . . . . . . . . . . . 200 8.3 Gravity and the Metric . . . . . . . . . . . . . . . . . . . . . . . 205 8.3.1 Building Intuition in flat space . . . . . . . . . . . . . . . 206 8.3.2 On to Angles . . . . . . . . . . . . . . . . . . . . . . . . . 207 8.3.3 Metrics on Curved space . . . . . . . . . . . . . . . . . . . 208 8.3.4 A first example . . . . . . . . . . . . . . . . . . . . . . . . 209 8.3.5 A second example . . . . . . . . . . . . . . . . . . . . . . 211 8.3.6 Some parting comments on metrics . . . . . . . . . . . . . 211 8.4 What is the metric of spacetime? . . . . . . . . . . . . . . . . . . 213 8.4.1 The Einstein equations . . . . . . . . . . . . . . . . . . . . 213 8.4.2 The Newtonian Approximation . . . . . . . . . . . . . . . 215 8.4.3 The Schwarzschild Metric . . . . . . . . . . . . . . . . . . 215 8.5 Experimental Verification of GR . . . . . . . . . . . . . . . . . . 217 8.5.1 The planet Mercury . . . . . . . . . . . . . . . . . . . . . 217 8.5.2 The Bending of Starlight . . . . . . . . . . . . . . . . . . 219 8.5.3 Other experiments: Radar Time Delay . . . . . . . . . . . 220 8.6 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . 221 9 Black Holes 227 9.1 Investigating the Schwarzschild Metric . . . . . . . . . . . . . . . 227 9.1.1 Gravitational Time Dilation from the Metric . . . . . . . 228 9.1.2 Corrections to Newton’s Law . . . . . . . . . . . . . . . . 228 9.2 On Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 9.2.1 Forming a black hole . . . . . . . . . . . . . . . . . . . . . 230 9.2.2 Matter within the Schwarzschild radius . . . . . . . . . . 231 9.2.3 The Schwarzschild radius and the Horizon . . . . . . . . . 232 9.2.4 Going Beyond the Horizon . . . . . . . . . . . . . . . . . 234 9.2.5 A summary of where we are . . . . . . . . . . . . . . . . . 236 9.3 Beyond the Horizon . . . . . . . . . . . . . . . . . . . . . . . . . 239 9.3.1 The interior diagram . . . . . . . . . . . . . . . . . . . . . 239 9.3.2 The Singularity . . . . . . . . . . . . . . . . . . . . . . . . 244 9.3.3 Beyond the Singularity? . . . . . . . . . . . . . . . . . . . 246 9.3.4 The rest of the diagram and dynamical holes . . . . . . . 246 9.3.5 Visualizing black hole spacetimes . . . . . . . . . . . . . . 249 9.4 Stretching and Squishing . . . . . . . . . . . . . . . . . . . . . . . 252 9.4.1 The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 9.4.2 The solution . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.4.3 The Differential equation . . . . . . . . . . . . . . . . . . 258 9.4.4 What does it all mean? . . . . . . . . . . . . . . . . . . . 259 9.4.5 Black Holes and the Schwarzschild Metric . . . . . . . . . 260 9.5 Black Hole Astrophysics and Observations . . . . . . . . . . . . . 261
CONTENTS 7 9.5.1 The observational evidence for black holes . . . . . . . . . 261 9.5.2 Finding other black holes . . . . . . . . . . . . . . . . . . 263 9.5.3 A few words on Accretion and Energy . . . . . . . . . . . 265 9.5.4 So, where does all of this energy go, anyway? . . . . . . . 267 9.6 Black Hole Odds and Ends . . . . . . . . . . . . . . . . . . . . . 268 9.6.1 A very few words about Hawking Radiation . . . . . . . . 268 9.6.2 Penrose Diagrams, or “How to put infinity in a box” . . . 269 9.6.3 Penrose Diagrams for Black holes . . . . . . . . . . . . . . 271 9.6.4 Some Cool Stuff . . . . . . . . . . . . . . . . . . . . . . . 276 9.7 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . 277 10 Cosmology 283 10.1 The Copernican Principle and Relativity . . . . . . . . . . . . . . 283 10.1.1 Homogeneity and Isotropy . . . . . . . . . . . . . . . . . . 284 10.1.2 That technical point about Newtonian Gravity in Homogeneous Space . . . . . . . . . . . . . . . . . . . . . . . . 284 10.1.3 Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . 285 10.2 Dynamics! (a.k.a. Time Evolution) . . . . . . . . . . . . . . . . . 287 10.2.1 Expanding and Contracting Universes . . . . . . . . . . . 287 10.2.2 A flat spacetime model . . . . . . . . . . . . . . . . . . . 289 10.2.3 On to the Einstein Equations . . . . . . . . . . . . . . . . 291 10.2.4 Negative Pressure, Vacuum Energy, and the Cosmological Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 10.3 Our Universe: Past, Present, and Future . . . . . . . . . . . . . . 294 10.4 Observations and Measurements . . . . . . . . . . . . . . . . . . 296 10.4.1 Runaway Universe? . . . . . . . . . . . . . . . . . . . . . 296 10.4.2 Once upon a time in a universe long long ago . . . . . . . 298 10.4.3 A cosmological ‘Problem’ . . . . . . . . . . . . . . . . . . 300 10.4.4 Looking for mass in all the wrong places . . . . . . . . . . 302 10.4.5 Putting it all together . . . . . . . . . . . . . . . . . . . . 305 10.5 The Beginning and The End . . . . . . . . . . . . . . . . . . . . 306
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6 CONTENTS<br />
8 General <strong>Relativity</strong> <strong>and</strong> Curved Spacetime 193<br />
8.1 A return to geometry . . . . . . . . . . . . . . . . . . . . . . . . . 194<br />
8.1.1 Straight Lines in Curved Space . . . . . . . . . . . . . . . 196<br />
8.1.2 Curved Surfaces are Locally Flat . . . . . . . . . . . . . . 198<br />
8.1.3 From curved space to curved spacetime . . . . . . . . . . 199<br />
8.2 More <strong>on</strong> Curved Space . . . . . . . . . . . . . . . . . . . . . . . . 200<br />
8.3 Gravity <strong>and</strong> the Metric . . . . . . . . . . . . . . . . . . . . . . . 205<br />
8.3.1 Building Intuiti<strong>on</strong> in flat space . . . . . . . . . . . . . . . 206<br />
8.3.2 On to Angles . . . . . . . . . . . . . . . . . . . . . . . . . 207<br />
8.3.3 Metrics <strong>on</strong> Curved space . . . . . . . . . . . . . . . . . . . 208<br />
8.3.4 A first example . . . . . . . . . . . . . . . . . . . . . . . . 209<br />
8.3.5 A sec<strong>on</strong>d example . . . . . . . . . . . . . . . . . . . . . . 211<br />
8.3.6 Some parting comments <strong>on</strong> metrics . . . . . . . . . . . . . 211<br />
8.4 What is the metric of spacetime? . . . . . . . . . . . . . . . . . . 213<br />
8.4.1 The Einstein equati<strong>on</strong>s . . . . . . . . . . . . . . . . . . . . 213<br />
8.4.2 The Newt<strong>on</strong>ian Approximati<strong>on</strong> . . . . . . . . . . . . . . . 215<br />
8.4.3 The Schwarzschild Metric . . . . . . . . . . . . . . . . . . 215<br />
8.5 Experimental Verificati<strong>on</strong> of GR . . . . . . . . . . . . . . . . . . 217<br />
8.5.1 The planet Mercury . . . . . . . . . . . . . . . . . . . . . 217<br />
8.5.2 The Bending of Starlight . . . . . . . . . . . . . . . . . . 219<br />
8.5.3 Other experiments: Radar Time Delay . . . . . . . . . . . 220<br />
8.6 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . 221<br />
9 Black Holes 227<br />
9.1 Investigating the Schwarzschild Metric . . . . . . . . . . . . . . . 227<br />
9.1.1 Gravitati<strong>on</strong>al Time Dilati<strong>on</strong> from the Metric . . . . . . . 228<br />
9.1.2 Correcti<strong>on</strong>s to Newt<strong>on</strong>’s Law . . . . . . . . . . . . . . . . 228<br />
9.2 On Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230<br />
9.2.1 Forming a black hole . . . . . . . . . . . . . . . . . . . . . 230<br />
9.2.2 Matter within the Schwarzschild radius . . . . . . . . . . 231<br />
9.2.3 The Schwarzschild radius <strong>and</strong> the Horiz<strong>on</strong> . . . . . . . . . 232<br />
9.2.4 Going Bey<strong>on</strong>d the Horiz<strong>on</strong> . . . . . . . . . . . . . . . . . 234<br />
9.2.5 A summary of where we are . . . . . . . . . . . . . . . . . 236<br />
9.3 Bey<strong>on</strong>d the Horiz<strong>on</strong> . . . . . . . . . . . . . . . . . . . . . . . . . 239<br />
9.3.1 The interior diagram . . . . . . . . . . . . . . . . . . . . . 239<br />
9.3.2 The Singularity . . . . . . . . . . . . . . . . . . . . . . . . 244<br />
9.3.3 Bey<strong>on</strong>d the Singularity? . . . . . . . . . . . . . . . . . . . 246<br />
9.3.4 The rest of the diagram <strong>and</strong> dynamical holes . . . . . . . 246<br />
9.3.5 Visualizing black hole spacetimes . . . . . . . . . . . . . . 249<br />
9.4 Stretching <strong>and</strong> Squishing . . . . . . . . . . . . . . . . . . . . . . . 252<br />
9.4.1 The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 254<br />
9.4.2 The soluti<strong>on</strong> . . . . . . . . . . . . . . . . . . . . . . . . . 257<br />
9.4.3 The Differential equati<strong>on</strong> . . . . . . . . . . . . . . . . . . 258<br />
9.4.4 What does it all mean? . . . . . . . . . . . . . . . . . . . 259<br />
9.4.5 Black Holes <strong>and</strong> the Schwarzschild Metric . . . . . . . . . 260<br />
9.5 Black Hole Astrophysics <strong>and</strong> Observati<strong>on</strong>s . . . . . . . . . . . . . 261
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