Notes on Relativity and Cosmology for PHY312 Donald Marolf Physics Department, Syracuse University c January 2003 2 Contents Table of Contents . .2 Preface 9 A letter of introduction . .9 How to use these notes . 10 Some parting comments . 11 credits . 12 Syllabus 13 0.1 Introduction . 13 0.1.1 A Small Warning . 14 0.1.2 Course Goals . 14 0.1.3 Course Objectives . 14 0.2 Administrative Info . 15 0.3 Coursework and Grading . 16 0.3.1 Creating your Project . 20 0.3.2 Project Checklist . 20 0.4 Some Suggestions for Further Reading . 22 1 Space, Time, and Newtonian Physics 25 1.1 Coordinate Systems . 26 1.2 Reference Frames . 28 1.3 Newtonian Assumptions about Space and Time . 31 1.4 Newtonian addition of velocities? . 33 1.5 Newton's Laws . 34 1.6 How can you tell if an object is in an inertial frame? . 35 1.7 Newton's other Laws . 36 1.8 Homework Problems . 38 2 Maxwell, E&M, and the Ether 43 2.1 The Basics of E & M . 43 2.1.1 Maxwell's Equations and Electromagnetic Waves . 45 2.2 The elusive ether . 48 2.2.1 The Aberration of Light . 49 2.2.2 Michelson, Morely, and their experiment . 51 3 4 CONTENTS 2.3 Homework Problems . 53 3 Einstein and Inertial Frames 55 3.1 The Postulates of Relativity . 55 3.2 Time and Position, take II . 56 3.3 Simultaneity . 58 3.4 Relations between events in Spacetime . 61 3.5 Time Dilation . 65 3.5.1 Rods in the perpendicular direction . 66 3.5.2 Light Clocks and Reference Frames . 68 3.5.3 Proper Time . 70 3.5.4 Why should you believe all of this? . 71 3.6 Length Contraction . 72 3.7 The Train Paradox . 74 3.8 Homework Problems . 77 4 Minkowskian Geometry 85 4.1 Minkowskian Geometry . 87 4.1.1 Invariants: Distance vs. the Interval . 87 4.1.2 Curved lines and accelerated objects . 90 4.2 The Twin paradox . 91 4.3 More on Minkowskian Geometry . 98 4.3.1 Drawing proper time and proper distance . 98 4.3.2 Changing Reference Frames . 99 4.3.3 Hyperbolae, again . 101 4.3.4 Boost Parameters and Hyperbolic Trigonometry . 102 4.4 2+1 dimensions: Aberration . 108 4.4.1 Stellar Aberration in Relativity . 108 4.4.2 More on boosts and the 2+1 light cone: the headlight effect112 4.4.3 Multiple boosts in 2+1 dimensions . 115 4.4.4 Other effects . 117 4.5 Homework Problems . 117 5 Accelerating Reference Frames . 121 5.1 The Uniformly Accelerating Worldline . 121 5.1.1 Defining uniform acceleration . 122 5.1.2 Uniform Acceleration and Boost Parameters . 124 5.1.3 Finding the Worldline . 126 5.2 The uniformly accelerated frame . 128 5.2.1 Horizons and Simultaneity . 128 5.2.2 Friends on a Rope . 131 5.2.3 The Long Rocket . 133 5.3 Homework Problems . 135 CONTENTS 5 6 Dynamics: Energy and ... 143 6.1 Dynamics, or, \Whatever happened to Forces?" . 143 6.2 Fields, Energy, and Momentum . 145 6.2.1 A word on Energy (E) . 145 6.2.2 A few words on Momentum (P) . 145 6.3 On to relativity . 147 6.3.1 Lasers in a box . 148 6.3.2 Center of Mass . 149 6.3.3 Mass vs. Energy . 149 6.3.4 Mass, Energy, and Inertia . 151 6.4 More on Mass, Energy, and Momentum . 153 6.4.1 Energy and Rest Mass . 153 6.4.2 Momentum and Mass . 154 6.4.3 How about an example? . 156 6.5 Energy and Momentum for Light . 157 6.5.1 Light speed and massless objects . 157 6.5.2 Another look at the Doppler effect . 157 6.6 Deriving the expressions . 159 6.7 Homework Problems . 164 7 Relativity and the Gravitational Field 167 7.1 The Gravitational Field . 168 7.1.1 Newtonian Gravity vs. relativity . 168 7.1.2 The importance of the field . 169 7.2 Some observations . 170 7.2.1 Free Fall . 171 7.2.2 The 2nd ingredient: The effects of gravity on light . 172 7.2.3 Gravity, Light, Time, and all that . 174 7.2.4 Gravity and Accelerating Frames . 176 7.3 The Equivalence Principle . 176 7.3.1 Gravity and Locality . 176 7.3.2 How Local? . 178 7.4 Going beyond locality . 180 7.4.1 A Tiny Tower . 182 7.4.2 The tall tower . 184 7.4.3 Gravitational time dilation near the earth . 184 7.4.4 The Global Positioning System . 186 7.5 The moral of the story . 187 7.5.1 Local frames vs. Global frames . 187 7.5.2 And what about the speed of light? . 189 7.6 Homework Problems . 189 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.