Notes on Relativity and Cosmology - Physics Department, UCSB
Notes on Relativity and Cosmology - Physics Department, UCSB Notes on Relativity and Cosmology - Physics Department, UCSB
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
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4 CONTENTS<br />
2.3 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . 53<br />
3 Einstein <strong>and</strong> Inertial Frames 55<br />
3.1 The Postulates of <strong>Relativity</strong> . . . . . . . . . . . . . . . . . . . . . 55<br />
3.2 Time <strong>and</strong> Positi<strong>on</strong>, take II . . . . . . . . . . . . . . . . . . . . . . 56<br />
3.3 Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58<br />
3.4 Relati<strong>on</strong>s between events in Spacetime . . . . . . . . . . . . . . . 61<br />
3.5 Time Dilati<strong>on</strong> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65<br />
3.5.1 Rods in the perpendicular directi<strong>on</strong> . . . . . . . . . . . . 66<br />
3.5.2 Light Clocks <strong>and</strong> Reference Frames . . . . . . . . . . . . . 68<br />
3.5.3 Proper Time . . . . . . . . . . . . . . . . . . . . . . . . . 70<br />
3.5.4 Why should you believe all of this? . . . . . . . . . . . . . 71<br />
3.6 Length C<strong>on</strong>tracti<strong>on</strong> . . . . . . . . . . . . . . . . . . . . . . . . . . 72<br />
3.7 The Train Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 74<br />
3.8 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . 77<br />
4 Minkowskian Geometry 85<br />
4.1 Minkowskian Geometry . . . . . . . . . . . . . . . . . . . . . . . 87<br />
4.1.1 Invariants: Distance vs. the Interval . . . . . . . . . . . . 87<br />
4.1.2 Curved lines <strong>and</strong> accelerated objects . . . . . . . . . . . . 90<br />
4.2 The Twin paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 91<br />
4.3 More <strong>on</strong> Minkowskian Geometry . . . . . . . . . . . . . . . . . . 98<br />
4.3.1 Drawing proper time <strong>and</strong> proper distance . . . . . . . . . 98<br />
4.3.2 Changing Reference Frames . . . . . . . . . . . . . . . . . 99<br />
4.3.3 Hyperbolae, again . . . . . . . . . . . . . . . . . . . . . . 101<br />
4.3.4 Boost Parameters <strong>and</strong> Hyperbolic Trig<strong>on</strong>ometry . . . . . 102<br />
4.4 2+1 dimensi<strong>on</strong>s: Aberrati<strong>on</strong> . . . . . . . . . . . . . . . . . . . . . 108<br />
4.4.1 Stellar Aberrati<strong>on</strong> in <strong>Relativity</strong> . . . . . . . . . . . . . . . 108<br />
4.4.2 More <strong>on</strong> boosts <strong>and</strong> the 2+1 light c<strong>on</strong>e: the headlight effect112<br />
4.4.3 Multiple boosts in 2+1 dimensi<strong>on</strong>s . . . . . . . . . . . . . 115<br />
4.4.4 Other effects . . . . . . . . . . . . . . . . . . . . . . . . . 117<br />
4.5 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . 117<br />
5 Accelerating Reference Frames ... 121<br />
5.1 The Uniformly Accelerating Worldline . . . . . . . . . . . . . . . 121<br />
5.1.1 Defining uniform accelerati<strong>on</strong> . . . . . . . . . . . . . . . . 122<br />
5.1.2 Uniform Accelerati<strong>on</strong> <strong>and</strong> Boost Parameters . . . . . . . . 124<br />
5.1.3 Finding the Worldline . . . . . . . . . . . . . . . . . . . . 126<br />
5.2 The uniformly accelerated frame . . . . . . . . . . . . . . . . . . 128<br />
5.2.1 Horiz<strong>on</strong>s <strong>and</strong> Simultaneity . . . . . . . . . . . . . . . . . . 128<br />
5.2.2 Friends <strong>on</strong> a Rope . . . . . . . . . . . . . . . . . . . . . . 131<br />
5.2.3 The L<strong>on</strong>g Rocket . . . . . . . . . . . . . . . . . . . . . . . 133<br />
5.3 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . 135
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