Notes on Relativity and Cosmology - Physics Department, UCSB

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