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197 Pages·2016·11.52 MB·English

2
Relativity for Poets
Benjamin Crowell
www.lightandmatter.com
Fullerton, California
www.lightandmatter.com
Copyright (cid:13)c 2015 Benjamin Crowell
rev. March 4, 2016
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Contents
I Space and time 9
1 Galilean relativity 11
1.1 Galileo versus Aristotle . . . . . . . . . . . . . . . 11
1.2 Frames of reference . . . . . . . . . . . . . . . . . . 16
1.3 Inertial and noninertial frames . . . . . . . . . . . 16
1.4 Circular motion . . . . . . . . . . . . . . . . . . . . 18
1.5 Addition of velocities . . . . . . . . . . . . . . . . . 19
1.6 The Galilean twin paradox . . . . . . . . . . . . . 21
1.7 The Galilean transformation . . . . . . . . . . . . . 22
1.8 Science and society: The Galileo affair . . . . . . . 23
2 Einstein’s relativity 27
2.1 Time is relative . . . . . . . . . . . . . . . . . . . . 27
2.2 The principle of greatest time . . . . . . . . . . . . 36
2.3 A universal speed limit . . . . . . . . . . . . . . . . 38
2.4 Einstein’s train . . . . . . . . . . . . . . . . . . . . 40
2.5 Velocities don’t simply add . . . . . . . . . . . . . 40
2.6 The light cone. . . . . . . . . . . . . . . . . . . . . 42
2.7 Science and society: Scientific revolutions . . . . . 44
3 The Lorentz transformation 47
3.1 Surveying space and time . . . . . . . . . . . . . . 47
3.2 The Lorentz transformation . . . . . . . . . . . . . 48
3.3 Correspondence principle . . . . . . . . . . . . . . 50
3.4 Scale, squish, stretch . . . . . . . . . . . . . . . . . 50
5
6 CONTENTS
3.5 No acceleration past c . . . . . . . . . . . . . . . . 51
3.6 Time dilation . . . . . . . . . . . . . . . . . . . . . 52
3.7 Length contraction . . . . . . . . . . . . . . . . . . 54
3.8 Not what you see . . . . . . . . . . . . . . . . . . . 55
3.9 Causality . . . . . . . . . . . . . . . . . . . . . . . 57
3.10 Doppler shift . . . . . . . . . . . . . . . . . . . . . 58
3.11 Optional: Proof that area stays the same . . . . . 61
4 The measure of all things 63
4.1 Operationalism . . . . . . . . . . . . . . . . . . . . 63
4.2 Galilean scalars and vectors . . . . . . . . . . . . . 64
4.3 Relativistic vectors and scalars . . . . . . . . . . . 65
4.4 Measuring relativistic vectors . . . . . . . . . . . . 66
II Matter and E = mc2 69
5 Light and matter 71
5.1 How much does matter really matter? . . . . . . . 71
5.2 How hard, how loud? . . . . . . . . . . . . . . . . . 72
5.3 Light . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 E = mc2 77
6.1 Energy shift of a beam of light . . . . . . . . . . . 77
6.2 Einstein’s argument . . . . . . . . . . . . . . . . . 79
6.3 Correspondence principle . . . . . . . . . . . . . . 81
6.4 Mass to energy and energy to mass . . . . . . . . . 82
6.5 The energy-momentum vector . . . . . . . . . . . . 83
6.6 Tachyons . . . . . . . . . . . . . . . . . . . . . . . 88
III Gravity 91
7 Newtonian gravity 93
7.1 Galileo and free fall. . . . . . . . . . . . . . . . . . 93
7.2 Orbits . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.3 Newton’s laws . . . . . . . . . . . . . . . . . . . . . 98
CONTENTS 7
7.4 Tidal forces . . . . . . . . . . . . . . . . . . . . . . 100
8 The equivalence principle 103
8.1 The equivalence principle . . . . . . . . . . . . . . 103
8.2 Chicken and egg . . . . . . . . . . . . . . . . . . . 105
8.3 Apparent weightlessness . . . . . . . . . . . . . . . 107
8.4 Locality . . . . . . . . . . . . . . . . . . . . . . . . 109
8.5 Gravitational time dilation . . . . . . . . . . . . . 110
8.6 Time dilation with signals . . . . . . . . . . . . . . 112
9 Curvature of spacetime 117
9.1 What is straight? . . . . . . . . . . . . . . . . . . . 117
9.2 The parable of the bugs . . . . . . . . . . . . . . . 120
9.3 Gravity as curvature . . . . . . . . . . . . . . . . . 122
9.4 Deflection of starlight . . . . . . . . . . . . . . . . 123
9.5 No extra dimensions . . . . . . . . . . . . . . . . . 125
9.6 Parallel transport . . . . . . . . . . . . . . . . . . . 126
10 Matter 129
10.1 No nongravitating matter . . . . . . . . . . . . . . 130
10.2 No repulsive gravity . . . . . . . . . . . . . . . . . 132
10.3 Finite strength of materials . . . . . . . . . . . . . 134
11 Black holes 137
11.1 Collapse to a singularity . . . . . . . . . . . . . . . 137
11.2 The event horizon . . . . . . . . . . . . . . . . . . 140
11.3 Detection . . . . . . . . . . . . . . . . . . . . . . . 142
11.4 Light cones . . . . . . . . . . . . . . . . . . . . . . 143
11.5 Penrose diagrams . . . . . . . . . . . . . . . . . . . 146
11.6 Tests of general relativity in weaker fields . . . . . 150
11.7 Speed of light . . . . . . . . . . . . . . . . . . . . . 151
12 Waves 155
12.1 No action at a distance . . . . . . . . . . . . . . . 155
12.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.3 Electromagnetic waves . . . . . . . . . . . . . . . . 158
12.4 Gravitational waves . . . . . . . . . . . . . . . . . 160
8 CONTENTS
IV Cosmology 163
13 For dust thou art 165
13.1 The Einstein field equation . . . . . . . . . . . . . 165
13.2 Conservation . . . . . . . . . . . . . . . . . . . . . 168
13.3 Change or changelessness? . . . . . . . . . . . . . . 170
13.4 Homogeneity and isotropy . . . . . . . . . . . . . . 172
13.5 Hubble’s law . . . . . . . . . . . . . . . . . . . . . 174
13.6 The cosmic microwave background . . . . . . . . . 176
13.7 A hot big bang . . . . . . . . . . . . . . . . . . . . 177
13.8 A singularity at the big bang . . . . . . . . . . . . 178
13.9 A cosmic calendar . . . . . . . . . . . . . . . . . . 183
14 Dark matter and dark energy 185
14.1 A difficult census . . . . . . . . . . . . . . . . . . . 185
14.2 Dark matter . . . . . . . . . . . . . . . . . . . . . . 186
14.3 Dark energy . . . . . . . . . . . . . . . . . . . . . . 188
Part I
Space and time
9