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This is a set of lecture notes for Ben Crowell's course Relativity for Poets at Fullerton College. It's a nonmathematical presentation of Einstein's theories of special and general relativity, including a brief treatment of cosmology.

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  • 2

  • Relativity for PoetsBenjamin Crowellwww.lightandmatter.com

  • Fullerton, Californiawww.lightandmatter.com

    Copyright c2015 Benjamin Crowell

    rev. December 5, 2015

    Permission is granted to copy, distribute and/or modify this doc-ument under the terms of the Creative Commons AttributionShare-Alike License, which can be found at creativecommons.org.The license applies to the entire text of this book, plus all theillustrations that are by Benjamin Crowell. All the illustrationsare by Benjamin Crowell except as noted in the photo creditsor in parentheses in the caption of the figure. This book canbe downloaded free of charge from www.lightandmatter.com ina variety of formats, including editable formats.

  • 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 Einsteins 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 Einsteins train . . . . . . . . . . . . . . . . . . . . 40

    2.5 Velocities dont 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 Einsteins 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 Newtons laws . . . . . . . . . . . . . . . . . . . . . 98

  • CONTENTS 7

    7.4 Tidal forces . . . . . . . . . . . . . . . . . . . . . . 100

    8 The equivalence principle 1038.1 The equivalence principle . . . . . . . . . . . . . . 1038.2 Chicken and egg . . . . . . . . . . . . . . . . . . . 1058.3 Apparent weightlessness . . . . . . . . . . . . . . . 1078.4 Locality . . . . . . . . . . . . . . . . . . . . . . . . 1098.5 Gravitational time dilation . . . . . . . . . . . . . 1108.6 Time dilation with signals . . . . . . . . . . . . . . 112

    9 Curvature of spacetime 1179.1 What is straight? . . . . . . . . . . . . . . . . . . . 1179.2 The parable of the bugs . . . . . . . . . . . . . . . 1209.3 Gravity as curvature . . . . . . . . . . . . . . . . . 1229.4 Deflection of starlight . . . . . . . . . . . . . . . . 1239.5 No extra dimensions . . . . . . . . . . . . . . . . . 1259.6 Parallel transport . . . . . . . . . . . . . . . . . . . 126

    10 Matter 12910.1 No nongravitating matter . . . . . . . . . . . . . . 13010.2 No repulsive gravity . . . . . . . . . . . . . . . . . 13210.3 Finite strength of materials . . . . . . . . . . . . . 134

    11 Black holes 13711.1 Collapse to a singularity . . . . . . . . . . . . . . . 13711.2 The event horizon . . . . . . . . . . . . . . . . . . 14011.3 Detection . . . . . . . . . . . . . . . . . . . . . . . 14211.4 Light cones . . . . . . . . . . . . . . . . . . . . . . 14311.5 Penrose diagrams . . . . . . . . . . . . . . . . . . . 14611.6 Tests of general relativity in weaker fields . . . . . 15011.7 Speed of light . . . . . . . . . . . . . . . . . . . . . 151

    12 Waves 15512.1 No action at a distance . . . . . . . . . . . . . . . 15512.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . 15512.3 Electromagnetic waves . . . . . . . . . . . . . . . . 15812.4 Gravitational waves . . . . . . . . . . . . . . . . . 160

  • 8 CONTENTS

    IV Cosmology 163

    13 For dust thou art 16513.1 The Einstein field equation . . . . . . . . . . . . . 16513.2 Conservation . . . . . . . . . . . . . . . . . . . . . 16813.3 Change or changelessness? . . . . . . . . . . . . . . 17013.4 Homogeneity and isotropy . . . . . . . . . . . . . . 17213.5 Hubbles law . . . . . . . . . . . . . . . . . . . . . 17413.6 The cosmic microwave background . . . . . . . . . 17613.7 A hot big bang . . . . . . . . . . . . . . . . . . . . 17713.8 A singularity at the big bang . . . . . . . . . . . . 17813.9 A cosmic calendar . . . . . . . . . . . . . . . . . . 183

    14 Dark matter and dark energy 18514.1 A difficult census . . . . . . . . . . . . . . . . . . . 18514.2 Dark matter . . . . . . . . . . . . . . . . . . . . . . 18614.3 Dark energy . . . . . . . . . . . . . . . . . . . . . . 188

  • Part I

    Space and time

    9

  • Chapter 1

    Galilean relativity

    1.1 Galileo versus Aristotle

    Once upon a time, there was an ornery man who liked to argue.He was undiplomatic and had a talent for converting allies intoenemies. His name was Galileo, and it was his singular misfor-tune to be correct in most of his opinions. When he was arguing,Galileo had a few annoying habits. One was to answer a perfectlysound theoretical argument with a contradictory experiment orobservation. Another was that, since he knew he was right, hefreely made up descriptions of experiments that he hadnt actu-ally done.

    Galileos true opponent was a dead man, the ancient Greekphilosopher Aristotle. Aristotle, probably based on generaliza-tions from everyday experience, had come up with some seem-ingly common-sense theories about motion.

    11

  • 12 CHAPTER 1. GALILEAN RELATIVITY

    The figure on the facing page shows an example of the kindof observation that might lead you to the same conclusions asAristotle. It is a series of snapshots in the motion of a rollingball. Time moves forward as we go up the page. Because theball slows down and eventually stops, it traces out the shape ofa letter J. If you get rid of the artistic details, and connect thedrawings of the ball with a smooth curve, you get a graph, withposition and time on the horizontal and vertical axes. (If we hadwanted to, we could have interchanged the axes or reversed eitheror both of them. Making the time axis vertical, and making thetop point toward the future, is a standard convention in relativity,like the conventional orientation of the compass directions on amap.)

    Once the ball has stopped completely, the graph becomes avertical line. Time continues to flow, but the balls position isno longer changing. According to Aristotle, this is the naturalbehavior of any material object. The ball may move becausesomeone kicks it, but once the force stops, the motion goes away.In other words, vertical lines on these graphs are special. Theyrepresent a natural, universal state of rest.

    Galileo adduced two main arguments against the Aristotelianview. First, he said that there was a type of force, friction, whichwas the reason that things slowed down. The grass makes africtional force on the ball. If we let the grass grow too high,this frictional force gets bigger, and the ball decelerates morequickly. If we mow the grass, the opposite happens. Galileodid experiments in which he rolled a smooth brass ball on asmooth ramp, in order to make the frictional force as small aspossible. He carried out careful quantitative observations of theballs motion, with time measured using a primitive water clock.By taking measurements under a variety of conditions, he showedby extrapolation from his data that if the ramp was perfectlylevel, and friction completely absent, the ball would roll forever.

    Today we have easier ways to convince ourselves of the sameconclusion. For example, youve seen a puck glide frictionlesslyacross an air hockey table without slowing down. Sometimes

  • 1.1. GALILEO VERSUS ARISTOTLE 13

  • 14 CHAPTER 1. GALILEAN RELATIVITY

    Galileos analysis may be hard to accept. Running takes us a lotof effort,