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"Christoph Schiller MOTION MOUNTAIN the adventure of physics relativity www.motionmountain.eu Christoph Schiller Motion Mountain The Adventure of Physics Relativity available free of charge at www.motionmountain.eu Editio vicesima secunda. Proprietas scriptoris © Christophori Schiller secundo anno Olympiadis vicesimae nonae. Omnia proprietatis iura reservantur et vindicantur. Imitatio prohibita sine auctoris permissione. Non licet pecuniam expetere pro aliquo, quod partem horum verborum continet; liber pro omnibus semper gratuitus erat et manet. Twenty-second edition, second printing, ISBN 978-300-021946-7. Copyright © 2009 by Christoph Schiller, the second year of the 29th Olympiad. This pdf file is licensed under the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Germany Licence, whose full text can be found on the website creativecommons.org/licenses/by-nc-nd/3.0/de, with the additional restriction that reproduction, distribution and use, in whole or in part, in any product or service, be it commercial or not, is not allowed without the written consent of the copyright owner. The pdf file was and remains free for everybody to read, store and print for personal use, and to distribute electronically, but only in unmodified form and at no charge. To Esther and Britta τῷ ἐμοὶ δαὶμονι Die Menschen stärken, die Sachen klären. Preface The present overview of relativity is the result of a threefold aim I have pursued since 1990: to present the basics of motion in a way that is simple, up to date and vivid. In the structure of modern physics, shown in Figure 1, relativity covers the two leftmost points. In order to be simple, the text focuses on concepts, while keeping mathematics to the necessary minimum. Understanding the concepts of physics is given precedence over using formulae in calculations. The whole text is within the reach of an undergraduate. Special relativity limits speeds by v ⩽ c; general relativity limits force by F ⩽ c 4 /4G. It is shown that within each domain, the principal equations follow from these limits. Basing the domains of physics on limit principles allows them to be introduced in a simple, rapid and intuitive way. This unusual way of learning physics should reward the curiosity of every reader – whether student or professional. In order to be up to date, the text is enriched by the many gems – both theoretical and empirical – that are scattered throughout the scientific literature. In order to be vivid, the text tries to startle the reader as much as possible. Reading a book on general physics should be like going to a magic show. We watch, we are astonished, we do not believe our eyes, we think, and finally – maybe – we understand the trick. When we look at nature, we often have the same experience. The text tries to intensify this by following a simple rule: on each page, there should be at least one surprise or provocation for the reader to think about. Numerous interesting challenges are proposed. Hints or answers to these are given in an appendix. Giving full rein to one’s curiosity and thought leads to the development of a strong and dependable character. The motto of the text, die Menschen stärken, die Sachen klären, a famous statement by Hartmut von Hentig on pedagogy, translates as: ‘To clarify things, to fortify people.’ Exploring any limit requires courage; and courage is also needed to abandon space and time as tools for the description of the world. Changing habits of thought produces fear, often hidden by anger; but we grow by overcoming our fears. The great adventures in life allow this: exploring love is one, exploring physics is another. Eindhoven and other places, 8 January 2009 “ Primum movere, deinde docere.* Antiquity ” Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Copyright © Christoph Schiller November 1997–January 2009 Advice for learners In my experience as a teacher, there was one learning method that never failed to transform unsuccessful pupils into successful ones: if you read a book for study, summarize every section you read, in your own words, aloud. If you are unable to do so, read the * ‘First move, then teach.’ In modern languages, the mentioned type of moving (the heart) is often called motivating; both terms go back to the same Latin root. 8 preface PHYSICS: Describing motion with action. (Unified) theory of motion Adventures: understanding everything, intense fun with thinking, catching a glimpse of bliss What are space, time and quantum particles? General relativity Adventures: the night sky, measuring curved space, exploring black holes and the universe, space and time How do everyday, fast and large things move? Quantum theory with gravity Adventures: bouncing neutrons, understanding tree growth Quantum field theory Adventures: building accelerators, understanding quarks, stars, bombs and the basis of life, matter, radiation How do small things move? What are things? Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Classical gravity Adventures: climbing, skiing, space travel, the wonders of astronomy and geology Special relativity Adventures: understanding time dilation, length contraction and E=mc2 c h, e, k G Quantum theory Adventures: death, sexuality, biology, enjoying art, colours in nature, all high-tech business, medicine, chemistry, evolution Galilean physics, electricity and heat Adventures: sport, music, sailing, cooking, using electricity and computers, understanding the brain and people F I G U R E 1 A complete map of physics: the connections are deﬁned by the speed of light c, the gravitational constant G, the Planck constant h, the Boltzmann constant k and the elementary charge e. Copyright © Christoph Schiller November 1997–January 2009 section again. Repeat this until you can clearly summarize what you read in your own words, aloud. You can do this alone in a room, or with friends, or while walking. If you do this with everything you read, you will reduce your learning and reading time significantly. In addition, you will enjoy learning from good texts much more and hate bad texts much less. Masters of the method can use it even while listening to a lecture, in a low voice, thus avoiding to ever take notes. Using this book Text in green, as found in many marginal notes, is a link that can be clicked in a pdf reader. Green links can be bibliographic references, footnotes, cross references to other preface 9 pages, challenge solutions or URLs of other websites. Solutions and hints for challenges are given at the end of each part. Challenges are classified as research level (r), difficult (d), standard student level (s) and easy (e). Challenges of type r, d or s for which no solution has yet been included in the book are marked (ny). A request The text is and will remain free to download from the internet. In exchange, please send me a short email on the following issues: — What was unclear? — What story, topic, riddle, picture or movie did you miss? — What should be improved or corrected? Feedback on the specific points listed on the www.motionmountain.eu/help.html web page is most welcome of all. You can also add feedback directly to www.motionmountain. eu/wiki. On behalf of all other readers, thank you in advance for your input. For a particularly useful contribution you will be mentioned – if you want – in the acknowledgements, receive a reward, or both. But above all, enjoy the reading. Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Challenge 1 s Copyright © Christoph Schiller November 1997–January 2009 R elativity In our quest to learn how things move, the experience of hiking and other motion leads us to discover that there is a maximum speed in nature, and that two events that happen at the same time for one observer may not for another. We discover that empty space can bend, wobble and move, we find that there is a maximum force in nature, and we understand why we can see the stars. Contents 15 1 Maximum speed, observers at rest, and motion of light Can one play tennis using a laser pulse as the ball and mirrors as rackets? 20 • Albert Einstein 21 • The speed of light as an invariant limit speed 22 • Special relativity with a few lines 24 • Acceleration of light and the Doppler effect 26 • The difference between light and sound 28 • Can one shoot faster than one’s shadow? 29 • The composition of velocities 31 • Observers and the principle of special relativity 32 • What is space-time? 36 • Can we travel to the past? – Time and causality 37 Curiosities about special relativity Faster than light: how far can we travel? 39 • Synchronization and time travel – can a mother stay younger than her own daughter? 39 • Length contraction 42 • Relativistic films – aberration and Doppler effect 44 • Which is the best seat in a bus? 45 • How fast can one walk? 48 • Is the speed of shadow greater than the speed of light? 48 • Parallel to parallel is not parallel – Thomas rotation 51 • A never-ending story – temperature and relativity 52 Relativistic mechanics Mass in relativity 52 • Why relativistic snooker is more difficult 55 • Mass is concentrated energy 55 • Collisions, virtual objects and tachyons 58 • Systems of particles – no centre of mass 60 • Why is most motion so slow? 61 • The history of the mass–energy equivalence formula of De Pretto and Einstein 61 • 4-vectors 62 • 4-momentum 65 • 4-force 66 • Rotation in relativity 67 • Wave motion 68 • The action of a free particle – how do things move? 69 • Conformal transformations – why is the speed of light constant? 70 Accelerating observers Acceleration for inertial observers 73 • Accelerating frames of reference 74 • Event horizons 78 • Acceleration changes colours 79 • Can light move faster than c? 80 • What is the speed of light? 81 • Limits on the length of solid bodies 82 Special relativity in four sentences Could the speed of light vary? 83 • What happens near the speed of light? 84 General rel ativit y: gravitation, maximum speed and maximum force Maximum force – general relativity in one statement 86 • The force and power limits 87 • The experimental evidence 89 • Deducing general relativity 90 • Spacetime is curved 95 • Conditions of validity of the force and power limits 96 • Gedanken experiments and paradoxes about the force limit 97 • Gedanken experiments with the power limit and the mass flow limit 102 • Hide and seek 105 • An intuitive understanding of general relativity 105 • An intuitive understanding of cosmology 108 • Experimental challenges for the third millennium 109 • A summary of general relativity 110 • Acknowledgement 111 The new ideas on space, time and gravit y Rest and free fall 112 • What is gravity? – A second answer 113 • What tides tell us about gravity 116 • Bent space and mattresses 118 • Curved space-time 120 • The speed of light and the gravitational constant 122 • Why does a stone thrown into the air fall back to Earth? – Geodesics 123 • Can light fall? 126 • Curiosities and fun challenges about gravitation 127 • What is weight? 132 • Why do apples fall? 132 Motion in general rel ativit y – bent light and wobbling vacuum 39 52 72 83 85 2 112 3 134 4 contents 134 13 154 162 Weak fields The Thirring effects 134 • Gravitomagnetism 136 • Gravitational waves 139 • Bending of light and radio waves 147 • Time delay 149 • Effects on orbits 149 • The geodesic effect 152 • Curiosities and fun challenges about weak fields 154 How is curvature measured? Curvature and space-time 158 • Curvature and motion in general relativity 159 • Universal gravity 160 • The Schwarzschild metric 161 • Curiosities and fun challenges about curvature 161 All observers – heavier mathematics The curvature of space-time 162 • The description of momentum, mass and energy 163 • Hilbert’s action – how things fall? 165 • The symmetries of general relativity 166 • Einstein’s field equations 166 • More on the force limit 169 • Deducing universal gravity 170 • Deducing linearized general relativity 170 • How to calculate the shape of geodesics 171 • Mass in general relativity 172 • Is gravity an interaction? 173 • The essence of general relativity 174 • Riemann gymnastics 175 • Curiosities and fun challenges about general relativity 177 5 Why can we see the stars? – Motion in the universe Which stars do we see? 178 • What do we see at night? 180 • What is the universe? 185 • The colour and the motion of the stars 186 • Do stars shine every night? 189 • A short history of the universe 191 • The history of space-time 194 • Why is the sky dark at night? 199 • Is the universe open, closed or marginal? 201 • Why is the universe transparent? 202 • The big bang and its consequences 203 • Was the big bang a big bang? 204 • Was the big bang an event? 204 • Was the big bang a beginning? 204 • Does the big bang imply creation? 205 • Why can we see the Sun? 206 • Why are the colours of the stars different? 207 • Are there dark stars? 208 • Are all stars different? – Gravitational lenses 208 • What is the shape of the universe? 210 • What is behind the horizon? 211 • Why are there stars all over the place? – Inflation 212 • Why are there so few stars? – The energy and entropy content of the universe 212 • Why is matter lumped? 214 • Why are stars so small compared with the universe? 214 • Are stars and galaxies moving apart or is the universe expanding? 214 • Is there more than one universe? 214 • Why are the stars fixed? – Arms, stars and Mach’s principle 215 • At rest in the universe 216 • Does light attract light? 216 • Does light decay? 217 Bl ack holes – falling forever Why study black holes? 218 • Horizons 218 • Orbits 221 • Hair and entropy 223 • Black holes as energy sources 225 • Curiosities and fun challenges about black holes 227 • Formation of and search for black holes 230 • Singularities 231 • A quiz – is the universe a black hole? 232 D oes space differ from time? Can space and time be measured? 234 • Are space and time necessary? 235 • Do closed timelike curves exist? 236 • Is general relativity local? – The hole argument 236 • Is the Earth hollow? 237 • Are space, time and mass independent? 238 General rel ativit y in ten points – a summary for the l ayman The accuracy of the description 241 • Research in general relativity and cosmology 242 • Could general relativity be different? 244 • The limits of general relativity 245 Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu 178 218 6 Copyright © Christoph Schiller November 1997–January 2009 233 7 240 8 246 Biblio graphy Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Copyright © Christoph Schiller November 1997–January 2009 contents Challenge hints and solu tions 269 276 14 Credits Image credits 276 Chapter 1 MA XIMUM SPEED, OBSERVERS AT R EST, AND MOTION OF LIGHT L Page ?? Ref. 1 Challenge 3 s ight is indispensable for a precise description of motion. To check whether a ine or a path of motion is straight, we must look along it. In other words, we use ight to define straightness. How do we decide whether a plane is flat? We look across it,** again using light. How do we measure length to high precision? With light. How do we measure time to high precision? With light: once it was light from the Sun that was used; nowadays it is light from caesium atoms. Light is important because it is the standard for undisturbed motion. Physics would have evolved much more rapidly if, at some earlier time, light propagation had been recognized as the ideal example of motion. But is light really a phenomenon of motion? Yes. This was already known in ancient Greece, from a simple daily phenomenon, the shadow. Shadows prove that light is a moving entity, emanating from the light source, and moving in straight lines.*** The Greek thinker Empedocles (c. 490 to c. 430 bce ) drew the logical conclusion that light takes a certain amount of time to travel from the source to the surface showing the shadow. Empedocles stated that the speed of light is finite. We can confirm this result with a different, equally simple, but subtle argument. Speed can be measured. And measurement is comparison with a standard. Therefore the perfect speed, which is used as the implicit measurement standard, must have a finite value. An infinite velocity standard would not allow measurements at all. In nature, the lightest entities move with the highest speed. * ‘Nothing is faster than rumour.’ This common sentence is a simplified version of Virgil’s phrase: fama, malum qua non aliud velocius ullum. ‘Rumour, the evil faster than all.’ From the Aeneid, book IV, verses 173 and 174. ** Note that looking along the plane from all sides is not sufficient for this check: a surface that a light beam touches right along its length in all directions does not need to be flat. Can you give an example? One needs other methods to check flatness with light. Can you specify one? *** Whenever a source produces shadows, the emitted entities are called rays or radiation. Apart from light, other examples of radiation discovered through shadows were infrared rays and ultraviolet rays, which emanate from most light sources together with visible light, and cathode rays, which were found to be to the motion of a new particle, the electron. Shadows also led to the discovery of X-rays, which again turned out to be a version of light, with high frequency. Channel rays were also discovered via their shadows; they turn out to be travelling ionized atoms. The three types of radioactivity, namely α-rays (helium nuclei), β-rays (again electrons), and γ-rays (high-energy X-rays) also produce shadows. All these discoveries were made between 1890 and 1910: those were the ‘ray days’ of physics. “ Fama nihil est celerius.* Antiquity ” Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Copyright © Christoph Schiller November 1997–January 2009 Challenge 2 s 16 1 maximum speed, observers at rest, and motion of light Jupiter and Io (second measurement) Earth (second measurement) F I G U R E 2 Rømer’s Sun Earth (first measurement) Jupiter and Io (first measurement) method of measuring the speed of light rain's perspective rain light's perspective light wind’s perspective wind Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu c c earth v Sun windsurfer c v v walker’s perspective human perspective windsurfer’s perspective α c c α v F I G U R E 3 The rainwalker’s or windsurfer’s method of measuring the speed of light α c Sun v v Copyright © Christoph Schiller November 1997–January 2009 Light, which is indeed extremely light, is an obvious candidate for motion with perfect but finite speed. We will confirm this in a minute. A finite speed of light means that whatever we see is a message from the past. When we see the stars,* the Sun or a person we love, we always see an image of the past. In a sense, nature prevents us from enjoying the present – we must therefore learn to enjoy the past. The speed of light is high; therefore it was not measured until the years 1668 to 1676, even though many, including Galileo, had tried to do so earlier. The first measurement * The photograph of the night sky and the milky way, on page 11 is copyright Anthony Ayiomamitis and is found..."

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