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"Christoph Schiller MOTION MOUNTAIN the adventure of physics motion without motion www.motionmountain.eu Christoph Schiller Motion Mountain The Adventure of Physics Motion Without Motion 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 everyday physics 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 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. This is especially important when discussing the contradictions between quantum theory and general relativity. These contradictions require resolution. This requirement leads us to the ultimate adventure in physics: finding the unified description of motion. It corresponds to the highest point in Figure 1. Researchers have not yet completed this task. Nevertheless, those parts which have already been completed are breathtaking enough to be worthwhile exploring. For example, space and time are not continuous, matter and vacuum do not differ, and points do not exist. 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. Research on unification has shown that matter cannot be distinguished from empty space at highest energies, and that space is not continuous. Research has also shown that particles are not points, but extended objects. 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 * ‘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 Quantum theory Adventures: death, sexuality, biology, enjoying art, colours in nature, all high-tech business, medicine, chemistry, evolution G 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 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 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. preface 9 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 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 Motion Withou t Motion: What Are Space, Time and Particles? Where through the combination of quantum mechanics and general relativity, the top of Motion Mountain is reached, and it is discovered that vacuum is indistinguishable from matter, that space, time and mass are easily confused, that there is no difference between the very large and the very small, and that a complete description of motion is possible. (Well, wait a few more years for the last line.) Contents 14 16 1 General rel ativit y versus quantum theory The contradictions 14 Does matter differ from vacuum? Planck scales 17 • Farewell to instants of time 19 • Farewell to points in space 21 • Farewell to the space-time manifold 23 • Farewell to observables and measurements 26 • Can space-time be a lattice? – A glimpse of quantum geometry 27 • Farewell to particles 28 • Farewell to mass 31 • Curiosities and fun challenges on Planck scales 34 • Farewell to the big bang 37 • The baggage left behind 38 Some experimental predictions Additional predictions 39 Nature at l arge scales – Is the universe something or nothing? Cosmological scales 43 • Maximum time 44 • Does the universe have a definite age? 45 • How precise can age measurements be? 45 • Does time exist? 47 • What is the error in the measurement of the age of the universe? 47 • Maximum length 51 • Is the universe really a big place? 51 • The boundary of space-time – is the sky a surface? 53 • Does the universe have initial conditions? 53 • Does the universe contain particles and stars? 54 • Does the universe contain masses and objects? 55 • Do symmetries exist in nature? 56 • Does the universe have a boundary? 57 • Is the universe a set? – Again 58 • Curiosities and fun challenges about the universe 59 • Hilbert’s sixth problem settled 60 • Does the universe make sense? 61 • Abandoning sets and discreteness eliminates contradictions 62 • Extremal scales and open questions in physics 63 • Is extremal identity a principle of nature? 63 The physics of love – a summary of the first five-and-a-half parts Summary on love and physics 75 Maximum force and minimum distance – physics in limit statements Fundamental limits to all observables Special relativity in one statement 77 • Quantum theory in one statement 78 • General relativity in one statement 79 • Deducing general relativity 80 • Deducing universal gravitation 83 • The size of physical systems in general relativity 83 • A mechanical analogy for the maximum force 84 Units and limit values for all physical observables Limits to space and time 86 • Mass and energy limits 87 • Virtual particles – a new definition 87 • Limits in thermodynamics 88 • Electromagnetic limits and units 88 • Vacuum and mass–energy – two sides of the same coin 89 • Curiosities and fun challenges about Planck limits 90 Upper and lower limits to observables Size and energy dependence 93 • Angular momentum, action and speed 93 • Force, power and luminosity 95 • Acceleration 95 • Momentum 96 • Lifetime, distance and curvature 96 • Mass change 96 • Mass and density 97 • The strange charm of the entropy bound 97 • Temperature 99 • Electromagnetic observables 99 • Curiosities and fun challenges about limits to observables 100 Limits to measurement precision and their challenge to thought Measurement precision and the existence of sets 102 • Why are observers 38 43 2 65 3 76 76 4 84 93 101 contents needed? 103 • A solution to Hilbert’s sixth problem 103 • Summary 104 105 5 13 108 114 116 119 122 124 125 127 132 The shape of points – extension in nature The starting point – vacuum and particles 106 • How else can we show that matter and vacuum cannot be distinguished? 107 The size and shape of elementary particles Do boxes exist? 109 • Can the Greeks help? – The limits of knives 109 • Are cross sections finite? 110 • Can one take a photograph of a point? 110 • What is the shape of an electron? 112 • Is the shape of an electron fixed? 113 • Summary of the first argument for extension 114 The shape of points in vacuum Measuring the void 115 • What is the maximum number of particles that fit inside a piece of vacuum? 115 • Summary of the second argument for extension 116 The large, the small and their connection Is small large? 117 • Unification and total symmetry 117 • Summary of the third argument for extension 119 Does nature have parts? Does the universe contain anything? 121 • An amoeba 121 • Summary of the fourth argument for extension 122 The entropy of black holes Summary of the fifth argument for extension 124 Exchanging space points or particles at Planck scales Summary of the sixth argument for extension 125 The meaning of spin Summary of the seventh argument for extension 127 Current research Conceptual checks of extension 128 • Experimental falsification of models based on extended entities 128 • Possibilities for confirmation of extension models 129 • Curiosities and fun challenges about extension 129 • An intermediate status report 131 • Sexual preferences in physics 131 • A physical aphorism 132 String theory – extensioin and a web of dualities Strings and membranes – why string theory is so difficult 133 • Masses and couplings 134 • Outlook 135 Unification (not yet avail able) The top of the mountain (not yet avail able) Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu 136 137 138 139 152 155 156 159 6 7 Postface Biblio graphy Challenge hints and solu tions Credits Image credits 155 Name index Subject index Copyright © Christoph Schiller November 1997–January 2009 Chapter 1 GENER AL R EL ATIVIT Y V ERSUS QUANTUM THEORY “ Man muß die Denkgewohnheiten durch Denknotwendigkeiten ersetzen.* Albert Einstein T he two stories told in the two parts of the path we have followed up to now, namely hat of general relativity and that of quantum field theory, are both beautiful and horoughly successful. Both are confirmed by experiments to the precision that is possible at present. We have thus reached a considerable height in our mountain ascent. The precision we have achieved in the description of nature is impressive, and we are now able to describe all known examples of motion. So far we have encountered no exceptions. On the other hand, the most important aspects of any type of motion, the masses of the particles involved and the strength of their coupling, are still unexplained. Furthermore, the origin of all the particles in the universe,** their initial conditions, and the dimensionality of space-time remain hidden from us. Obviously, our adventure is not yet complete.*** This last part of our hike will be the most demanding. In the ascent of any high mountain, the head gets dizzy because of the lack of oxygen. The finite amount of energy at our disposal requires that we leave behind all unnecessary baggage and everything that slows us down. In order to determine what is unnecessary, we need to focus on what we want to achieve. Our aim is the precise description of motion. But even though general relativity and quantum theory are extremely precise, we carry a burden: the two theories and their concepts contradict each other. We first list these contradictions. The contradictions In classical physics and in general relativity, the vacuum, or empty space-time, is a region with no mass, no energy and no momentum. If matter or gravitational fields are present, space-time is curved. The best way to measure the mass or energy content of space-time is to measure the average curvature of the universe. Cosmology tells us how we can do this. Cosmological measurements reveal an average energy density of the ‘vacuum’ of * ‘One needs to replace habits of thought by necessities of thought.’ ** The photograph on page 11 shows an extremely distant, thus extremely young, part of the universe, with its large number of galaxies (NASA). *** Nevertheless, nothing of what follows if of interest for biology, chemistry, medicine, geology, or engineering, neither for their work nor for their examinations. ” Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu Copyright © Christoph Schiller November 1997–January 2009 Ref. 1 Page ?? general rel ativit y versus quantum theory 15 E/V ≈ 1 nJ/m3 . Ref. 2 Page ?? Page ?? (1) However, quantum field theory tells a different story. A vacuum is a region with zeropoint fluctuations. The energy content of a vacuum is the sum of the zero-point energies of all the fields it contains. Indeed, the Casimir effect ‘proves’ the reality of these zeropoint energies. Their energy density is given, within one order of magnitude, by E 4πh =3 V c ∫ ν max 0 ν 3 dν = πh 4 ν . c 3 max (2) The approximation is valid for the case in which the cut-off frequency ν max is much larger than the rest mass m of the particles corresponding to the field under consideration. Particle physicists argue that the cut-off energy has to be at least the energy of grand unification, about 1016 GeV= 1.6 MJ. That would give a vacuum energy density of Motion Mountain – The Adventure of Physics available free of charge at www.motionmountain.eu E ≈ 1099 J/m3 , V (3) Ref. 3 Page ?? Page ?? which is about 10108 times higher than the experimental limit deduced from spatial curvature using general relativity estimates. In other words, something is slightly wrong here. General relativity and quantum theory contradict each other in other ways. Gravity is curved space-time. Extensive research has shown that quantum field theory, which describes electrodynamics and nuclear forces, fails for situations with strongly curved space-times. In these cases the concept of ‘particle’ is not precisely defined. Quantum field theory cannot be extended to include gravity consistently, and thus to include general relativity. Without the concept of the particle as a discrete entity, we also lose the ability to perform perturbation calculations – and these are the only calculations possible in quantum field theory. In short, quantum theory only works because it assumes that gravity does not exist. Indeed, the gravitational constant does not appear in any consistent quantum field theory. On the other hand, general relativity neglects the commutation rules between physical quantities discovered in experiments on a microscopic scale. General relativity assumes that the classical notions of position and momentum of material objects are meaningful. It thus ignores Planck’s constant ħ, and only works by neglecting quantum effects. Measurements also lead to problems. In general relativity, as in classical physics, it is assumed that arbitrary precision of measurement is possible – for example, by using finer and finer ruler marks. In quantum mechanics, on the other hand, the precision of measurement is limited. The indeterminacy principle yields limits that result from the mass M of the apparatus. The contradictions are most evident in relation to time. According to relativity, time is what is read from clocks. But quantum theory says that precise clocks do not exist, especially if gravitation is taken into account. What does ‘waiting 10 minutes’ mean, if the clock goes into a quantum-mechanical superposition as a result of its coupling to space-time geometry? In addition, quantum theory associates mass with an inverse length via the Compton wavelength; general relativity associates mass with length via the Schwarzschild radius. Copyright © Christoph Schiller November 1997–January 2009 16 1 general rel ativit y versus quantum theory Page ?? Ref. 4, Ref. 5 Ref. 6, Ref. 7 Similarly, general relativity implies that spac..."

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