ULTIMATE HORIZONS probing the limits of the universe

T H E F R O N T I E R S Helmut Satz C O L L E C T I O N U T M T H R Z N U T M T H R Z N LTIMATE HORIZONS UL IMATE HORIZONS ULTI ATE HORIZONS ULTIMA E HORIZONS ULTIMATE ORIZONS ULTIMATE HO IZONS ULTIMATE HORI ONS ULTIMATE HORIZO S ULTIMATE HORIZONS LTIMATE HORIZONS UL IMATE HORIZONS ULTI ATE HORIZONS ULTIMA E HORIZONS ULTIMATE ORIZONS ULTIMATE HO IZONS ULTIMATE HORI ONS ULTIMATE HORIZO S ULTIMATE HORIZONS ULTIMATE HORIZONS Probing the Limits of the Universe 123 THE FRONTIERS COLLECTION Series editors Avshalom C Elitzur Université Grenoble I Centre Équation, Labo Verimag, Gières, France e-mail: avshalom.elitzur@weizmann.ac.il Laura Mersini-Houghton Department of Physics & Astronomy, University of North Carolina, Chapel Hill, North Carolina, USA e-mail: mersini@physics.unc.edu T Padmanabhan Inter University Centre for Astronomy and Astrophysics (IUC), Pune University Campus, Pune, India e-mail: paddy@iucaa.ernet.in Maximilian Schlosshauer Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences, Portland, Oregon, USA e-mail: schlossh@up.edu Mark P Silverman Department of Physics, Trinity College, Hartford, Connecticut, USA e-mail: mark.silverman@trincoll.edu Jack A Tuszynski Department of Physics, University of Alberta, Edmonton, Alberta, Canada e-mail: jtus@phys.ualberta.ca Rüdiger Vaas University of Giessen, Giessen, Germany e-mail: ruediger.vaas@t-online.de For further volumes: http://www.springer.com/series/5342 THE FRONTIERS COLLECTION Series Editors A C Elitzur L Mersini-Houghton T Padmanabhan M Schlosshauer M P Silverman J A Tuszynski R Vaas The books in this collection are devoted to challenging and open problems at the forefront of modern science, including related philosophical debates In contrast to typical research monographs, however, they strive to present their topics in a manner accessible also to scientifically literate non-specialists wishing to gain insight into the deeper implications and fascinating questions involved Taken as a whole, the series reflects the need for a fundamental and interdisciplinary approach to modern science Furthermore, it is intended to encourage active scientists in all areas to ponder over important and perhaps controversial issues beyond their own speciality Extending from quantum physics and relativity to entropy, consciousness and complex systems—the Frontiers Collection will inspire readers to push back the frontiers of their own knowledge Helmut Satz ULTIMATE HORIZONS Probing the Limits of the Universe 123 Helmut Satz Fakultät für Physik Universität Bielefeld Bielefeld Germany This work appears in a parallel German edition ‘‘Gottes unsichtbare Würfel’’, published by C H Beck Verlag ISSN 1612-3018 ISSN 2197-6619 (electronic) ISBN 978-3-642-41656-9 ISBN 978-3-642-41657-6 (eBook) DOI 10.1007/978-3-642-41657-6 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013953242 Ó Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) In memory of my mother who dared to venture into the unknown in search of a better life for her sons Preface Confronted with the choice between paradise and knowledge, man, according to the Bible, chose knowledge Were these really alternatives? It came to be that the gaining of knowledge and the wider horizon outside the garden of Eden brought to many as much pleasure and satisfaction as any paradise they could imagine Humans have always wanted to explore the world they live in, and they have always wanted to know what lies beyond the horizons that limit their view The search for richer pastures, better climates, easier communication—all these certainly played a part in this, but behind it all there was an inherent human sense of curiosity This curiosity triggered a journey starting some 200,000 years ago in a remote corner of Africa and has driven us to navigate all the oceans, to conquer the entire Earth, to probe the heavens and to penetrate ever more deeply into interstellar space, to study ever more distant galaxies At the other end of the scale, high-energy particle accelerators allow us to resolve the structure of matter to an ever higher degree, to look for its ultimate constituents and study how they interact with each other to form our world Are there limits, is there an end to this drive, at the large scale as well as at the small? In the last hundred years, modern physics and cosmology have shown that there exist regions forever beyond our reach, hidden from us by truly ultimate horizons These regions we can access in our imagination only; we can speculate what they are like and whether perhaps some sign of their existence, some indication of their nature can ever reach our world Such hidden regions exist in those remote parts of the universe where, from our point of view, space expands faster than the speed of light Closer to us, they are found in black holes, where gravity is strong enough to retain even light within its horizon of ultimate attraction And in the realm of the very small, quarks remain forever confined to their colorful world of extreme density; they can never be removed from it But given the Big Bang origin of the universe, our world in its very early stages was immensely hot and dense; and given the spectrum of all the particles created in high-energy collisions, we can try to reconstruct ever earlier stages The evolution of the universe, with cooling and expansion, then defines horizons in time, thresholds through which the universe had to pass to reach its present state What were the earlier stages like? vii viii Preface Although it is not possible to transmit information across the ‘‘event horizons’’ that form the borders of these forbidden regions, still sometimes strange signals may appear, providing us with hints of the existence of those other worlds Such striking phenomena can become possible through quantum effects; ‘‘Hawking– Unruh’’ radiation provides one example expected to arise in a variety of cases, whenever there exists an event horizon And looking at the multitude of ‘‘elementary’’ particles produced in high-energy accelerators, we can speculate that they originally came from a simpler, more symmetric world, which in the course of the evolution experienced transitions, like the freezing of water or the magnetization of metals, to form the many-faceted and less symmetric world we see today The aim of this book is to tell the story of how the different horizons, on Earth and in the heavens, on large and on small scales, now and in the past, were discovered and used to define our view of the world It is a story of the evolution of this view, which started before ‘‘science,’’ and which is much more than just ‘‘something for scientists.’’ It started with philosophers wondering what matter was made of, and how; with sailors daring to find out if the world ends somewhere; with astronomers trying to determine our position among the stars, to estimate the size of the Earth by looking at the Sun and using the newly developed geometry With Edgar Allan Poe, the Big Bang appeared in literature before it was commonplace in physics and cosmology; and aspects of both black holes and wormholes were part of the stories of Lewis Carroll before they became significantly appreciated in science Many of the ideas, even today’s, have come up here and there in the course of time The ways of treating them, and the tools used for that were different, of course, and changed over the centuries But what remained was that desire to see what lies beyond, and to find out whether there is a limit to what we can reach and understand We begin by looking at the various horizons partitioning our world and then show how different forbidden regions arise in the universe, and when and how they can emit signatures as testimony to their presence and their nature The mysterious light emerging from an event horizon, or the equally mysterious clusters in a new and strange ether, they may well remain all that we can ever see of what is hidden beyond the ultimate horizons This book is not meant to give a systematic presentation of the recent developments in physics or cosmology Its aim is to tell a story that began a long time ago and that will certainly not come to an end very soon And it covers developments that sometimes, as in the age of Vasco da Gama and Columbus, or in the time of Einstein, Planck, Bohr and Heisenberg, revolutionize the world in two or three decades At other times, between Ptolemy and Copernicus, it takes a millennium to add a couple of epicycles to the accepted scheme of things The problem is, in the words of the renowned Austrian theorist Walter Thirring, that ‘‘to something really new, you have to have a new idea,’’ and that does not happen so very often It does not suffice to play on the keyboard of the available theoretical formalisms; this just leads to many melodies and not to any convincing and lasting new harmony Preface ix I have tried to present things in a way not needing any mathematics That is, as I indicate in the section on Notation, a two-sided issue Even Einstein sometimes presented the special theory of relativity in terms of people on a train versus people on the ground It can be done, and it is indeed helpful to convey the basic ideas For a full understanding of the ultimate conclusions, however, mathematics becomes essential To travel a middle road, I have at times added inserts, in which some aspects of the basic mathematical formulation are indicated But I hope that the presentation remains understandable even if you skip these One unavoidable aspect appears if one tries to present things in as readable a way as possible: some points and concepts are mentioned more than once Although strictly speaking logical, the reminder ‘‘as already discussed in the previous Chapter’’ is in fact often not what the reader wants; it seems better to just briefly recall the idea again So I offer my apologies for a number of repetitions And another apology is probably also needed When forced to choose between scientific rigor and simplifying an idea enough to make it understandable, I generally took the latter path I thought it better to try to have readers follow my train of thought, even if they will later need corrections, than to lose them in technical details they cannot follow My inspiration here were the words of the great Danish physicist Niels Bohr, who noted that Wahrheit (truth) and Klarheit (clarity) are complementary: the more precisely you enforce one, the less precise the other becomes Finally, it is my pleasure to express sincere thanks to all who have helped me with this endeavor Obvious support came from my colleagues here in Bielefeld, in Brookhaven, at CERN, in Dubna and elsewhere They have been of crucial importance in forming my view of things And last, but far from least, profound thanks go to my wife, who has patiently borne with me during all these years Bielefeld, May 2013 Helmut Satz Contents Horizons 1.1 The Horizon of Accessibility 1.2 Forbidden Rooms in the Universe 1.3 Ultimate Constituents 1.4 The End of the Earth 1.5 The Roof of Heaven 12 The 2.1 2.2 2.3 2.4 2.5 Vanishing Stars The Speed of Light Why Is the Sky Dark at Night? The Big Bang Cosmic Inflation The Absolute Elsewhere 19 19 29 33 37 38 The 3.1 3.2 3.3 3.4 Secret Glow of Black Holes The Escape Velocity Tidal Effects The Sea of Unborn Particles Invisible Light on the Horizon 43 43 48 51 54 The 4.1 4.2 4.3 4.4 4.5 Visions of an Accelerating Observer Gravity and Acceleration A Total End of Communication The Temperature of the Vacuum Lightning in Empty Space Quantum Entanglement 59 61 63 64 66 67 The 5.1 5.2 5.3 5.4 5.5 5.6 Smallest Possible Thing Why Does the Sun Shine? The Strong Nuclear Interaction The Weak Nuclear Interaction The Quarks The Standard Model The Confinement Horizon 71 77 78 84 88 95 98 xi 154 The Last Veil of the three great areas, is still missing: it is, as someone has noted, a name, an idea in search of a theory The only apparently firm result incorporating all three of the mentioned constants of nature, and the Boltzmann constant as well, is that of the Hawking radiation of a black hole, predicted to be kT = c3 /(8πG M), where M is the mass of the hole And this radiation is, for reasons we have mentioned, hidden for many, many eons below the microwave background radiation But the basic question remains: What happens when the density of constituents becomes so great that position—momentum uncertainties are essential—what happens at scales below the Planck length rPl = (hG/c3 )1/2 ? What happens at times shorter than the Planck time, tPl = rPl /c? Then we need something, to be called quantum gravity The evolution of the universe before inflation, in the Planck era, requires such a combination of micro and macro worlds Speculations in this field of thought have led to some exciting shadows behind the veil, shadows which, in the future, may or may not turn out to be a reflection of reality 8.4 Hyperspace The stage for one such dream is obtained by adding further space dimensions We live in a world of three dimensions in space and one in time Let us put the time aside for a moment In our three space dimensions, we can imagine a simpler world of only two space dimensions, occupied by “flat” creatures living there They could never see “out” of their two-dimensional world, just as we are constrained to our three space dimensions But they could check whether their world is distorted, just as we can check whether space in our three-dimensional world is “curved”, for example as an effect of gravity as described in general relativity If the two-dimensional world were really flat, two parallel light beams would never cross But if it contained a center of severe gravity somewhere, that would make the surface seem to “bulge” there, and now two light beams would cross at the bottom of the bulge (see Fig 8.4) Evidently this is quite similar to the deviation of the starlight observed by Sir Arthur Eddington in his celebrated confirmation of Einstein’s general relativity Quite generally, planar geometry would fail within the bulge: the angles of triangles would add up to more than 180◦ , just as they on the surface of a globe So the flat people, within their 2-d world, could detect curvature And to help them understand such curvature, they could imagine that their world is embedded in a larger one, of three dimensions: in hyperspace The additional dimension is, from their point of view, purely hypothetical, it is not real, they can never “enter” it, and any light beam will pass in the two dimensions of their real space But if they take the hypothetical third dimension to be flat, then their 2-d world becomes a curved surface in that 3-d space, as illustrated in Fig 8.4 We used a similar method in Chap 3, to illlustrate the effect of a strong magnet on a metallic coin; adding another, “fictitious” dimension often helps to illustrate the modifications of space due to force So let us return to the picture of the 2-d world in 3-d hyperspace In Fig 8.5 we illustrate a beam of light passing from point A to 8.4 Hyperspace 155 (a) (b) Fig 8.4 The effect of space curvature as observed by “flat” creatures in a two-dimensional world (a) and as seen when embedded in a hyperspace of three dimensions (b) point B in the 2-d world—it always remains “in” its 2-d surface, also when that becomes curved The travel time of the beam is thus the length of the curved path divided by the speed of light If, however, it became possible by some miraculous means to construct a tunnel through the hyperspace, then the path and hence the length of time needed for the passage would be correspondingly shortened For the inhabitants of the 2-d world, the result would be most striking For them, the speed of light is determined by its value in their curved surface world So the light signal through the tunnel would appear to have travelled from A to B at superluminous speed And if point B were outside the Hubble radius for point A, out of reach by “normal” means, the hyperspace tunnel would allow a signal to get there So then A could transgress its ultimate spatial horizon! Moreover, this scheme is quite general: if any two points in our own world, say on Earth and on a distant star X, are a hundred light years apart in “real” space, but somehow only one meter in a hyperspace, a tunnel through the latter would allow almost instantaneous communication with that star The crucial question thus is whether such tunnels through hyperspace are merely a figment of the imagination of science fiction writers, or whether they could exist in the world defined by the laws of physics hyperspace A B space Fig 8.5 A beam of light (solid red line) passing from point A to point B in a 2-d world A possible tunnel (pink tube) through the third hyperspace dimension would clearly provide a shortcut for its passage (dashed red line) 156 8.5 The Last Veil Cosmic Connections The search for tunnels of this kind has therefore triggered much study in the past decades The first possibility in physics was introduced in 1935 by Albert Einstein and Nathan Rosen; it is today known as the Einstein–Rosen bridge and provides a tunnel between two distinct universes Something like such a tunnel had appeared much earlier in a more poetic form, in Lewis Carroll’s Alice in Wonderland: Alice passed through a rabbit hole from her normal environment into another, fantastic world In general relativity, wormholes, as these paths through hyperspace are generally referred to now, are indeed possible solutions of the equations of general relativity But these solutions suffer from a number of difficulties that severely hinder their application as useful tunnels of passage They connect regions showing some form of singularity, and even that only for a brief instant of time, as a fluctuation; moreover, they are generally of almost vanishing thickness, so that no Alice would fit through This has led to investigations of conditions needed to keep them open for longer and for larger dimensions, to move them out of the range of quantum fluctuations One puzzle we have already encountered was what made the Big Bang bang— why did the early universe expand? In particular, what forces were responsible for the extremely rapid expansion leading to inflation, what could cause the increasing expansion still observed today, and are the two related? It has been suggested that the origin of the presently observed expansion is a novel medium filling all of space, dark energy Normally the expansion of a medium leads to a reduction of its pressure; for dark energy, the opposite holds, so that the more it expands, the greater the pressure becomes This would account for the fact that with increasing “size”, the expansion of the universe accelerates further What could this mysterious medium be? Its only function is to make the universe expand—it is not subject to any of the other forces, in particular it is not affected by gravity In any case, cosmologists have proposed that if somehow the interior of a wormhole could be filled with such dark energy, its negative pressure might hold the wormhole open wider and for a longer time So the same force that leads to the expansion inherent in the Big Bang might also allow apparently superluminous cosmic connections between different regions of space and thus permit us to transgress the ultimate horizons of our conventional world This has led to a curious problem, also seemingly more fiction than science If achievable, such hyperspace tunnels allow not only superluminous travel in space, they also make possible travel in time In fact, simply the possibility of travel faster than the speed of light allows communication with the past, since an absolute future and an absolute past are definable for an observer only based on a finite universal speed of light For a superluminous observer, there will be events for which past and future become inverted And this causes severe problems with our concepts of causality If I could travel back in time and kill my parents before I am born, how can my existence be explained? So the existence of both superluminous travel in general and wormholes in particular are difficult to accommodate in a world based on chronology and causality Omar Khayyam, with whom we opened this chapter 8.5 Cosmic Connections 157 Fig 8.6 The fractional contributions of media in the universe to the overall energy required to account for the present accelerated expansion dark energy 22% dark matter 4% 74% visible matter looking for what might lie beyond the ultimate horizons, provided already a thousand years ago a poetic exclusion of time travel into the past, The Moving Finger writes; and, having writ, Moves on: nor all thy Piety nor Wit Shall lure it back to cancel half a line So one aim of a future quantum theory must be the clarification of these problems In the words of Stephen Hawking, “we must make the world safe for historians” One thing we have to keep in mind is that possible solutions of the equations of general relativity are not per se also solutions of the general laws of physics As we have noted, the bridge between quantum physics and gravity does not yet exist, and once it is found, its results may destroy the bridge of Einstein and Rosen and all related considerations Quantum physics might rule out singularities in space and in time, and thereby also rule out the possibility of wormholes of any kind What we presently see behind that veil may well therefore be illusions, distorted shadows of the real world And another aspect also has to be borne in mind, while we are waiting for quantum gravitation To formulate a theory of everything, this quantum gravitation, once it exists, still has to be made to match the quantum field theory of the standard model, or whatever extensions beyond, for the interactions of quarks and leptons Nevertheless, there remain also less speculative issues that require an answer from a quantum gravity yet to come We have seen that the expansion of the universe is determined by its density, specifically its energy density The latest measurements of the accelerating expansion allow an estimate of the present energy density, and this estimate creates several problems We can determine fairly well all the visible matter in the universe, stars, galaxies, interstellar gas and the like All this accounts for only % of what is needed And in fact gravity estimates of galaxies indicate that besides the visible matter, there must be a large amount of invisible dark matter—matter which cannot be seen, but still contributes to the gravitational pattern of galaxies In contrast to black holes, this dark matter neither emits nor absorbs electromagnetic radiation And even if we include the amount required here, if we add up all matter we can detect in the universe, this only amounts to some 26 % of the energy density needed to get the acceleration; more than three quarters is still missing (see Fig 8.6) The remainder exists presumably in the form of the dark energy introduced above as the origin for the Big Bang expansion in the first place It cannot be seen, it is not subject to gravity, it just makes the space of the world expand 158 The Last Veil But once we have a final theory, a theory of everything: what we mean by final, by everything? We have seen that, given quantum mechanics, the structure and the spectrum of atoms, for example, of the helium atom, could be calculated But this does not account for the behavior of liquid helium, with superfluidity and the like To understand “everything”, there is not only the problem of the large and the small That problem is the issue for reductionism, in size, in structure and in interaction of the constituents But there is also the problem of the few and the many A new field of study has appeared in recent decades, emergent behavior, based on the observation that a system of many interacting components may lead to forms of collective behavior not derivable from the two-body interaction In some simpler cases—we have seen that in the case of spontaneous symmetry breaking and the resulting phase transitions—the given form of the two-body interaction allows the calculation of the equation of state of the macroscopic system Lars Onsager showed this for the case of a two-dimensional spin system, the Ising model But in general, the transition from dynamics to thermodynamics, in particular to the critical behavior found in phase transitions, is not at all straightforward In the physics of strong nuclear interactions, quantum chromodynamics provides another instance of this problem Energetic scattering processes, involving short-distance two-body interactions, are calculable, and the results agree with the corresponding experimental observations up to remarkable precision And through numerical studies, the spectrum, the masses, charges, etc of all the hadrons have also been calculated However, for the thermodynamics of quarks and gluons, as we saw in Chap 6, much still has to be clarified The confinement–deconfinement transition, from hadronic matter to a quark–gluon plasma, has been established; but the nature of the deconfined medium, the quark– gluon plasma, in the region hopefully accessible to nuclear collision experiments, that remains yet to be understood Keeping these aspects in mind, we see that once we have determined the interaction form in the standard model, the behavior of bulk matter in the relevant regime is still another issue Is the Higgs transition the point where the baryon asymmetry of our universe was born? What are the possible states of matter above the transition point? And the same question of course reoccurs for any still earlier transitions What is the equation of state, the phase diagram for matter in the grand unification regime? For experimental studies, the answers to these questions seem to lie well beyond accessibility But once the theoretical problems on a local, few-body level are solved, we can at least try to see what collective behavior this might imply We have mentioned emergent behavior Difficult though it may be to extend the few-body dynamics of specified constituents to the statistical limit, it still presumes that the nature of these constituents and the form of their interaction remain relevant, are determining for the result However, the ultimate idea of emergent behavior transcends such a reductionist basis Percolation theory, for example, provides a basis for galaxy formation, for magnetization, for quark deconfinement—but so it does also for the spread of forest fires, for the behavior of animal swarms, for the distribution of subterranean oil deposits So there seem to emerge general patterns of behavior that not depend on the nature of the constituents and their form of 8.5 Cosmic Connections 159 interaction These present us with yet another quite striking and conceptually very different form of grand unification Some of what today still remains behind different ultimate horizons may then obtain a visible shape through future studies; mirages might become reality But in spite of the monumental achievements of theoretical physics, astrophysics and cosmology, the ultimate test of science is and must continue to be a comparison with observable nature It is a great challenge to find out what the possible worlds are for the most general theoretical schemes However, there is no law of nature stating that what is possible must also exist So in the end, ultimate horizons are the visible borders that define for us the limits of the real in the sea of the imaginable A Notes on Notation Each subject of human endeavor seems to have a language particularly suited to it More than a thousand years ago, Charlemagne, Carolus Magnus, leader of the Holy Roman Empire in the tenth century, is supposed to have said that I speak Spanish to God, Italian to the ladies, French to the men, and German to the horses Even today, the horses seem to appreciate German Mathematics, it was said, is the language God uses if he wants to speak to humans This may well be true, although he is most likely more polyglot and able to communicate as well through music or poetry Nevertheless, if he really wants to make a point, it seems that he does resort to mathematics How else can we understand that the arrangement of the petals on all flowers follow a sequence published around 1200 A.D by the Italian mathematician Leonardo da Pisa, now known as Fibonacci, to describe the growth of a population of rabbits Just for fun, we recall it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … It was known already in antiquity, related to the proportio divina, the divine proportion between different length scales Today it is a popular item on mathematics tests: how does it go on? What law governs the pattern? And for the more advanced: what is the limit obtained for two successive Fibonacci numbers, 13/8, 21/13, 34/21, …? These are not artifacts only of the human mind—they are found, for example, in the arrangements of the centers of sunflowers, up to 144/89 or 233/144 Here we only want to recall some mathematical notation helpful for us Multiple products are generally written as powers n, so that × × becomes 23 Since the cosmos involves very large and the microcosmos very small numbers, powers of ten provide a convenient formulation Thus one thousand becomes 103 , million 106 , with the exponent counting the number of zeros following the one: 103 = 1, 000, and so on Similarly, one thousandth becomes 10−3 , one millionth 10−6 , etc.; here the negative exponent counts the number of places between the decimal point and the one: 10−3 = 0.001 If we want to concentrate only on the exponent, we consider the logarithm: log(106 ) = This form is referred to as the logarithm of the base of 10, since it counts the powers of 10 Another, perhaps even more frequently arising form is the natural logarithm, ln Its base is the Euler number e ◦ 2.718 , so that ln x = means x = e3 ; it is named after the Swiss mathematician Leonhard Euler H Satz, Ultimate Horizons, The Frontiers Collection, DOI: 10.1007/978-3-642-41657-6, © Springer-Verlag Berlin Heidelberg 2013 161 162 Notes on Notation and can be written in the form 1 1 e=1+ + + + + 1×2 1×2×3 1×2×3×4 Together with π, it is perhaps the most important number in mathematics, and it has an immense number of applications in a diverse number of fields Just one illustration: Any rate of growth of a population that is proportional to the number of its members becomes exponential, i.e., it has the form N (x) = N (0)e x Thus if at a given time a country has N inhabitants, and if the population increase (births minus deaths) is % per year, the overall population will grow with time t as N (t) = N e0.01t , where t is measured in years This means that the population has doubled after some 70 years In daily use, it is convenient to have names for the powers of ten, just as our common thousand or million, but more systematic So a thousand meters becomes a kilometer, a thousand liters a kiloliter, a thousand volts a kilovolt The most commonly used prefixes here are kilo mega giga tera peta (k) (M) (G) (T) (P) 103 106 109 1012 1015 while at the other end of the scale we have centi milli micro nano pico femto (c) (m) (µ) (n) (p) (f) 10−2 10−3 10−6 10−9 10−12 10−15 The common measure of energy (or mass, following Einstein) in elementary particle physics is the electron volt (eV); it is the amount of energy a single electron gains if its potential energy level is increased by one volt The mass of an electron is roughly 0.5 MeV, that of a pion some 140 MeV, and that of proton about GeV This is to be compared to the collision energies of present day accelerators: the Large Hadron Collider (LHC) at the European Organization for Nuclear Research CERN in Geneva is today the most energetic such facility, with a top energy of some TeV for proton–proton collisions If this energy, in a collision, were totally converted into pions, it could lead to some ten thousand such mesons, indicating why Notes on Notation 163 these interactions result in multiparticle production And when the LHC is used for nuclear collisions, using lead nuclei instead of protons as projectiles, the collision energy reaches the PeV regime, with correspondingly higher multiparticle production This is, as we noted above, one of the reasons one hopes to make primordial matter in such collisions We have at various times made use of the basic constants of nature; let us therefore list here their numerical values The constant connecting the energy of mechanics with the temperature of thermodynamics is the Boltzmann constant, eV , K with the approximate equality sign ◦ we indicate that the value is given only with the accuracy of the number of significant digits shown (a = 1.3728 becomes a ◦ 1.37) The universal speed of light in vacuum is m c ◦ 3.0 × 105 s Newton’s constant of gravity is k ◦ 8.6 G ◦ 6.7 × 10−11 m3 kg s2 Finally, Planck’s constant is given by h ◦ 4.1 × 10−15 eV s Here one should also mention the often-used reduced Planck’s constant, = h/2 π Finally, we note, for excursions into the literature, that in particle physics it is often convenient to measure velocities in multiples of the speed of light, energies in units of the (reduced) Planck constant per second, and temperatures in units of the Boltzmann constant This is commonly abbreviated by c = = k = B Further Reading While there does not seem to be any one book adressing specifically the different horizons emerging in our attempt to find the limits of the universe, there exist quite a few excellent general coverages of specific topics, far too many to list here The following are just those I found particularly understandable and illuminating Cosmology, Gravity and Black Holes: Schutz, B.: Gravity from the Ground Up Cambridge University Press, Cambridge (2003, in press) Smolin, L.: Three Roads to Quantum Gravity Weidenfeld and Nicolson, London (2000) Thorne, K.S.: Black Holes and Time Warps W W Norton & Co., New York (1994) Rees, M., Begelman, M.: Gravity’s Fatal Attraction, 2nd edn Cambridge University Press, New York (2010) Interactions and Elementary Particles: Davies, P.C.W.: The Forces of Nature, 2nd Edn Cambridge University Press, Cambridge (1986) Close, F.: Particle Physics: A Very Short Introduction Oxford University Press, Oxford (2004) Symmetry: Close, F.: Lucifer’s Legacy: The Meaning of Asymmetry Oxford University Press, Oxford (2000) Zee, A.: Fearful Symmetry Macmillan Publishing, New York (1986) H Satz, Ultimate Horizons, The Frontiers Collection, DOI: 10.1007/978-3-642-41657-6, © Springer-Verlag Berlin Heidelberg 2013 165 Author Index A Alfonso X of Castile, 13 Anderson, Carl, 51 Aristarchos, 13 Aristotle, 11, 20 Aston, Francis William, 78 B Becquerel, Henri, 82, 83 Bell, John, 67 Berra, Yogi, 82 Bohr, Niels, 75 Boltzmann, Ludwig, 108 Born, Max, 75 Bose, Satyendranath, 76 Brahe, Tycho, 14 Broglie, Louis de, 75 Brout, Robert, 96 Bruno, Giordano, 17, 19, 34 Buridan, Jean, 133, 134 Bux, Bastian Balthasar, 51 C Cabbibo, Nicola, 112 Carroll, Lewis, 156 Casimir, Hendrik, 53 Cassini, Giovanni Domenico, 20 Cavendish, Henry, 45 Chadwick, James, 73 Columbus, Christopher, 10, 12 Copernicus, Nicolaus, 13 Coulomb, Charles Augustin de, 23 Creutz, Michael, 111 Curie, Pierre, 128 D Dalton, John, 71 Darwin, Charles, 78 Democritus, 71 Descartes, Ren´, 20 Dias, Bartolomeu, 10 Dicke, Robert, 33, 34 Digges, Thomas, 17, 40 Dirac, Paul, 51 Doppler, Christian, 30 E Eanes, Gil, 10, 12 Eddington, Arthur, 32, 78, 154 Einstein, Albert, 26, 27, 32, 46, 61, 67, 78, 156 Ende, Michael, 51 Englert, Franỗois, 96 F Faissner, Helmut, 97 Fermi, Enrico, 76, 84 Feynman, Richard, 79, 116 Fizeau, Hippolyte, 22 Foucault, L´eon, 33 Friedmann, Alexander, 33 Fritzsch, Harald, 91 G Galilei, Galileo, 14, 15, 27,33 Gama, Vasco da, 10, 12 Gauss, Carl Friedrich, 104 Gell-Man, Murray, 88, 91 Ghazali, Al, 133 Glashow, Sheldon, 95 H Satz, Ultimate Horizons, The Frontiers Collection, DOI: 10.1007/978-3-642-41657-6, © Springer-Verlag Berlin Heidelberg 2013 167 168 Goldstone, Jeffrey, 94, 135 Gross, David, 91 Guralnik, Gerald, 96 Guth, Alan, 37 H Hagedorn, Rolf, 112 Hagen, Richard, 96 Hales, Thomas C., 104 Harriot, Thomas, 104 Harrison, David, 68 Hawking, Stephen, 50, 54, 147 Heisenberg, Werner, 52, 65, 75, 134 Helmholtz, Hermann von, 78 Henry the Navigator, 1, 9, 11, 12 Higgs, Peter, 96 Hooke, Robert, 25 Hubble, Edwin, 30 Hunefer, 59 Huygens, Christiaan, 21 I Ising, Ernst, 130 J Joyce, James, 88 K Kelvin, Lord (William Thomson), 78 Kepler, Johannes, 20, 104 Khayyam, Omar, 156 Kibble, Tom, 96 L Laplace, Pierre-Simon de, 45 Lederman, Leon, 84, 93 Lee, Tsung-Dao, 115 Leibnitz, Gottfried Wilhelm, 131 Lemaitre, Georges, 33 Lenz, Wilhelm, 130 Leucippus, 71 Li, Keran, 115 Lucretius, 8, 9, 72, 77, 88, 90, 119, 153 Luther, Martin, 14 M Magellan, Fernando, 10, 12 Matsui, Tetsuo, 121 Maxwell, James Clerk, 25 Mendeleev, Dmitri, 72 Author Index Michell, John, 43 Michelson, Albert, 26 Morley, Edward, 26 N Nambu, Yoichiro, 94, 134, 135 Ne’eman, Yuval, 88 Newton, Isaac, 15, 131 Nishijima, Kazuhiko, 88 O Ockham, William of, 133 Oersted, Hans Christian, 25 Olbers, Heinrich, 19, 29, 30 Onsager, Lars, 131, 134, 158 P Parisi, Giorgio, 112 Pascal, Blaise, 25 Pati, Yogesh, 97 Pauli, Wolfgang, 56, 75, 82, 84 Peebles, Jim, 34 Penrose, Roger, 50 Penzias, Arno, 33 Perkins, Donald, 80 Perl, Martin, 86 Pessoa, Fernando, Planck, Max, 74 Podolsky, Boris, 67 Poe, Edgar Allan, 19, 29 Politzer, David, 91 Powell, Cecil, 80 Ptolemy, Claudius, 13 R Raleigh, Sir Walter, 104 Reinhold Bertlmann, 67 Richer, Jean, 21 Richter, Burt, 93 Rindler, Wolfgang, 63 Rømer, Ole, 20, 21 Rosen, Nathan, 67 Rutherford, Ernest, 72 S Salam, Abdus, 95, 97 Sauter, Friedrich, 65 Schrödinger, Erwin, 75 Schwartz, Melvin, 84 Schwarzschild, Karl, 46 Schwinger, Julian, 66 Author Index Steinberger, Jack, 84 169 T Thomson, J J., 72 Ting, Sam, 93 Torricelli, Evangelista, 25 Weinberg, Steven, 95, 140 Wheeler, John, 50 Wilczek, Frank, 91 Wilkinson, David, 34 Wilson, Kenneth, 92, 111 Wilson, Robert, 33 U Unruh, William, 61 Y Yukawa, Hideki, 79 V Vivaldo, Guido and Ugolino de, 12 Z Zach, Franz Xaver von, 45 Zel’dovich, Yakov, 50 Zweig, George, 88 W Index A Accessibility horizon, Action at a distance, 23 Age of the universe, 36 Annihilation, 52, 98, 99 Antimatter, 83 B Baryon, 82 Beta-decay, 83 Boson, 76 Bottom, 93 Bottomonia, 122 C Charm, 93 Charmonia, 93, 122 Chiral symmetry, 137 Close packing, 104 Color, 90 Computer methods, 111 Computer simulation, 111, 132 Confinement, 90 Confinement horizon, 98 Continuous symmetry, 126, 135 Copernican principle, 148 Cosmological constant, 32 Critical behavior, 111 Critical points, 110 Curie point, 128 D Dark energy, 37, 157 Dark matter, 37, 157 Deconfinement, 105 Degenerate ground states, 133 Dirac sea, 51, 52 Discrete symmetry, 126 Doppler effect, 30, 34, 36 E Electricity, 22 Electromagnetism, 22 Electron, 72 Electroweak interaction, 95, 150 Electroweak transition, 142 Emergent behavior, 158 Entanglement, 67 Entropy, 109, 132 Escape velocity, 43 Ether, 25 Event horizon, Exclusion principle, 75 F Fermion, 76 Flavor, 90 G Gauge boson, 139 General theory of relativity, 27, 32, 41 Global symmetries, 138 Gluon, 90, 111 Goldstone particle, 94 Grand unification, 140, 150 Gravitation, 15 Gravity, 15 H Satz, Ultimate Horizons, The Frontiers Collection, DOI: 10.1007/978-3-642-41657-6, © Springer-Verlag Berlin Heidelberg 2013 171 172 H Hadron, 82 Hadrosynthesis, 118 Hawking radiation, 55 Hawking temperature, 56 Heavy ions, 117 Higgs boson, 142, 143, 145, 146 Higgs field, 142 Higgs mechanism, 143 Higgs transition, 142 High-energy nuclear interactions, 116 Horizon problem, 35 Hubble’s law, 31, 36 Hyperspace, 47, 154 I Inflation, 37 Inside-out cascade, 100 Interaction over a distance, 24 Ising model, 130, 131 J Jupiter, 20 L Latent heat, 113 Latent heat of deconfinement, 113 Leptons, 85 Little bang, 114 Local symmetries, 138 M Magnetism, 23 Magnetization, 131 Meson, 79 Muon, 84 N Neutrino, 84 Nucleon, 73 Nucleosynthesis, 118 Nucleus, 115 O Order parameter, 131 Index P Pair creation, 52 Percolation, 106 Periodic table in year 1869, 73 Periodic table of the elements, 72 Pion, 79 Planck constant, 74 Positron, 51 Q Quantum chromodynamics, 91, 108 Quantum gravity, 49, 146, 153, 154, 157 Quark confinement, 91 Quark–gluon plasma, 111 Quarkonia, 122 Quarks, 88 R Radioactivity, 83 Redshift, 30 Rindler horizon, 63 S Schwarzschild radius, 46 Schwinger effect, 66 Singularity, 49 Special theory of relativity, 27, 41 Speed of light, 19 Spin wave, 135 Spontaneous symmetry breaking, 129, 142 Standard model, 95, 151 Strong interaction, 78 Symmetry, 125, 126 Symmetry breaking, 128 T Theory of everything, 140, 146, 158 Tidal effects, 48 U Uncertainty principle, 52 Unruh radiation, 65 W Wave–particle duality, 75 Weak nuclear interaction, 84, 86 Wormholes, 156 ... Constituents Moreover, given the expansion of the universe, the strange world of the quarks was not always a feature only of the very small If we let the film of the evolution of the universe run backwards... concluded that the speed of light is indeed finite and that the 22 is the time it needs to traverse the diameter of the orbit of the Earth around the Sun To obtain the actual value of the speed of light... Contents Horizons 1.1 The Horizon of Accessibility 1.2 Forbidden Rooms in the Universe 1.3 Ultimate Constituents 1.4 The End of the Earth 1.5 The Roof of Heaven