Ultimate Horizons: Probing the Limits of the Universe 364241656X, 9783642416569

Table of contents :
Preface
Contents
1 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
2 The Vanishing Stars
2.1 The Speed of Light
2.2 Why Is the Sky Dark at Night?
2.3 The Big Bang
2.4 Cosmic Inflation
2.5 The Absolute Elsewhere
3 The Secret Glow of Black Holes
3.1 The Escape Velocity
3.2 Tidal Effects
3.3 The Sea of Unborn Particles
3.4 Invisible Light on the Horizon
4 The Visions of an Accelerating Observer
4.1 Gravity and Acceleration
4.2 A Total End of Communication
4.3 The Temperature of the Vacuum
4.4 Lightning in Empty Space
4.5 Quantum Entanglement
5 The Smallest Possible Thing
5.1 Why Does the Sun Shine?
5.2 The Strong Nuclear Interaction
5.3 The Weak Nuclear Interaction
5.4 The Quarks
5.5 The Standard Model
5.6 The Confinement Horizon
6 Quark Matter
6.1 Quarks Become Deconfined
6.2 Collective Behavior
6.3 The Ultimate Temperature of Matter
6.4 The Little Bang
6.5 Universal Hadrosynthesis
6.6 How Hot is the Quark--Gluon Plasma?
7 Hidden Symmetries
7.1 The Ising Model
7.2 Shadow Particles
7.3 Local Symmetries
7.4 Primordial Equality
8 The Last Veil
8.1 Ultimate Horizons in Time
8.2 Ultimate Horizons in Space
8.3 The End of Determinacy
8.4 Hyperspace
8.5 Cosmic Connections
A Notes on Notation
B Further Reading
Author Index
Index

Citation preview

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

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THE FRONTIERS COLLECTION

Series editors Avshalom C. Elitzur Université Grenoble I Centre Équation, Labo. Verimag, Gières, France e-mail: [email protected] Laura Mersini-Houghton Department of Physics & Astronomy, University of North Carolina, Chapel Hill, North Carolina, USA e-mail: [email protected] T. Padmanabhan Inter University Centre for Astronomy and Astrophysics (IUC), Pune University Campus, Pune, India e-mail: [email protected] Maximilian Schlosshauer Institute for Quantum Optics and Quantum Information Austrian Academy of Sciences, Portland, Oregon, USA e-mail: [email protected] Mark P. Silverman Department of Physics, Trinity College, Hartford, Connecticut, USA e-mail: [email protected] Jack A. Tuszynski Department of Physics, University of Alberta, Edmonton, Alberta, Canada e-mail: [email protected] Rüdiger Vaas University of Giessen, Giessen, Germany e-mail: [email protected]

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  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?

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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 do 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

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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

1

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 . . . . . . . . .

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Smallest Possible Thing . . . . . . Why Does the Sun Shine? . . . . The Strong Nuclear Interaction . The Weak Nuclear Interaction. . The Quarks . . . . . . . . . . . . . . . The Standard Model . . . . . . . . The Confinement Horizon . . . .

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Quark Matter . . . . . . . . . . . . . . . . . . . . . 6.1 Quarks Become Deconfined . . . . . . . . 6.2 Collective Behavior . . . . . . . . . . . . . . 6.3 The Ultimate Temperature of Matter . . 6.4 The Little Bang. . . . . . . . . . . . . . . . . 6.5 Universal Hadrosynthesis . . . . . . . . . . 6.6 How Hot is the Quark–Gluon Plasma? .

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Last Veil. . . . . . . . . . . . . . Ultimate Horizons in Time . Ultimate Horizons in Space The End of Determinacy . . Hyperspace . . . . . . . . . . . . Cosmic Connections . . . . .

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Notes on Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

Horizons

Beyond the horizon, behind the Sun, at the end of the rainbow, life has only begun. Bob Dylan

We live in a finite world. Even from the highest mountain or from an airplane, our view always ends at a horizon, beyond which we cannot see. Moreover, horizons are elusive. We see them, we’re surrounded by them, we try to reach them, and when we get “there”, they have moved to somewhere else. Yet they always confront us with the challenge to find out what lies beyond; at all times humans have wondered that. And nowhere is the challenge quite as present as at the sea, where water and sky touch in that sharp horizontal line. Already more than three thousand years ago, on the eastern shores of the Mediterranean Sea, the Phoenicians built navigable sailing vessels (Fig. 1.1), and they were familiar with astronomical orientation. Their ships explored the entire Mediterranean and passed beyond the limits of their world, the pillars of Hercules, today’s Strait of Gibraltar. A thousand years ago, the ships of the Vikings set out into the unknown northern seas and reached what turned out to be a new continent. And the systematic exploration of all the lands beyond all the horizons began when the Portuguese sailors of Henry the Navigator dared to find out if the Earth ended somewhere. The inquisitive curiosity to discover if and how the known world continues—this was surely one of the driving forces that made mankind conquer the whole Earth and go on beyond. Once all earthly horizons were surpassed, the sky became the limit, receding back further and further. At first, man could only look up, then telescopes gave him the power to see further, and today, there are human footsteps on the moon and our probes in space penetrate ever more distant stellar regions. Are there still regions in the universe which will remain forever beyond our reach? Each horizon forms a boundary not only in space, but also in time. If in ancient times a traveller saw a distant mountain range at the horizon, he knew that it would take many hours to see what might lie on the other side. His horizon of vision, of cognition, thus had a spatial dimension in miles and a temporal one in hours,

H. Satz, Ultimate Horizons, The Frontiers Collection, DOI: 10.1007/978-3-642-41657-6_1, © Springer-Verlag Berlin Heidelberg 2013

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1 Horizons

Fig. 1.1 Phoenician sailing vessel

determined by his walking speed. This temporal limit also inspired men to find ways to to transcend it faster. A horse could help to bring our traveller more quickly to the mountains, and for ages that was the solution. Stage coaches defined the travel time and comfort. Postal relay stations were established, where tired riders and exhausted horses could be replaced, and in this way, news was distributed with remarkable speed (Fig. 1.2). Such post rider systems existed already in ancient Egypt, Persia and

Fig. 1.2 Post rider in 1648, announcing the end of the Thirty Years’ War in Europe

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Horizons

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China three thousand years ago, and in the Roman empire, post rider relays could cover 300 km in a twenty-four hour period. Post riders and post carriages determined the speed of communication until the nineteenth century, and it was the Pony Express that brought the American West within reach. More than 400 horses and over ten days were needed to transport a bag of letters from coast to coast. If we combine the spatial and the temporal aspects of horizons, we obtain an interesting new form of limit.

1.1

The Horizon of Accessibility

For illustration, let’s go back to the time of the post riders, with a 300 km per day coverage. In that case, to send a message to some person in a place 900 km away would have taken at least three days. For that length of time, the person to be reached was simply beyond our accessibility horizon. Of course, the longer we are willing to wait, the greater becomes the region with which we can communicate. The resulting partition of space and time into accessible and inaccessible regions is shown in Fig. 1.3. It is, however, a relative thing—the size of the region accessible to us after a given time also depends on the speed of the messenger; the faster the messenger, the further back the horizon recedes. Today’s means of transportation reduce the days, weeks or months of former times to just a matter of hours. A hundred years ago, a trip from Europe to the Far East meant many weeks on a steamboat; today it takes ten hours or less by plane. In fact, if it comes down to simply exchanging information with the “other side of the mountain”, we don’t need a messenger; telephones can do that almost instantaneously, and satellite stations connect us to all parts of the Earth. For communication, our temporal

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distance to far away regions has thus become largely a matter of how fast we can transmit a signal there. But we know that there is a limit to the speed with which we can transfer information: the speed of light, some 300,000 km/s. There is no way to send a signal faster than that. Learning this was certainly one of the crucial steps in our study of nature and the universe. On Earth, the effect of the finite speed of light is practically negligible. To send a radio message half way around the globe (over a distance of 20,000 km) takes about 1/15 of a second, so for everyday purposes, it’s almost instantaneous. But the stars we see are very far away, and with the given finite speed of light, that really matters. What is happening here and now can be known in distant stellar worlds only much later, and what we know of them is their remote past. The light of the stars that we see now was emitted millions of years ago, and we don’t know if these stars still exist today, and if they do, where they are. So there are horizons seemingly beyond our reach. Nevertheless, also that inaccessibility seems to be just a question of time. If we wait long enough, even the light that distant starts emit now will eventually arrive on Earth. Just as we could define an accessibility radius for the post rider, we can also do this for radio signals travelling at the speed of light. Then, a place 900 km away was out of reach for us for three days; here and now, we have regions we cannot communicate with for some fractions of a second. What is different, however, besides the sheer scale of things, is that by going from man to horse to train to plane, the messenger speed increased, and so did the range to the horizon at a given time; its size was relative. For the radio signal, on the other hand, travelling with the speed of light, no further speed-up is possible. This is the end of the line, the ultimate horizon at any specific time, or in physics terminology, the event horizon. Whatever lies beyond this horizon is out of our reach—with that reach defined in terms of both space and time. In astronomical dimensions, the size of the space-time region beyond our reach of course grows considerably. Given the present human life span, a star 100 light years away cannot today send us a signal we will live to receive, nor can we send it one which it will get in our lifetime. This, however, is our personal problem; our great-grandchildren could in principle receive the signal sent today from that star. So if we consider the ultimate accessibility limit given by the speed of light, shown in Fig. 1.4, we can label the accessible region as “future”, the inaccessible one as “elsewhere”. The distant star * is now in the “elsewhere”, we have no way of reaching it. But if we wait a while — quite a while in fact — then in the future a radio beam from our position will reach it, and its signal will reach us. So the existence of the event horizon means our contact with the world around us is a question of space and time. The further away something is in space, the longer the time needed to send it a signal, or to receive one it sent. It is the event horizon that forms the border between future and elsewhere. What is now at some point outside of our reach, in the elsewhere, will in the future become accessible for us. But there are instances where this is no longer true. Today’s physics and cosmology provide a more stringent form of limit: a truly final horizon, the absolute event horizon. It defines those hidden regions of the world with which no communication

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The Horizon of Accessibility

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distance [billion km]

will ever be possible for us, not now and not at any time in the future. From such disconnected regions, we can never ever receive a signal, no matter how long we wait and what tools we use. How is that possible? This question leads to some of the most striking phenomena encountered in our present view of the world. Not being able to communicate with a region of the universe must mean that light from “there” can never reach us. This can happen either if there are regions which are somehow moving away from us faster than the speed of light, or if there are regions which do not allow light to leave them. Both in fact exist. Was our universe always there? If not, how old is it? Modern cosmology tells us of a beginning, a Big Bang about 14 billion years ago, producing immensely hot and dense primordial matter, which has subsequently expanded to become our universe. The time of the Big Bang is specified, but spatially it is not defined: 14 billion years ago it began “everywhere”, the primordial world was not a hot little sphere, which then exploded. That means that if there were, at that time, regions far away from where our part of the world started, then they could not, until today, send us a signal. Light emitted by them has simply not yet had the time to reach us. The world that we see is a result obtained by combining the speed of light and the age of the universe. Anything beyond the limits that this defines is simply outside of our reach: we have no sign of it. But this is still the observable world now. The longer we wait, the more of the primordial world will become visible—or so it seems; the light from more distant stars is “on its way to us”. But while we are waiting, the universe does not hold still. Recent astronomical observations have shown that it is in ever increasing expansion. If this expansion is rapid enough, there will be stars whose light can never reach us, which will remain forever beyond our horizon. And some of the stars from which we are presently receiving light will eventually, through the expansion of the universe, be pushed beyond our event horizon: they will fade away and be gone for us.

6

1 Horizons

But this cosmic event horizon is still “ours”; a distant galaxy we can see will have its own cosmic event horizon, which will reach further out than ours. In other words, our accessible worlds will overlap in part, but they will not be identical. And at our horizon, or at that of any other galaxy, absolutely nothing happens. Its again that elusive thing: the closer we get to it, the further away it moves. Besides these fleeting limitations to our outreach, there are, however, also more definite ones. In many old fairy tales, there is a castle with many rooms. You may visit them all, except one, which you should never ever enter: if you do, you will suffer a horrible fate. It turns out that this can also happen in outer space.

1.2

Forbidden Rooms in the Universe

If you enter a black hole, you will never come out again to tell what you saw and what happened to you. At the horizon of the black hole, if you try to avoid falling into it, you will certainly experience some rather unpleasant effects. And this will not just be your fate—it will happen to anyone who would dare to try. Black holes are “dead” stars of huge mass, but small size. A star starts its career as a gaseous cloud, which gravity contracts more and more. When it has become compact enough, the fusion of hydrogen to helium lets it shine, but eventually all the fuel is burnt and gravity compresses the remaining ashes of the stellar mass to an ever smaller sphere. At the end we have an object of such a high gravity that it pulls everything in its vicinity into its range of attraction, even light. Since no signal from such a black hole can reach the outside, it appears to be completely decoupled from our world. We can never see what is inside, and for anything within its interior, we are behind an insurmountable event horizon. Thus, in the vast expanses of space, of the cosmos, there are indeed regions remaining forever beyond our horizon. But also at the other end of the scale, in the microcosmos, in the very small, we find an ultimate limit. Just as there is an end to our reach in the limit of large scales, there is one as we try to divide things into ever smaller entities. Since antiquity, man has tried to picture the complex world we find around us as the result of a combined effort of many identical, simple building blocks, interacting according to basic laws. Complexity thus is thought to be a random child of simplicity, evolving through patterns defined on a higher level. This “reductionism” has been immensely successful in understanding the structure of matter. Depending on how the building blocks are packed, we have solids, liquids or gases; their constituents are molecules arranged in decreasing orderliness. The molecules themselves are made of atoms, which in turn consist of positively charged nuclei surrounded by negatively charged electrons, bound by electromagnetic forces to form electrically neutral entities. If we heat the system enough, or apply a very strong electric field, such as a stroke of lightning, the atoms break up into their charged constituents, forming a fourth state of matter, the plasma. Our view of the states of matter, with solids, liquids, gases and plasmas, thus agrees very well with that of antiquity, having earth, water, air and fire (Fig. 1.5). And already in antiquity

1.2

Forbidden Rooms in the Universe

7

Fig. 1.5 The four states of matter in antiquity: fire, air, water, earth

the philosophers, in the Greek and as well as in the Hindu–Buddhist world, thought it necessary to have a fifth form, a quintessence, as a stage for the others, a medium in which they exist: the void, empty space. The existence of different states of matter leads to features very reminiscent of horizons. For a trout, the surface of the water forms its horizon of existence, apart from short leaps up to catch flies; the shore as well is a definite limit to its living space. In general, the boundary surfaces between the different states of matter (air– water, water–ice and so on)—in physics terminology: phase boundaries—separate worlds of different structure. In ice, the molecules are arranged by firm bonds to form an orderly crystal pattern, a regular lattice with a periodic structure and of welldefined symmetry. In water, that lattice is no longer present; the bonds soften and become flexible. They now allow the molecules to move around in any direction, yet still restrain them to a rather small spatial volume. In the gaseous state, the bonds dissolve completely and we now have a system of balls colliding and scattering off each other, but otherwise free to move around in the entire container. So the same basic constituents in different order patterns give rise to the different states of matter, and the boundaries between such states form horizons between worlds of

8

1 Horizons

different order. But such horizons are again of fleeting nature, they can be shifted, lakes can dry up, land can become flooded. And in all these cases, however, the states remain divisible into their constituents; we can isolate such a constituent and consider it individually. In fact, we can continue with the division, breaking up the molecule into atoms, the atom into a nucleus and electrons. Nuclei in turn consist of nucleons, that is, protons and neutrons; by binding different numbers of these, we obtain the nuclei of the different elements, from hydrogen to uranium and even heavier transuranium elements, artificially created by man. For this binding, strong nuclear forces come into the game, overcoming the electric repulsion between the positive protons. Also these basic constituents of matter can in fact exist in vacuo: electrons, nuclei, protons and neutrons can be isolated and have a mass and a size. So in a way they are the true building blocks of matter; however, the experimental study of the forces between individual nucleons has shown that they are not really the end of the line.

1.3

Ultimate Constituents

If we collide two protons, such a collision produces a multitude of similar particles. It is not that the protons are “broken up”: they are also still there, in addition to all the other newly created ones. An understanding of such interactions ultimately led to further substructure: a nucleon is a bound state of three quarks, bound by an extremely strong nuclear force—bound so strongly that an infinite amount of energy would be needed to split a nucleon into quarks. So we can never isolate a single quark. The Roman philosopher Lucretius had concluded over two thousand years ago that the ultimate constituents of matter should not have an independent existence, that they can only exist as parts of a larger whole. And indeed this feature is today the basic property of quarks, whose bound states form our elementary particles (Fig. 1.6). The quarks are forever confined to their world, quite different from ours, a world that does not have a vacuum, in which there is no empty space, in which they always remain in close contact with their neighbors. They can never escape from this world of exteme density, just as nothing can ever escape from the interior of a black hole.

Confinement Horizon

Matter

Atoms

Nucleus & Electrons

Fig. 1.6 The chain of reduction for the structure of matter

Protons & Neutrons

Quarks

1.3 Ultimate Constituents

9

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 until we get to times close to the Big Bang, we find galaxies being compressed, less and less empty space existing, matter reaching ever greater densities. And when we are close enough to the beginning, the overall density of the entire universe will be higher than that inside a single nucleon, there will be no more void, and the universe will consist of primordial quark matter. The world as we know it, clusters of material in empty space, is gone; one of the primordial temporal horizons of the universe is thus the birth of the vacuum. Human imagination has carried us back even further than that. Electrons and quarks still have intrinsic masses, and so, following again Lucretius, we can ask where they came from. We can picture an even younger universe, in which such masses did not yet exist, only energy. The appearance of intrinsic masses thus defines yet another, even earlier horizon of the nascent universe. So wherever we look, be it on Earth or in space, on large or on small scales, now or in the past, even back to the very beginning: we always seem to encounter horizons, and beyond these, further horizons. We have always been searching for the last horizon, and the perseverance in keeping up this search is perhaps one of the features that made mankind what it is today. Is there an end to our search? Before turning to the stellar dimensions of the cosmos beyond what we can see, or to the microcosmos at scales below what we can see, it seems natural to look at the world around us and remember how its limits were discovered.

1.4

The End of the Earth

Around 1400 A.D., this end had a name: Cape Bojador, the cape of fear, the cape of horrors, the cape of no return. That is where you might risk falling off the face of the Earth, and of all the horrible things that could happen to those who went to sea in the days of old, that was the worst. They had to face a multitude of dangers. Uncounted men did not return, uncounted mothers and wifes wept for sons and husbands. “How much of the salt in the sea comes from the tears of Portugal?” asked the great Portuguese poet Fernando Pessoa. Cliffs, storms, killer waves, sea serpents, giant octopuses and other monsters of the deep—more horrifying than all these was the thought of falling over the edge of the Earth (Fig. 1.7), of disappearing into nothing, without a grave, without a cross, without the blessings of the church. Somewhere the world must presumably end, and one should not really sail that far. From our modern point of view, Cape Bojador is the western tip of Africa; but then the world looked different. In the year 1419, the Portuguese Prince Henrique, Infante of Portugal and “Henry the Navigator” for posterity, became governor of the Algarve, and he dedicated his life to finding out what was beyond Bojador. First, he had collected all reports about the approach to the unknown regions, to establish a theoretical basis for further action. At the same time, he supported the development of a new type of ship, the caravelle, which in matters of navigation was a great

10

1 Horizons

Fig. 1.7 Sailing off the edge of the Earth

improvement over all other vessels existing at the time. Finally, in the year 1423, Henry gave the orders to sail south and check reality. Fifteen times, ships set out to see what, if anything, was to be found beyond Cape Bojador. They either returned without being able to tell anything about the beyond (“the horror made us turn back”), or they were never heard of again. Finally, in 1423, on his second try, captain Gil Eanes and his brave crew succeeded: they sailed around the end of the Earth and thereby showed that this it was not. The subsequent events are well-known: Following his course, Bartolomeu Dias reached the Cape of Good Hope in 1488, and noted that the coast of Africa there turned north again. Given this information, Vasco da Gama left Portugal in 1497 with the aim of reaching India. This turned out to be quite straightforward: in Malindi, in what today is Kenya, he met the Arab nautical expert Ahmed ibn Majid, who provided him with maps and a local pilot. And some weeks later, on May 18, 1498, the Portuguese fleet reached the Malabar coast of India, where Vasco da Gama proceeded to present his credentials and royal Portuguese greetings to the Raja of Calicut. Some years earlier, in 1492, Christopher Columbus, in the service of the Spanish crown, had reached “West India”, on the other side of the Earth. In spite of considerable evidence to the contrary, such as the lack of cities and the failure of the natives to understand the Arab interpreters of the Spanish fleet, Columbus insisted all his life that it was India that he had found. But when Fernando Magellan

1.4

The End of the Earth

11

not much later sailed from Europe around Cape Horn, the southern tip of what was in fact the “new” American continent, continued westward and finally returned via India, it was clear to all: the Earth is a globe. There is no mystical border, beyond which unknown forces operate. The Earth as a flat disk of finite size: even in the time of Henry the Navigator that was actually more of a maritime legend of old than accepted reality. As early as four centuries before Christ, Aristotle had argued that the Earth must be a sphere, since viewed from the coast first the hull and only later the sails of departing ships would disappear. Moreover, the shadow of the Earth at a lunar eclipse was always circular. And in spite of intermediate objections, this knowledge was not forgotten. The Earth as a flat disk from which you could fall off: in educated circles that was never very credible. The most influencial theologian of the middle ages, Thomas Aquinas, summarized the situation 200 years before Henry the Navigator quite precisely: Astrologus demonstrat terram esse rotundam per eclipsim solis et lunae. The astronomer proves through solar and lunar eclipses that the Earth is round.

Even the size of the terrestrial sphere was quite well known. More than 200 years before Christ, the Greek astronomer Eratosthenes had used solar measurements in Egypt to determine it. He compared the positions of the Sun precisely at noon in the city of Syene (today’s Assuan) with that in Alexandria. The two cities lie on the same longitude, so that they do not have a time shift. He noted that when the Sun was at the zenith, directly overhead, in Syene (point a in Fig. 1.8), in Alexandria (point b) it was an angle α of 7.2◦ off the zenith line (i.e., a line orthogonal to the surface of the Earth). Simple geometry shows that α is also the angle between the lines from the center of the Earth to Syene and to Alexandria, respectively. The observed angle of 7.2◦ is just 1/50 of the full circle of 360◦ , so that 50 times the distance L between the two cities would

Fig. 1.8 Eratosthenes’ determination of the Earth’s circumference

sun

α L a b

α earth

12

1 Horizons

give the circumference of the Earth. The separation distance had been determined by royal step-markers of the Egyptian court, men who would walk from one city to the other in steps of as equal a length as possible. They had found the distance between the two cities to be 5,000 stadia, about 750 km. The full circumference of the Earth must thus be 50 times that distance, 50 × 750 = 37,500 km. Today’s measurements give 40,000 km for the polar circumference, attesting to both the logical reasoning of Eratosthenes and the precision of the royal step-markers. So, all that was known at the time of Henry the Navigator, but it was theory. 200 years before Vasco da Gama and Columbus, in 1291, the brothers Ugolino and Guido de Vivaldo from Genoa in Italy had left their city on board two well-armed ships, the Allegranza and the Sant’Antonio, along with a crew of 300 men, with the aim of reaching India via the Atlantic. So the idea of such a passage had also been around for a while—theirs was the first known try. The Genoese sailed south along the Maroccan coast, and the last message from them came from a place about a hundred miles before Bojador. Nothing was ever heard of them again. Many things can interfere between our ideas and the real world, and the early explorers—Gil Eanes, Vasco da Gama, Christopher Columbus, Fernando Magellan—had established where they matched. Their achievements were a crucial step in making observation, not contemplation, the way to determine our ultimate picture of the world. After them, our terrestrial world was finite, was a sphere. For mankind ever after, that was not theory, not thinking, not imagination, but reality.

1.5

The Roof of Heaven

1.5

The Roof of Heaven

13

And that inverted Bowl we call The Sky, Whereunder crawling, coop’t we live and die,

wrote the Persian astronomer, mathematician and, last but not least, poet Omar Khayyam around 1100 after Christ. Is the sky indeed something like a roof over the Earth, and if so, what is above that roof? The idea of a “firmament” above us, on which the Sun, the Moon and the stars are attached, ran into problems from the beginning, because up there everything is in motion. So not only Sun and Moon would have to move along fixed tracks on the firmament, but all the planets as well. Once it was established that the Earth was a sphere, the geocentric view of the world meant that it was the stationary center surrounded by concentric moving spheres. The Earth is the center of the universe, and all the heavenly bodies are attached to spheres around it. These in turn rotate in different directions and with different rotation speeds. God lives behind the last and largest of the spheres and, as “prime mover”, keeps them rotating. This is indeed a task for a god. While it is quite easy to picture the Sun on one sphere around the Earth, and the Moon on another, to account for their positions relative to us, precision measurements of the relative Sun–Moon positions began to pose problems, and the relative motions of the planets led to immense complexity. Thus, as seen by a terrestrial observer, the planets, such as Mars, did loops in the sky… Nevertheless, astronomers of the time were up to the task. The culminating geocentric scenario was developed by Claudius Ptolemy, a Roman citizen of Greek origin living in Alexandria, Egypt, in the first century A.D.; his work is generally known by its Arab title Almagest, since it was preserved, as were many other Greek works, in Arab translation. In this picture, the planets still move around the Earth, but in order to account for their observed orbits, they perform smaller circles (epicycles) around a larger circular path . The entire world is still surrounded by a rotating firmament, on which the most distant “fixed stars” are attached. The final pattern traced out by the heavenly bodies is a beautifully intricate pattern, shown in Fig. 1.9. Complex as it is, the corresponding tables did allow remarkably accurate predictions of stellar positions and remained in good service for over a thousand years. But with time and further observations, things became more and more involved and apparently ad hoc: the epicycles of Ptolemy had to be determined specifically for each planet, the center of the large circle was shifted from the Earth, and more. The complexity of the formalism had become so great that King Alfonso X of Castile , who was a great patron of astronomy in the eleventh century and had a compilation made of Ptolemy’s works, based on Arab translations, is supposed to have said that “if the Lord Almighty had consulted me before embarking on Creation, I would have suggested something simpler”. Hence it seemed not unreasonable to step back and ask if there might not be a more appropriate way to account for the observed. This is where Nicolaus Copernicus came in, around 1510 A.D., when he proposed the Sun as the center of the observable stellar world. He did acknowledge some hints from antiquity; the Greek astronomer Aristarchos of Samos had suggested a heliocentric universe already more than two centuries B.C. Aristarchos had estimated the Sun to be much larger and heavier than the Earth, and thought it more reasonable

14

1 Horizons

Fig. 1.9 The orbit of Mars around the Earth, according to Ptolemy

for the smaller body to circle around the larger. But Copernicus now developed a mathematical model, in which the different planets circled around the Sun in different distances and moreover rotated around their own axes. It was still a world of spheres, with a final outer sphere for the fixed stars, centered at the Sun and containing within it the circular orbits of the planets. In the aesthetic and religious thinking since antiquity, circles and spheres were considered as the symbol of universal harmony, and so their use as a basis seemed natural to Copernicus. Nevertheless, the Earth was now no longer the stationary center, the fixed point of the universe. It rotates about its own axis once a day and around the Sun once a year. In its time, the model of Copernicus did not receive serious criticism and was apparently received favorably even by the Roman clergy. This does not imply, however, that it was accepted in the present sense. It was rather considered an abstract construct, a mathematical scheme to calculate the motion and position of the heavenly bodies, and even at that, it was not perfect. It was left for Johannes Kepler to replace the circular orbits by ellipses to obtain precise agreement. And for much of the common world, a heliocentric universe with a rotating Earth was simply nonsense. Martin Luther is quoted as saying about Copernicus “that fool is turning astronomy upside down…”. Johannes Kepler, some hundred years later, had one great advantage: he had access to detailed astronomical measurements by Galileo Galilei and by Tycho Brahe. Developments in telescope construction had made these possible and so provided a solid empirical basis requiring a precise mathematical description. Kepler, as well as Galileo, considered the heliocentric universe as the true description of the cosmos, not just a model to compute the positions of planets. As a result, strong protest came from both the catholic and the protestant churches. Moreover, his work was carried out during the time of the 30 years’ War between the two christian fractions in

1.5

The Roof of Heaven

15

Germany, and Kepler, refusing to take sides, had to flee several times from persecution. Nevertheless, he remained deeply religious. For posterity, he remains, perhaps above all, a brilliant mathematician and thus able to construct a mathematical theory to account for the data he had obtained, known today as Kepler’s laws of planetary motion. These laws described with great precision the elliptical orbits of the planets around the Sun, without, however, explaining why they moved in this way. Kepler believed that there must be some force of the Sun, acting over large distances and counterbalancing a centrifugal outward push, to keep the planets in orbit. At his time, that was speculation—to be made into a physical theory almost 80 years later, by Isaac Newton, who wanted to explain as well as describe. The required abstraction was that the same forces that act on Earth also govern the motion in the heavens. On Earth, “falling bodies” were a common phenomenon, rain fell from clouds, apples fell from trees, arrows and cannonballs rose and then fell. Correcting some Aristotelean misconceptions, Galileo Galilei had already established that the falling of all objects follows a universal law: the distance a body has fallen grows with the square of the time and is the same no matter what the mass of the body is. To be sure, a feather falls slower than a stone, but this is because it tends to “float” in the air. A stone the weight of a feather falls in the same time the same distance as a heavier stone. The observations of Galileo soon led to what is today called classical mechanics— the beginning of physics as we now understand it. Isaac Newton, in his celebrated Philosophiae Naturalis Principia Mathematica formulated the theory describing the effect of forces on material bodies and on their motion. In antiquity, the natural state of a body was thought to be “at rest”; any motion seemed to require some action on the body, a cause for getting it to move. Galileo, and following him more succintly Newton, replaced this by noting that rest means something different for someone on a boat floating on a river and for an observer on the banks of the river. So a first kind of relativity principle appeared: all states of constant relative motion with respect to each other are equivalent, none is more natural than the other. Or, in Newton’s terms, a body in uniform motion will remain that way unless acted upon by some force. That introduced the concept of force as the agent resulting in a change in the state of being of anything, as the reason for acceleration, as the origin of action and reaction. One immediate outcome of this was the theory of gravitation, of the forces between celestial bodies. Gravity was the first universal force to be encountered by humans. To be sure, there were many other forces, of wind, of the sea, of an ox pulling a plow, of a bowstring shooting an arrow. But they were dependent on time, circumstance and cause, whereas gravity was always there, everywhere and at all times. A stone released would fall to the ground, in the same way, no matter who released it, where and when. There seemed to be a mysterious attraction of things to the Earth. It was Newton’s great achievement to relate this everyday force to that determining celestial structure and motion. Newton’s theory of gravitation states that a massive object attracts any other massive object with a force that is proportional to the product

16

1 Horizons

Fig. 1.10 The Copernican picture of a universe with an ultimate horizon, a final outer firmament holding the stars

of their masses and inversely proportional to the square of their separation distance, M1 M2 r2 where M1 and M2 are the masses, r their separation, and G Newton’s universal constant of gravitation. The force of gravity is always attractive, and it acts over immense distances without any apparent connection between the interacting objects, and, as it seemed, instantaneously. It holds the Earth and the other planets in orbits around the Sun, with the centifugal force of their motion just balancing the attraction of gravity. In the same way, it binds the Moon to the Earth. We know today that it is this force that holds galaxies together and that determines the large-scale structure of our universe. And yet it is the same force that determines the change of motion of the objects of our everyday world, the falling of apples, the rising of airplanes, the orbits of the satellites providing our communication. Gravity is thus the most universal force in the world, operative from our human scale to that of the entire universe. So, at this point, astronomers had a consistent theoretical explanation for the structure and motion of the observable world: the Earth, the Moon, the Sun, the other planets and their moons. The Sun is its center, and the force holding everything in place in the heavens, gravity, is the same force giving mass and weight to all objects on Earth, making apples fall from trees and preventing stones from jumping into the sky. Behind all this, there still was the the outer sphere, holding the fixed stars (Fig. 1.10), and beyond that sphere…what was there? In Greek philosophy, nothing, infinite and eternal nothing. But off and on, the possibility of a universe without a last sphere was brought up. Instead, beyond the solar system, there could be an infinity filled homogeneously with fixed stars; such a scenario had been considered by the F=G

1.5

The Roof of Heaven

17

English astronomer Thomas Digges in 1576. Thoughts of this kind were always on the verge of being heretic, in the eyes of the church. The Italian philosopher Giordano Bruno not only believed that the universe is infinite, but that it is filled with an infinity of worlds just like our own. This was clearly in violent contradiction to the dogma of one world made by one creator according to the scripture. And so on February 17, 1600, Giordano Bruno was burned at the stake in Rome.

2

The Vanishing Stars

Were the succession of stars endless, then the background of the sky would present us a uniform luminosity—since there could be absolutely no point, in all that background, at which there would not exist a star. Edgar Allan Poe, Eureka, 1848

In spite of Giordano Bruno’s fate, the limits of the universe continued to occupy the minds of many scientists and philosophers. Is there indeed some ultimate celestial sphere? And if so, what is in that forbidden “room” beyond it? The existence of a final firmament, to which the fixed stars are attached, did in fact answer one rather curious question. Why is the sky dark at night? If there were no such sphere, if instead a world of stars continues on and on, homogeneously, with the same density, forever outward, then every spot in the sky will be filled with shining stars, some closer, some further out, and further yet. Copernicus insisted on a fixed outer sphere with a finite number of stars and thus avoided the problem. Kepler had realized the difficulty and therefore also ruled out the possibility of an infinite universe. Still the question kept reappearing and is today known as Olbers’s paradox, after the German astronomer Heinrich Olbers, who formulated it most succinctly in 1823. It is an excellent illustration of how a well-posed question can lead to progress in thinking and understanding. To answer it, however, we first have to address one of the basic issues of physics: what is light?

2.1

The Speed of Light

But what and how great should we take the speed of light to be? Is it instantaneous perhaps, or momentary? Or does it require time, like other movements? Could we assure ourselves by experiment which it may be?

H. Satz, Ultimate Horizons, The Frontiers Collection, DOI: 10.1007/978-3-642-41657-6_2, © Springer-Verlag Berlin Heidelberg 2013

19

20

2 The Vanishing Stars

The question had been around for quite a while when Galilei, in his Renaissance treatise on the Two New Sciences, made his alter ego Salviati ask it. Already Aristotle had complained more than 300 years before Christ that Empedocles says that the light from the Sun arrives first in the intervening space before it comes to the eye or reaches the Earth.

He, Aristotle, was sure that this was completely wrong, that “light is not a movement”, and his belief dominated western thinking for almost 2,000 years. The speed of light is infinite—even great scientists and philosophers like Johannes Kepler and René Descartes were more than convinced of that. Descartes said that “it is so certain, that if one could prove it false, I am ready to confess that I know nothing at all of philosophy”. Galilei, of course, proposed the right way to resolve also this issue: experiment. He even tried it himself, but at that time terrestrial techniques were not up to the task. A distant assistant had to cover and uncover a lamp, and Galilei tried to measure the time it took him to see that. He correctly noted that light did travel faster than sound. But to determine its speed, one needed longer times and hence larger distances, and these were then to be found only in astronomical domains. The problem was, in fact, twofold. Is the speed of light finite, and if so, what is its value? The first question was answered several decades later by Ole Rømer, a truly multitalented man from Aarhus in Denmark. His real name would have been Ole Pedersen, but with so many Pedersens around, his father had started to call himself Rømer, after the island of Rømø, where they came from. Ole had studied physics, mathematics and astronomy in Copenhagen and eventually married the daughter of his professor there. In between, he had worked for King Louis XIV in Paris and took part in the design of the fountains of Versailles. After this interlude, he returned to Denmark for an appointment as “royal mathematician”, where he introduced the first national system of weights and measures, as well as the Gregorian calender. And besides all this, he became Chief of the Copenhagen Police, responsible for the installation of the first street lights there. In Paris, he had worked as an assistant for the astronomer Giovanni Domenico Cassini, and Cassini had made a remarkable observation. The planet Jupiter, fifth around the Sun and largest of all, had a Moon, called Io (named after a nymph seduced by the Roman god Jupiter, in his Greek avatar form of Zeus), which circled around it approximately once every 42 h, in contrast to the 28 days our earthly Moon takes for its orbit. That meant that seen from the Earth, there would be many “eclispses” of Io at any stage of the Earth ’s orbit around the Sun; the geometry is shown in Fig. 2.1. One could thus measure the time at which Io disappears behind Jupiter, and do this for a series of eclipses. This provided a determination of the time between successive eclipses, giving a prediction for the next. And the striking observation first made by Cassini was that the onset of an eclipse fell more and more behind schedule the further away the Earth was from Jupiter. Cassini was not sure, but thought that perhaps light takes some time to reach us. Eventually, he seems to have rejected this conclusion. Rømer, instead, combined a number of different measurements, extrapolated them to eliminate interference

2.1 The Speed of Light Fig. 2.1 Ole Rømer’s basis for the determination of the speed of light

21 Io Jupiter

b

Sun Earth a

effects, and found that the delay in time of eclipse onsets seen from the point of greatest Earth—Jupiter separation (point a) compared to those seen from the smallest distance (point b) was about 22 min. From this he now concluded that the speed of light is indeed finite and that the 22 min 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 from these measurements, the size of the orbit of the Earth around the Sun had to be known. How far did light have to travel in these 22 min it took between the two extreme points? This distance, divided by 22 min, would then be the speed of light. The relevant information to determine the distance from Earth to Sun was actually available at that time, due mainly to the studies of Cassini. The first numerical value for the speed of light, however, was apparently obtained by the Dutch physicist Christiaan Huygens in 1678, two years after Ole Rømer had announced his conclusions. Kepler had, in this “third law” of celestial motion, concluded that the time for a planet to orbit the Sun was related to the distance between this planet and the Sun; from this, the relative distances of all planets from the Sun were known. In particular, the distance between Mars and the Sun was found to be about 1.5 times that of the Earth and the Sun. To arrive at an actual value for the Earth–Sun distance, some astronomical distance had to be measured in terrestrial units, and this “calibration” had in fact been carried out by Cassini and his collaborator Jean Richer. They measured simultaneously the position of Mars relative to the fixed star background, Cassini in Paris and Richer in French Guiana. This gave them an angle and a known terrestrial distance, the 4,000 km between Paris and Guiana, and geometry then determined the distance between Mars and Earth. They found it to be about 73 million km. At the point of closest approach of Mars and Earth, that led to 146 million km for the distance between Earth and Sun. Since light travelled, according to Rømer, twice that distance in 22 min, Huygens noted that its speed must be about 220,000 km/s. This result, obtained over 300 years ago by a combination of logical thinking, abstraction and rudimentary measurements, is certainly one of the great achievements of the

22

2 The Vanishing Stars θ

stationary mirror

stationary mirror

d detector rotating mirror

light source

2θ rotating mirror

detector light source

Fig. 2.2 The mirror arrangement used by Fizeau and Foucault for a terrestrial determination of the speed of light

human mind; it is only about 25 % too low according to today’s precision value, measured using radio signals between space craft positioned in the solar system. The first terrestrial measurements were carried out in Paris by Hippolyte Fizeau and Léon Foucault around 1850, improving the attempt of Galileo by reflecting light in a clever arrangement of mirrors. Foucault, with his celebrated pendulum, had in fact also provided for the first time direct proof of the rotation of the Earth around its axis. But he now modified an older apparatus devised by Fizeau to measure on Earth the time light needs to go from one point to another. The set-up is illustrated in Fig. 2.2. Two mirrors are placed as far apart as possible, at a distance d; they now play the role of Galileo and his assistant. One of the two mirrors is rotating at a speed ω, the other is stationary. A beam of light is directed at the rotating mirror, and that reflects it to the stationary one. When it now returns to the rotating mirror, it has travelled between the two mirrors a total distance 2d. During the travel time, the rotating mirror has turned an angle θ, so it reflects the beam back not at the source of light, but at a detector placed at an angle 2θ away. Knowing d, θ and the rotation speed ω gives the speed of light as c = 2d ω/θ. The results of Fizeau and Foucault were within 1 % of the present value, 299,792,458 km/s. So, the light from the Sun did have to travel through the intermediate space before reaching the Earth, as Empedokles had supposed 2,500 years ago. But what is this light travelling through what we think is empty space? What is it that is moving at 300,000 km/s? This question led to another basic and universal phenomenon of the inanimate world: electromagnetism. Initially, electricity and magnetism entered as two quite separate and distinct features. The first appearance of electricity in the life of humans was lightning, for a long time thought to express the wrath of the gods in a frightful way, and beyond human understanding. A more mundane version was observed by the ancient Egyptians, more than 3,000 years ago; they were familiar with electric fish which could produce remarkable bolts of electricity to stun their prey. This source of electricity was supposedly used already in those days for the treatment of neural illnesses. In ancient Greece, it was noted that rubbing amber with a catskin made it attract feathers and other light objects—and it was this feature that gave the name to the mysterious force, with elektron as the word for amber in ancient Greek.

2.1 The Speed of Light

23

But it took still more than 1,500 years until these various and seemingly unrelated phenomena began to be understood, and only in the last 100 years has electricity dramatically changed human life. Magnetism was more well-defined from the beginning. Several millennia ago it was noticed in China that a certain kind of stone attracts iron, and if suspended by a string, it would orient itself along a north–south axis. Making use of this, the ancient Chinese constructed the first magnetic compass for navigation. In ancient Greece, Thales of Milos described the effect, and since the stones showing such behavior there came from a province called Magnesia, he called it magnetic. In English, it became “leadstone” and finally “lodestone”, presumably because it could be used to lead travellers in the desired direction. Both electricity and magnetism became part of natural science only less than 300 years ago. It was discovered that there exist two different forms of electricity, arbitrarily denoted as positive and negative; each form could be produced by rubbing, for example, and each kind can exist on its own. If two metal balls were prepared to have different “charges”, like and like repelled each other, while positive and negative showed attraction—both by invisible means across the distance of their separation. Charles Augustin de Coulomb in France showed in 1785 that these reactions followed a law very similar to that proposed by Newton for the equally invisible action at a distance provided by gravity (Fig. 2.3). Coulomb’s law gives for the electric force q1 q2 F=K 2 , r

Moon

Earth

+

N

S

S

N

_

Fig. 2.3 Three forms of action at a distance: the gravitational attraction between the Earth and the Moon, the electric attraction between positive and negative charges, and the magnetic attraction between opposite poles, accompanied by the repulsion between like poles

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2 The Vanishing Stars

where q1 and q2 measure the amount of charge on each ball and r their separation; the constant K plays the role of Newton’s universal constant of gravitation, except that it is now positive (repulsion) for like and negative (attraction) for unlike charges. While positive and negative electric charges could exist independently and could be produced separately, magnets were curious animals. They had a north pole and a south pole, and given two magnets, north and south attracted each other, while north/north or south/south meant repulsion. But there was no way to get just one pole. Cut a magnet in two in the middle, and you had two new magnets, each with its north and its south pole. And until today, physicists are still wondering if there isn’t some way to create a monopole. The magnetic force was not quite of the inverse square form encountered in Coulomb’s law of electric interaction or Newton’s law of gravity, since each pair of magnets experienced both attraction, between the opposite poles, and repulsion, between the equal poles. Nevertheless, the interaction between two magnets, as well as that between metals and magnets, was again by some invisible means over the distance of separation. So both electric and magnetic interactions showed a mysterious feature already encountered in the case of gravitation: an interaction over a distance, without any apparent connection between the interacting objects. How such an interaction could arise was something that had puzzled people at all times. Was there some invisible medium filling all of space to provide a connection? The beginning of an answer was provided by the British physicist Michael Faraday, who proposed that each charge would be surrounded by an electric field, radiating out starlike lines of force emerging from the source in all directions (Fig. 2.4). And this field would “feel” the presence of other charges and react accordingly: the lines of force would bend either towards the other charge or away from it, depending on the sign.

Fig. 2.4 Lines of force emerging from isolated sources of positive and negative electricity (top) and from neighboring like and unlike sources (bottom)

2.1 The Speed of Light

25

Moreover, in the early 1800s, Hans Christian Oersted in Copenhagen discovered that there was a strange connection between electricity and magnetism. It was known that certain materials—today’s conductors —allow a rapid spreading of electric charge: they result in the flow of an electric current between opposite charges, forming an electric circuit. Now Oersted observed that a magnet would align itself in a direction orthogonal to the line of current flow, as if the current had created magnetic lines of force around its flow axis. So one could imagine unending lines of force corresponding to magnetic fields, closed loops having neither beginning nor end. This would explain why cutting a magnet in two simply produced two magnets, and did not yield an isolated pole. In the course of the nineteenth century, extensive studies showed that electric and magnetic forces are indeed closely intertwined: electric currents produced magnetic fields and moving magnets induce electric currents. This suggested a unified theory of electromagnetic fields, and it was the great British physicist James Clerk Maxwell who created it, with his famous equations. Through Maxwell, electricity and magnetism were unified to electromagnetism. And in addition, he provided the basis for an understanding of how the interaction of electromagnetic sources could occur over distances. Maxwell showed that a changing electric field generates a magnetic field, just as a changing magnetic field would through induction create an electric field. So the combination of the two, electromagnetic fields, now gained an independent existence, without the need of currents or magnets. And one simple solution of Maxwell’s equations was that of travelling waves, like an excitation travelling down a string, or a wave travelling across a pool of water. The action over a distance could thus occur through the exchange of electromagnetic signals in the form of such waves. They propagate through space at a fixed speed, which can be measured and was found to be the familiar speed of light. The fundamental question what is light? was therefore now answered: it is an electromagnetic wave travelling through space, and the different colors of light simply correspond to different possible wavelengths. Beyond the range of visible light, we recognize today electromagnetic radiation on both sides, with radio waves of longer wavelength (beyond the infrared) and X-rays of shorter wavelength (beyond the ultraviolet). And in a way, it also answered the question of how distant charges could interact: through the exchange of an electromagnetic signal. But the answer was not really complete. If distant charges communicated by electromagnetic waves travelling between them: what was being excited to form such waves? In our everyday world, it can be a string, the surface of water, the density of air. But what is it in empty space that is vibrating? And so the ether entered the world of physics, an invisible medium filling all of the so-called empty space. This satisfied those who thought that truly empty space was “unnatural”, such as the French philosopher Blaise Pascal, who believed that “nature abhors a vacuum”. When Evangelista Torricelli in Italy succeeded in removing all the air from a vessel, Pascal noted that the absence of air does not mean empty. For light, the ether was first introduced by Robert Hooke, in 1665; he pictured a pulse of light like a stone thrown into a pool of water, with concentric waves spreading out. Just as a tsunami wave is formed by an earthquake at the bottom of the sea far out in the

26

2 The Vanishing Stars

ocean and then travels towards some shore, so a change in the electromagnetic state somewhere would be communicated across space to a distant receiver in the form of an electromagnetic tsunami wave in the ether. This ether turned out to be one of the most-travelled dead-end roads of physics. From the time of Hooke to the time of Einstein, a great number of well-known physicists tried their hand at it, and always with rather limited success. Is the ether stationary, or is it comoving with stars? Is there an ether-wind due to the Earth moving through it? Is matter perhaps only a form of vortices in the ether? The presence of an ether resolved the puzzle of an action at a distance, but to do so, it had to be a material substance and yet, at the same time, not seriously affect the motion of the stars. One of the most celebrated experiments to find it was carried out in the 1880s by the American physicists Albert Michelson and Edward Morley. If light was travelling through the ether everywhere at its fixed speed, then it would have to be slower if measured in the direction of the Earth’s motion than if perpendicular to it. They devised an interferometer constructed such as to have two beams of light, one along and one perpendicular to the motion of the Earth, travel the same distance and by means of a mirror arrangement meet again at a given point (see Fig. 2.5) The slowing effect of the Earth’s motion would throw them out of phase, so that a valley in the wave of one would hit a peak in that of the other beam, causing interference. Much to their frustration, Michelson and Morley found no effect whatsoever; all waves arrived completely in phase. No matter how they positioned their apparatus, the speed of light seemed always to be exactly the same. So there was no evidence for any form of ether, and after numerous attempts to find a way out, it was finally banned from physics by Albert Einstein, almost 20 years later. It is now definitely ruled out, at least as far as electromagnetism is concerned.

N

mirror 1 E

W S

M

light source

mirror 2

detector

Fig. 2.5 The Michelson–Morley experiment to detect the presence of an ether. A beam of light is directed at a partially transmitting mirror M, from where part of it is reflected to mirror 1 and then on to the detector, another part to mirror 2 and then to the detector. The direction from mirror 1 to the detector is chosen to be north-south, that from the light source to mirror 2 east-west, and both mirrors 1 and 2 are equidistant from the central mirror M. The motion of the Earth (east-west) relative to the ether was predicted to modify the speed of the corresponding light beam and thereby lead to interference patterns between the two beams arriving at the detector

2.1 The Speed of Light

27

However, even today it is not so clear what the role of a cosmological constant or dark energy is; we shall return to these somewhat ether-like ideas later on. Maxwell’s equations implied a unique speed for electromagnetic waves travelling through empty space, the universal speed of light. This is in fact much more dramatic than it seems at first sight: such a behavior is simply not in accord with our everyday experience. A car moving at 100 km/h, as seen by a stationary observer, has a relative speed of only 70 km/h for someone moving in the same direction at 30 km/h. And two cars, both travelling at 100 km/h in the same direction, are not moving at all relative to each other. If someone in the compartment of a moving train drops a coin, it falls straight down: train, passenger and coin, though all are travelling at high speed for an observer on the ground, are at rest relative to each other. Light is not like that. If a stationary and a moving observer measure the same beam of light, they both find the same value for its speed. No matter how fast you move, the speed of light you measure is always that 300,000 km/s. By moving faster, you can neither start to catch up with a light beam, nor run away from it. And ten different observers, all moving at different speeds, find that, although their relative speeds differ, that of a given light beam is always the same universal value. In the framework in which Newton formulated his laws, this was simply impossible. In a fixed space with a universal time, the speed of light would change for observers moving at different speeds. To make a constant speed of light possible, the ideas of space and time had to be fundamentally modified. To keep a universal speed of light, the scales for distance and time must become dependent on the observer. Let me measure the speed of light in a laboratory here on Earth, and an astronaut measures it in a space ship moving at high speed relative to the Earth: if we both get the same result, than his standard meter and his standard second, as seen by me here on Earth, must have taken on different values than mine—and they do. The resulting milestone in physics was Albert Einstein’s theory of relativity, more exactly, the special theory of relativity. The “special” is an a posteriori modification, indicating that it holds in a restricted spatial region of the universe only. The extension to the entire cosmos, including the role of gravity, followed 10 years later with the general theory of relativity, and again it was Einstein who did it. To formulate his special theory of relativity, Einstein combined a principle proposed by Galileo Galilei 400 years earlier with the recently discovered universal speed of light. Galileo had insisted that the laws of physics be the same for all observers in uniform motion relative to each other. In other words, if I measure the time it takes a stone to fall to the ground from a height of one meter, once in the laboratory and once on a high speed train, the results should be identical. Einstein realized that if this was to hold and at the same time a universal speed of light was to be maintained for all observers in uniform relative motion, our ideas of space and time would have to be modified, space and time would have to be related, and their scales have to depend on the speed of the observer (see Box 1). In Newton’s world, there was a unique time, the same everywhere, and one could talk about two events occurring at the same time. In a relativistic world, synchronization over large distances is not possible, and what is first for one observer, may be later for another.

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2 The Vanishing Stars

Another striking result of relativity theory was the conclusion that no material body could ever move at the speed of light. According to Newton’s law of force, an increase of force must increase the acceleration of a mass and hence eventually bring its speed to arbitrarily high values, faster than the speed of light. Einstein showed that in the regime in which relativistic effects cannot be neglected, that is, at speeds lower but comparable to that of light, Newton’s law becomes modified. Only part of the force serves to increase the speed; an ever larger fraction goes into increasing the mass, the inertia of the accelerated body. In our everyday world, the speeds encountered are so far below that of light that we can safely ignore the speed corrections and work with a speed-independent inertial mass. But in modern high-energy particle accelerators, such as the Large Hadron Collider at the European Laboratory for Nuclear Research CERN in Geneva, Switzerland, one brings protons to speeds 95 % of the speed of light, and then the effective mass of these particles is more than three times their mass at rest. And so it is evident that we can never bring a material body to move at the speed of light—it would require an infinite force to do that. No massive object can ever catch up with a beam of light in empty space; light remains the fastest agent in the universe. Box 1. Relativistic Motion If an observer moving in a spaceship at a high speed v with respect to a laboratory on Earth finds that the speed of light is the same as ours, it must mean that from our point of view his length measure is shorter than ours, or his clock runs slower than ours, or both. Actually, it is indeed both: a given length d0 , a standard meter, has that value for us here as well as for the observer in his moving space ship. But his moving meter stick, as seen by us, becomes shortened to the length d,  d = d0 1 − (v/c)2 , where c denotes again the speed of light. And a fixed time interval t0 on the spaceship clock, if we measure it from here on Earth, appears dilated to become to a longer interval t, t0 . t= 1 − (v/c)2 Evidently, the faster the space ship moves, the greater is the effect, both in the contraction of the length scales and the dilation of the time scales. As a consequence, Newton’s law of force becomes modified as well; it now reads m0 a, F= 1 − (v/c)2 so that the inertial mass m 0 of a body at rest is at speed v increased to m0 m= . 1 − (v/c)2

2.1 The Speed of Light

29

At low speed, as long as we can ignore the (v/c)2 , we recover both the speed-independent inertial mass m 0 and Newton’s force law F = m 0 a. If we consider the force F to be gravity, we see from the relativistic form of Newton’s law that the inertial mass of a body, i.e., its resistance to a force, is not its rest mass, but rather a mass including the energy of motion. Einstein formulated this in his celebrated relation between mass and energy, E = mc2 , which means in particular that energy offers an inertial resistance to any force. Even photons, which have no rest mass, will thus be affected by gravity as if they had a mass determined by their energy. So we can weigh the photons trapped in a container: an empty container is lighter than one containing a gas of photons. So we now know that the light from the stars we see today has been travelling for many years, waves of electromagnetic energy moving through an empty space containing no ether, at a speed of some 300,000 km/s, no matter who measured it. We are therefore prepared to return to the puzzle we had started with.

2.2

Why Is the Sky Dark at Night?

The paradox is today named after Heinrich Olbers; he was not the first to realize it, Kepler did earlier and concluded that the succession of stars is not endless. With Edgar Allan Poe, the problem entered the literary world, leading to pictures that a century later became science, such as an expanding universe starting from a Big Bang. As an earthly illustration of the problem, one can consider an infinite forest: wherever you look horizontally, your line of vision hits a tree. Olbers, in 1823, did state most clearly the assumptions which had led to the paradox: • The universe is infinite in all directions and has existed forever as it is now. • The stars are distributed with the same density throughout the universe, they have existed forever, and they have a finite size and brightness. Given these conditions, the whole sky should be as bright as a typical star; it should never get dark at night. So something must be wrong somewhere, and that something leads us directly to the forefront of modern cosmology and its view of the origin of the universe. If the age of the universe is finite, if there was a Big Bang starting everything a certain number of years ago, then the universe we can see today will also be of finite size, because light has only had those years to travel. To be sure, the numbers are huge, but they are not infinite. Moreover, the stars had to form sometime after the Big Bang, so their number is also finite. In other words, a finite age of the universe allows us to see only a finite spatial part of it, and in that part only a finite number of stars can have appeared since the Big Bang. That is why the sky is dark at night—a

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late answer to Heinrich Olbers, requiring both a finite speed of light and a Big Bang origin of the universe. A simple question can lead you a long way… But how can we be sure that this view of things is really correct? The origin of the universe, in fact the question whether it has an origin, has been the subject of much dispute, scientific, philosophical and religious. There are two main reasons why today most scientists tend to believe in the Big Bang theory—but let us approach them slowly and step by step. A well-known effect in the physics of everyday phenomena is that the pitch of a sound you hear is modified if the source of the sound is moving. The sound of a race car engine seems higher pitched as the car approaches and lower as it moves away, leading to a characteristic tonal flip as it moves past you. In earlier days, the change in tone of the whistle of a passing railroad engine was the typically cited example. The phenomenon is known as the Doppler effect, after the Austrian physicist Christian Doppler. The tone you hear is caused by sound waves of a certain wavelength, and when the source of the sound approaches you, the distance between wave peaks, the wavelength, is shortened, giving a higher sound, and when it moves away, it becomes longer and hence results in a lower sound. The same “Doppler effect” also occurs for light waves, so that one can in fact check if a given far-away star is stationary or moving. Stars emit light of certain characteristic wavelengths (“spectral lines”), and if this light is Doppler-shifted when it arrives at the telescope on Earth, its source must be moving. Let’s say a star is emitting light of a fixed wavelength λ0 , as measured by an observer stationed on that star. For an observer moving away from  the star with a speed v, that light will appear to have a longer wavelength λ = λ0 / 1 − (v/c)2 , i.e., it will be shifted in the direction from blue towards red, it will experience a redshift. The American astronomer Edwin Hubble, working in the 1920s at the Mount Wilson Observatory in California, had studied the light from very distant stars. From measurements of redshifts it was already known that they all seem to be moving away from us at different speeds. Hubble made the striking observation that the further away they are, the faster they recede. The Doppler shift, and hence the speed of the stars’ motion, was rather well measurable—the crucial factor for reaching Hubble’s conclusion was the determination of the distance of the stars in question. To measure the distance of fairly nearby objects in the sky, such as planets, one could use the parallax method employed by Cassini and Richer to determine the distance between Mars and the Earth. Howevever, for the very remote stars Hubble was after, the parallax angle became for too minute to be measurable. The solution came through the extension of a very simple phenomenon. The brightness of a given light source decreases the further one is away from it. Since light is emitted spherically from its source, the light incident on a given surface becomes less and less with distance. The size of the spheres grows as d 2 , with d denoting that distance, and therefore the light per area decreases as 1/d 2 . So if we know the original brightness of the source and its apparent brightness at some distance, then the difference between the two measurements determines d. Now it so happened that the inherent brightness of the stars Hubble was studying, the so-called Cepheid variables, had recently been determined; they were what astronomers today call standard candles. Measuring their apparent luminosity as observed at Mount Wilson, Hubble had at least a good

2.2 Why Is the Sky Dark at Night?

31

estimate of their distance, enough to show him that their speed of recession v became ever greater, the further they were from Earth, with d measuring that distance. The law v = H0 d was named after him, as was the crucial constant H0 . By today’s measurement, his value of H0 was off a bit, but the idea was right and changed our view of the universe. In fact, no matter where he looked, the stars appeared to move away in every direction, so it seemed that the whole universe was expanding. Could that be the case? In Box 2 we look in a little more detail why one might think that. Box 2. The Expansion of Space To simplify matters, we take space to have only two dimensions instead of three, a “flat” world. Consider three stars in this world, numbered 1–3, positioned at an arbitrary starting time t = 0 as shown in Fig. 2.6, with a separation distance d0 between 1 and 2, as well as between 2 and 3. Now let us assume that the space in this world expands with time t by a factor Rt in each direction, so that any distance s0 at t = 0 becomes st = Rt s0 at time t. The separation between stars 1 and 2 thus becomes dt = Rt d0 , and so their speed of separation is dt − d0 (Rt − 1) = d0 = Ht d0 , t t defining Ht = (Rt − 1)/t as our “Hubble” constant. The relation tells us that the rate of separation grows with the initial separation distance d0 . To check that this is really true, we can look at the speed of◦ separation of points 1 and 3, which are initially further apart, namely r0 = 2d0 , as obtained from the triangle relation r02 = d02 + d02 . The rate of separation of 1 and 3 thus becomes ◦ rt − r0 (Rt − 1) vt (13) = = r0 = Ht r0 = 2Ht d0 → 1.4Ht d0 . t t vt (12) =

The separation velocity is thus a factor 1.4 larger than that between the closer stars 1 and 2. We have so far not said how the expansion of space takes place. If it happens at a constant rate, with Ht = H0 t + 1, we get the time-independent form v = H0 d of what is now known as Hubble’s law, with H0 for the Hubble constant. From Fig. 2.6 it is also directly evident that stars 1 and 3, compared to 1 and 2, have to separate by a larger distance in the same time interval and hence must have a higher speed of separation. At this point, we can also clarify a little what is meant by the acceleration of the expansion. The crucial feature is the scale factor Rt , defining how much a meter stick expands in a given time t. For Rt = H0 t + 1, the expansion rate is constant: the stick expands in one minute the same amount now as next year.

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If the expansion increases with time, for an accelerating expansion, the meter stick will grow more in one minute next year than it does now—or less, for a decelerating expansion. The same forms as discussed here in two dimensions hold of course as well in a three-dimensional space. Hubble’s discovery came really at a very opportune moment. The most up-to-date theory of the universe had just appeared at that time, in 1916: Albert Einstein’s general theory of relativity, linking the effect of the force of gravity to the nature of space and time. A ball tied to a string will fly in a circle—but if you only look at its motion, it could just as well be rolling freely in a curved container. The role of the force can thus be replaced by force-free motion in a curved space. Near massive stellar objects, such as the Sun, the force of gravity would in this way distort the surrounding space to such an extent that even a ray of light passing near it would be deflected from its straight-line path. Einstein’s theory was tested in celebrated observations during a solar eclipse in 1919, carried out by the British astronomer Arthur Eddington and his collaborators, and these showed that the positions of stars whose light passed close to the Sun appeared in fact shifted by the amount predicted by Einstein, bringing him world-wide aclaim. However, at the time he formulated his theory of gravitation, the general belief was that the universe was static, neither expanding nor contracting, and so Einstein needed some force to counteract the attractive force of gravity acting on all the matter in the universe. For this, there was no immediate candidate, and the problem has remained somewhat enigmatic until today. Einstein reluctantly solved it by introducing a “cosmological fluid”, filling the entire universe uniformly and providing the pressure needed to balance gravity. It had to have rather strange properties—not affecting any phenomena in the universe, other than gravity, so that it remained undetectable in all other ways. And it had to be tuned very precisely in order to just balance gravity. In a sense, it was a late counterpart of the ether introduced earlier to provide a medium for electromagnetic waves, and this presumably made it particularly undesirable to Einstein. And when Hubble discovered that the universe was in fact expanding, Einstein called his introduction of a cosmological constant, as the fluid is now generally denoted, his biggest blunder. Had he stuck to his original equations, without such a constant, he

Fig. 2.6 The separation of stars due to the expansion of space, starting from a given initial time t = 0 (black) to a final time t (blue)

3

dt

* 3

*

2

d0

* * r * r0

t

2

d0

dt

1

*

1

2.2 Why Is the Sky Dark at Night?

33

could have in fact predicted the expansion of the universe before it was discovered. Today, cosmologists are not so sure if it really was a blunder—dark energy, which we will encounter later in the context of an inflation scenario for the Big Bang, this dark energy may well turn out to be the modern version of Einstein’s cosmological constant, or even of the ether of still earlier times. In any case, in 1922 a Russian theorist, Alexander Friedmann, presented a general solution of Einstein’s equations and showed that they can readily accommodate expanding or contracting universes. And when Hubble a little later found his expansion, the scene was set.

2.3

The Big Bang

The theory itself was initiated in 1927 by Georges Lemaitre, who had studied mathematics and physics at the University of Louvain in Belgium and at the same time prepared for Catholic priesthood; with success on both counts: he received his doctorate in physics in 1920 and was ordained as a priest in 1923. In 1926, when Einstein’s equations had just been seen to describe so well the forces in and the structure of the universe, Lemaitre independently derived Friedmann’s expanding solution and used it to account for the observations of Hubble: he concluded that our visible universe is continuously expanding. Looking the other direction in time, it must then have originated in a very dense, hot, energetic “primordial medium”, which led to the creation of our world. For the Catholic priest Lemaitre, such a creation must have seemed very natural, even though it was a long way away from the dogma applied to Giordano Bruno or Galileo Galilei. But Einstein apparently was not so happy with the results of Lemaitre; “your calculations are correct, but your physics is abominable”, he was supposed to have written to him. Nevetheless, over the years the Big Bang theory continued to gain support, and the perhaps decisive step came in 1964, when the American astronomers Arno Penzias and Robert Wilson discovered what is now known as the cosmic background radiation. It is present throughout the universe as a direct relic of the Big Bang, and it can be measured in the different regions of the sky. Its discovery is one of the truly serendipitous findings of science. Penzias and Wilson were working for the Bell Telephone Company, and they were trying to establish a viable method of microwave communication, by reflecting such signals off high-up balloon satellites. This required the elimination of all other interfering sources of radiation, up to a remarkable precision. Even the detector was cooled to a temperature of a few kelvin, to prevent its “heat” from producing radiation. And when they had eliminated all known sources, including bird droppings on the antennas, there still remained a mysterious background radiation of some three kelvin. It was there day and night, and in all directions. From some friends they heard that in nearby Princeton University, Robert Dicke and collaborators were finishing work on background radiation produced by and remaining from the Big Bang. Penzias and Wilson got in touch with them, discussed their findings and concluded that they had indeed found this left-over

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flash of the Big Bang. Their work was published in 1965 in Astrophysical Journal Letters, in the same issue as the theoretical work of Robert Dicke, Jim Peebles and David Wilkinson, predicting that a form of primordial light should still exist today. So there is more to consider than just the light from the stars. While the Big Bang, in the absence of air, could of course not really “bang”, it did “flash”, leading to the emission of light, and this light is still there as the microwave background radiation observed by Penzias and Wilson. The primordial matter initially was a medium of interacting constituents, a plasma of quarks, electrons, photons and more. Eventually, as the medium expanded and cooled, the quarks combined to form protons and neutrons, and these in turn combined with electrons to form electrically neutral atoms. From this time on, from the decoupling era, about 300,000 years after the Big Bang, the photons were “on their own”; in the absence of any charged constituents, they no longer interacted with the medium, and they don’t interact with each other. From their point of view, the universe contained nothing but light passing freely into the expanding space. From our point of view, the photons of the microwave background radiation are the most primordial signals of the Big Bang we can ever get. Before decoupling, the plasma of charged constituents was opaque to light, so from this plasma we cannot get any direct information. The time of decoupling, of the formation of electrically neutral atoms, is thus for us an ultimate horizon in time—there is no way we can get any direct information from earlier times. When the microwave background radiation was emitted, that is, when the photons became decoupled from any matter, they formed a gas of an effective temperature of about 3,000 K. As a result, the wavelength of the radiation was in the yellow part of our spectrum, so that then the sky was not dark at all—it was in fact bright yellow. But the universe kept on expanding, by about a factor 1,000 since the age of formation of atoms. Since its volume increased, its density of energy became lower and lower, and this in turn meant that its effective average temperature also decreased. Through the expansion, the hot universe of the decoupling era has by today cooled down to about 3 K. As a result, the wavelength of the radiation became longer and longer, so that with about 7 cm it is now in the microwave region, far below the visible range. In a way then, the sky is dark at night for us only because we cannot see this microwave radiation remaining from the Big Bang. If we could put on the right kind of glasses, we could see the glow of the sky at night…a glow not of stars everywhere, but the afterglow of the Big Bang itself. At this point we should note that the light of the stars is, of course, also affected by the expansion of the universe. The Doppler effect that we mentioned above will “redshift” that light, move it to ever longer wavelengths. So up there, in addition to the cosmic background radiation, there is more light than just that of the stars we see. The stars that are moving away from us emit light of wavelengths beyond our visibility range—again, we would have to put on special glasses to see the light from all those stars pushed away from us at an increasing rate by the expansion of space. This redshift is thus an additional reason for the darkness of the night sky. However, the afterglow of the Big Bang also leads to a striking problem. The microwave radiation we receive today from different regions of the sky was emitted in the decoupling era from regions of the universe that had no causal communication,

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which were outside each other’s event horizon. The reason for this is that decoupling occurred so early in the evolution of the universe and hence so long ago, with immense expansion since then. Two markers separated by a distance of 1 km appear to an observer 1 km away from each marker to subtend an angle of 60∞ (see Fig. 2.7). For an observer 10 km away from each marker, the angle has decreased to only a little more than 10∞ . At decoupling, only regions separated by no more than 300,000 lightyears could communicate with each other, and if they are now 1010 light-years away, they appear to us only some fraction of a degree apart in the sky. We have here for the moment neglected the expansion of the universe, which additionally enhances the effect. In other words, if we measure the microwave background radiation at a certain angle in the sky, and then at another angle only a few degrees away, the sources of the two radiation measurements had no chance to communicate at the emission time. So why do both show the same temperature? The microwave radiation we observe was emitted from millions of sources, of spatial regions, which up to decoupling had no way to “tune” their radiation. It is like a gigantic orchestra, without a conductor and with many, many musicians who have no possibility of getting in tune—yet they all end up playing the same melody. If the decoupling of photons and matter, due to the formation of electrically neutral atoms, had occurred at different times in different regions, the temperature of the background radiation should be correspondingly different. But all regions behaved as if some imaginary omnipotent conductor had lowered his baton and indicated “decouple now”. This horizon problem is one of the big puzzles of today’s cosmology, and it is not really resolved to everyone’s satisfaction, in spite of some very interesting proposals. We will soon have a look at one of them in a little more detail. 1 km

Now

60

t

observer x 5 km

Last Scattering 11.5

observer

Big Bang

Fig.2.7 The radiation emitted from a fixed spatial region covers an ever smaller angle of observation with time (left). As a result, the microwave background radiation we receive today comes from regions that were causally disconnected at decoupling time (right)

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But first let us dwell a little more on the cooling process, which since decoupling has brought the temperature of the microwave background radiation from its initial 3,000 K to today’s 3 K. The frequency of light emitted from a hot body decreases as its temperature is lowered. This occurs through the interaction of the light with the atoms of the system, which maintain the temperature of the medium. They exist in various states of excitation, and correspondingly emit and absorb photons on moving from one state to another. As the medium is cooled, the atoms absorb more high frequency photons and emit more low frequency photons, leading to an overall shift towards lower frequencies, i.e., longer wavelengths, for the radiation. How then can a “cosmic redshift” occur in the universe, where there is so much empty space and so few atoms to regulate the temperature? The origin of the cosmic cooling is a little bit like the Doppler effect we encountered earlier for waves emitted by moving sources. We see those waves “stretched” in wavelength as the source is moving away from us. A similar thing happens to a solitary wave travelling through an expanding space—the distance between crest and valley in the wave becomes stretched, the wavelength longer, the more the space expands. And if the space has expanded by a factor thousand since the emission of the cosmic light, the frequency of the light has decreased by this factor and the wavelength increased. So the cosmic redshift does not tell us that the source of the radiation is locally moving, but rather that the space through which it travels is expanding. The expansion of the universe is encoded in “Hubble’s law”, stating that the velocity of a distant galaxy, relative to Earth, is proportional to its distance from Earth. The crucial scale factor is the “Hubble constant” H0 , for which the best present value is about 22 km/s per million light-years. So a galaxy one million light-years away is receding from us at a velocity of 22 km/s, while one two million lightyears is receding at 44 km/s. If the expansion of the universe takes place at constant acceleration, the inverse of the Hubble constant gives us the age of the universe: 13.8 billion years; the details are shown in Box 3. Box 3. The Age of the Universe Hubble’s law, v = H0 d, gives us the recession velocity v of a distant star, with d specifying its distance from Earth and H0 the Hubble constant. For constant acceleration, i.e., for a constant rate of expansion of space, v = d/t0 , where t0 is the time since the Big Bang, assuming both the star and the Earth were effectively born shortly afterwards. Compared to the present distance, the separation of star and Earth at their birth are negligible, d = 0 shortly after the Big Bang. So it follows that v = d/t0 = H0 d, and from this that t0 = 1/H0 is the time since the Big Bang, the age of the Universe. Many aspects, both observational and theoretical, enter the determination of the expansion and its time dependence. One crucial feature is the overall mass of the universe. If it is large enough, gravity could eventually stop the expansion and the

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universe will start to contract again. The result would then be a final “Big Crunch”. If the mass of the universe is sufficiently small, the expansion can overcome gravity and the acceleration will increase with time. The critical boundary between the two extremes results in constant acceleration. The overall mass contained in the universe is not easily determined, since in addition to the visible content there is a large amount of invisible dark matter, which manifests itself only through gravity. And then, even more elusive, there is most likely an overall background of dark energy, which permeates the entire universe and hence affects its expansion rate. According to recent results that led to the award of the 2011 Nobel prize in physics, the vote goes to an acceleration increasing with time and hence assigns an important role to the mysterious dark energy. In any case, when all is said and done, the best value for the age of the universe today remains at about 14 billion years. It is perhaps interesting to elaborate here a little on the nature of the expansion following the Big Bang. First, we should, however, note that the “reason”, the initial cause of the bang, is not really known. One very impressive attempt to describe the very early stages of the universe was first proposed by the American cosmologist Alan Guth in 1980.

2.4

Cosmic Inflation

Whenever we measure something, we need a reference, a “zero”. The height of a mountain is measured “above sea level”, the depth of the ocean floor “below sea level”. Mount Everest is the highest mountain on Earth only if we use the average sea level as reference, giving it a height of some 8,800 m. The volcano Mauna Kea on Hawaii rises more than 10,000 m above the floor of the ocean at its position— so it is indeed the tallest mountain on Earth. But let us now imagine a dammed river: on the high side, upstream, the level of water is quite different than on the low side, downstream. And this difference in water levels corresponds to a difference in potential energy that can be used, for example, to create electricity by the water rushing down the damm. So the transition from one level to the other can happen very abruptly, and it can liberate energy. In cosmic inflation, the entire universe we can see today was a small bubble of extremely hot matter just after the Big Bang, small enough to be causally connected and in uniform thermal equilibrium; its ground state, the reference point, was far above ours today. The bubble expanded, cooled and thereby was driven to a critical point, over the dam, down the waterfall. In this process, the space of the medium expanded dramatically in an extremely short time, and its new reference point became our physical vacuum of today. Since the medium had been in equilibrium before, it remained uniform even after the expansion of space had broken it up into causally disconnected regions. So that is why, according to inflation theory, we measure the same microwave radiation from all parts of the sky: before inflation, the sources not able to communicate with each other at decoupling time were originally all in the same pot, in which they could adjust to each other’s tune, and this information was conserved in the transition. Moreover, in descending from the upper to the lower level, energy was liberated, and this energy, “dark energy”

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in today’s terminology, permeates the entire universe; it drove and continues to drive the expansion of the universe. But even cosmic inflation can only show that, given certain conditions, a hot expanding early universe can be formed in an extremely short time. It does not explain the origin of these conditions, so that, for the time being, the beginning of the world seems well beyond our science. The subsequent evolution depends, as we mentioned, on the strength of gravity, on the overall mass of the universe. The Big Bang provided the expansion, gravity counteracts this, dark energy may modify it, and, whatever the final verdict on the role of the different components, the universe continues to expand. This expansion is not an “explosion”, throwing debris into some empty space. Rather, space was made in the Big Bang, and it is space itself that is expanding. So a better analogy for our present universe is that of raisins in a cake dough, after some time in the oven. As the cake “rises”, any given raisin notes that all its neighboring raisins are moving further and further away. And the dough between the raisins, that is “space”. For the concept of expansion, it does not matter how much dough there is or if there is an end to it. Similarly, in the Big Bang, the primordial matter as such was not localized at some point in space. We can only see that part of the universe from which light has been able to reach us in 14 billion years, and that part was indeed localized. Whatever more there was (and now is), we simply cannot tell. But we can speculate that there is more; we can’t see it now, but it seems that if we, mankind, wait long enough, light from there will arrive, so that we, taken generically, should be able to see it then, at sufficiently much later time. Unfortunately (or fortunately, depending on your point of view), that is not true. We can use Hubble’s law to see how far away a distant star has to be at present so that for us it is moving away at the speed of light. Using the value of the Hubble constant given above, we find that the critical distance is 14 billion light-years. A star further away from us than that is now moving, relative to us, faster than the speed of light, and any signal it may send will never be able to reach us. So there is an absolute cosmic horizon.

2.5

The Absolute Elsewhere

For us and all our descendents, the universe presently further away than 14 billion light-years is forever beyond any communication; we cannot send “them” a signal, nor ever receive one from “there”. Our world thus remains in principle bounded by the “Hubble sphere” with a radius of 14 billion light-years. But this specific limit applies only to us here on Earth. A distant star will have its own Hubble sphere, and that will cover a different region of space—which may or may not have an overlap with ours. There is more “out there”, but our capability to communicate with it has an absolute limit. But, you may say, how can something move faster than the speed of light? And indeed, nothing can “outrun” a light beam. The new feature entering in cosmic dimensions is theexpansion of space. The far-away star will emit a light beam, and,

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measured on that star, it will start on its path towards us with exactly the universal speed of light. The problem is that while it is travelling, the space of the universe expands, and if this expansion is fast enough, the light beam will never reach the Earth. So Hubble’s law is not saying that the distant star is “running away” from us; from the point of view of other stars near it, it is stationary. And whatever region the light beam passes on its way, any observer there will see it moving with the universal speed of light. The light beam is thus a little like a worm crawling through the expanding cake, from one raisin towards another. Any observer it passes will see it crawling with its standard worm speed, but in the meantime, the rising cake stretches the space it has to traverse, and if the cake rises fast enough, the poor worm will never reach the next raisin. Even during inflation, it was space that was undergoing the abrupt expansion—on a sufficiently small local level, nothing was moving faster than the speed of light. If the Hubble constant were really a constant, our Hubble sphere would have been the limit of our universe since the Big Bang. Slight time variations of H0 even now are under discussion by the experts, and immediately after the Big Bang, as we saw, there may well have been a very short period of a much more rapid “inflationary” expansion. For our overall picture, we will skip over the evolution of the very early universe and assume that our Hubble radius has been “in effect” almost since the Big Bang. That means that any part of the universe beyond our Hubble horizon shortly after the Big Bang was then and ever afterwards outside our world, unreachable for us. It was then expanding away from us faster than the speed of light, and has continued to accelerate since then. What about a star formed just inside our Hubble sphere not long after the Big Bang? The expansion of the space environment of that star proceeded, as seen by us, with an effective speed slightly less than that of light, and so the light of the star could still eventually reach the Earth. But the expansion rate continued to increase, and shortly afterwards became greater than that of light. From our point of view, at that instant the light of the star went off, it disappeared from our world. But the light it had emitted before crossing our Hubble limit continued to travel through the expanding space. And when it finally reaches us, its source star is far, far away outside our world, in our absolute elsewhere. To find out how far, we ask if a light signal was sent out from Earth shortly after the Big Bang, how far has it travelled in the time since then, until today? For a static universe, that distance would be the speed of light times the age of the universe: about 14 billion light-years. But the expansion makes the distance much larger, as our worm discovered above inside the expanding cake dough. Taking the expansion rate to be that of constant acceleration, the light beam has travelled three times the static distance since the Big Bang: 42 billion light-years; the calculation is shown in Box 4.

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Box 4. How Far Has Light Travelled Since the Big Bang? For a static universe, the distance travelled by a light beam between an initial time ti and a final time tf would be d = c(tf − ti ), where c is the speed of light. But the universe expands in that time interval by a factor (tf /ti )2/3 , as is predicted by an acceleration not changing with time. In this case, the stretching of space makes the travel time longer, so that the distance now becomes  tf dt (tf /ti )2/3 = 3ctf d=c ti

if we take the initial time ti = 0 to be that of the Big Bang. So when the light reaches us, it has travelled three times the distance it would traverse in a stationary world; one unit ctf for “local” travel, two units thanks to the expansion of space. As a result, the most distant stars we see are now much further away from us than the speed of light times the age of the universe. When the light we receive from them today was emitted by them, they were much closer to us than they are today, just inside our Hubble sphere. But during the time of travel of the light beam, the universe expanded, and so our distance to them today is, as we just saw, a combination of the time of travel of the light and the expansion of the universe during that time. The most distant star whose light we see today is therefore now 42 billion light-years away from us. Provided it still exists, of course…this we can never find out. From a philosophical point of view, this form of an ultimate spatial limit, of an ultimate horizon of our universe, is really quite satisfying. The “last outer sphere” in older cosmologies always led to a number of unanswerable questions. What is the origin of such an ultimate sphere? What is it made of, what happens if a signal sent by us hits it? And finally, the forbidden question: what is behind this last limit, this end of the universe? In today’s cosmology, the limit exists only in the eye of the beholder. At that imagined surface in space 14 billion light-years away from us, there is nothing special, no discontinuity, no great wall of any kind, and there is no reason to expect that beyond this limit, things are different. Only we can no longer check that. The limit exists for us, for our eyes only, not for other observers in far away parts of the universe. The world according to Thomas Digges, some 450 years ago, is also ours today, except that we now know it had a beginning and that our probing must reach an end. Box 5. The Doll in the Doll In the mechanics of Newton, instantaneous interactions over large distances were implicitily considered possible—and from our present view, this means that effectively the speed of light was taken to be infinite. For a vast range of natural phenomena, this assumption is satisfied to high precision: as long as

2.5 The Absolute Elsewhere

things move with a velocity much less than that of light, Newtonian mechanics remains correct. It is only when particles move with velocities close to that of light, as they do for example in today’s large particle accelerators, that relativity theory, more specifically, Einstein’s special theory of relativity, becomes the correct description. The resulting relativistic mechanics contains Newton’s non-relativistic mechanics as the limiting form obtained for small velocities. The mechanics of special relativity in turn remain correct only as long as the force of gravity remains comparatively weak. A light beam on Earth is not measurably bent downward, and for the motion of a particle in one of the mentioned accelerators, the effect of gravity can also be totally ignored. It is only on a cosmic scale, for forces between galaxies or light passing massive stars, that the deformation of space through interaction plays a role. At that point, Einstein’s general theory of relativity gives the relevant explanation. In the limit of small scales and weak gravity, it gives the special theory as an excellent approximation. So in a way, it’s like the Russian babushka dolls: the biggest, general relativity, contains a smaller one, special relativity, and this turn contains a still smaller one, Newtonian mechanics.

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3

The Secret Glow of Black Holes

But oh, beamish nephew, beware of the day. If your snark be a Boojum! For then you will softly and suddenly vanish away, And never be met with again! Lewis Carroll, The Hunting of the Snark

There are many curious things in our universe, but black holes must be among the most curious. You cannot see them, you cannot hear them, you cannot feel them, and if you ever meet one, you won’t be able to tell anyone about it afterwards. In fact, there will be no afterwards for you. So a black hole is one of those rooms in our universe that you should never even think of entering. The idea that such things might exist in our rational world was first announced in 1783 by John Michell, a natural philosopher in England, educated at Queens’ College, Cambridge, and in his later years parish priest in the small community of Thornhill in West Yorkshire. As was often the case in natural science, he had the right vision, even though his details were faulty. If a stone is thrown up into the air, it falls back to Earth. The faster it is thrown, the higher it rises. When it leaves our hand, it has energy of motion, kinetic energy, and when it comes to a stop somewhere up there, it has no more of that, but lots of potential energy, which it converts back into motion by falling down. So how fast does it have to be thrown in order to not fall back down? In our modern age of space ships and satellites, that is almost an everyday question.

3.1

The Escape Velocity

A bullet shot upwards from the surface of the Earth has to have a certain speed so that it does not return; the idea was already known at the time of Newton. Just as objects of different mass fall the same distance in the same time (barring air resistance and such), the escape velocity from Earth is the same for all objects, about 11 km/s (if you want to follow the derivation, see Box 6). This is the velocity a bullet has to

H. Satz, Ultimate Horizons, The Frontiers Collection, DOI: 10.1007/978-3-642-41657-6_3, © Springer-Verlag Berlin Heidelberg 2013

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have to escape Earth once it has left the gun; its kinetic energy is then sufficient to overcome the potential energy provided by the force of gravity pulling it back. Box 6. The Escape Velocity On the surface of a star of mass M and radius R, a body of mass m is attracted by the force of gravity Mm , R2 where G is the universal constant of gravitation. As a result, it has a negative potential energy F=G

Mm . R To escape from the star, it has to be shot upward with a speed v sufficient to have the kinetic energy V = −G

1 2 mv , 2 which is needed to overcome the potential energy of gravitational attraction. From T =

Mm 1 2 mv = G . 2 R one finds that the escape velocity is  2G M . vescape = R Using the known values for the mass and the radius of the Earth, this gives a terrestrial escape velocity vescape ◦ 11 km/s. Applying this argumentation (incorrectly) to light, one finds that to restrain it from escaping from a star, the stellar mass and radius have to satisfy 2G M , R where c is the velocity of light. We know today that this derivation is not right, but we also know that c2