ULTIMATE HORIZONS probing the limits of the universe

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 hori

THE FRONTIERS COLLECTION

Inter University Centre for Astronomy and Astrophysics (IUC),

Pune University Campus, Pune, India

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 theforefront of modern science, including related philosophical debates In contrast totypical research monographs, however, they strive to present their topics in amanner accessible also to scientifically literate non-specialists wishing to gaininsight into the deeper implications and fascinating questions involved Taken as awhole, the series reflects the need for a fundamental and interdisciplinary approach

to modern science Furthermore, it is intended to encourage active scientists in allareas to ponder over important and perhaps controversial issues beyond their ownspeciality Extending from quantum physics and relativity to entropy, conscious-ness and complex systems—the Frontiers Collection will inspire readers to pushback the frontiers of their own knowledge

Helmut Satz

ULTIMATE HORIZONS Probing the Limits of the Universe

123

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)

This work appears in a parallel German edition ‘‘Gottes unsichtbare Würfel’’, published by

C H Beck Verlag

who dared to venture into the unknown

in search of a better life for her sons

Confronted with the choice between paradise and knowledge, man, according tothe Bible, chose knowledge Were these really alternatives? It came to be that thegaining of knowledge and the wider horizon outside the garden of Eden brought tomany as much pleasure and satisfaction as any paradise they could imagine.Humans have always wanted to explore the world they live in, and they havealways wanted to know what lies beyond the horizons that limit their view Thesearch for richer pastures, better climates, easier communication—all these cer-tainly played a part in this, but behind it all there was an inherent human sense ofcuriosity This curiosity triggered a journey starting some 200,000 years ago in aremote corner of Africa and has driven us to navigate all the oceans, to conquer theentire Earth, to probe the heavens and to penetrate ever more deeply into inter-stellar 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 anever higher degree, to look for its ultimate constituents and study how they interactwith each other to form our world Are there limits, is there an end to this drive, atthe large scale as well as at the small?

In the last hundred years, modern physics and cosmology have shown that thereexist 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 theyare like and whether perhaps some sign of their existence, some indication of theirnature can ever reach our world

Such hidden regions exist in those remote parts of the universe where, from ourpoint of view, space expands faster than the speed of light Closer to us, they arefound in black holes, where gravity is strong enough to retain even light within itshorizon of ultimate attraction And in the realm of the very small, quarks remainforever confined to their colorful world of extreme density; they can never beremoved from it But given the Big Bang origin of the universe, our world in itsvery early stages was immensely hot and dense; and given the spectrum of all theparticles created in high-energy collisions, we can try to reconstruct ever earlierstages The evolution of the universe, with cooling and expansion, then defineshorizons in time, thresholds through which the universe had to pass to reach itspresent state What were the earlier stages like?

vii

Although it is not possible to transmit information across the ‘‘event horizons’’that form the borders of these forbidden regions, still sometimes strange signalsmay appear, providing us with hints of the existence of those other worlds Suchstriking 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 ‘‘ele-mentary’’ particles produced in high-energy accelerators, we can speculate thatthey originally came from a simpler, more symmetric world, which in the course

of the evolution experienced transitions, like the freezing of water or the netization of metals, to form the many-faceted and less symmetric world we seetoday

mag-The aim of this book is to tell the story of how the different horizons, on Earthand in the heavens, on large and on small scales, now and in the past, werediscovered and used to define our view of the world It is a story of the evolution ofthis view, which started before ‘‘science,’’ and which is much more than just

‘‘something for scientists.’’ It started with philosophers wondering what matterwas 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 thesize 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 com-monplace in physics and cosmology; and aspects of both black holes and worm-holes were part of the stories of Lewis Carroll before they became significantlyappreciated in science Many of the ideas, even today’s, have come up here andthere in the course of time The ways of treating them, and the tools used for thatwere different, of course, and changed over the centuries But what remained wasthat 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 thenshow how different forbidden regions arise in the universe, and when and how theycan emit signatures as testimony to their presence and their nature The mysteriouslight emerging from an event horizon, or the equally mysterious clusters in a newand strange ether, they may well remain all that we can ever see of what is hiddenbeyond the ultimate horizons

This book is not meant to give a systematic presentation of the recent opments in physics or cosmology Its aim is to tell a story that began a long timeago and that will certainly not come to an end very soon And it covers devel-opments that sometimes, as in the age of Vasco da Gama and Columbus, or in thetime of Einstein, Planck, Bohr and Heisenberg, revolutionize the world in two orthree decades At other times, between Ptolemy and Copernicus, it takes a mil-lennium to add a couple of epicycles to the accepted scheme of things Theproblem is, in the words of the renowned Austrian theorist Walter Thirring, that

devel-‘‘to do something really new, you have to have a new idea,’’ and that does nothappen so very often It does not suffice to play on the keyboard of the availabletheoretical formalisms; this just leads to many melodies and not to any convincingand lasting new harmony

I have tried to present things in a way not needing any mathematics That is, as Iindicate in the section on Notation, a two-sided issue Even Einstein sometimespresented 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, mathematicsbecomes essential To travel a middle road, I have at times added inserts, in whichsome aspects of the basic mathematical formulation are indicated But I hope thatthe presentation remains understandable even if you skip these

One unavoidable aspect appears if one tries to present things in as readable away as possible: some points and concepts are mentioned more than once.Although strictly speaking logical, the reminder ‘‘as already discussed in theprevious Chapter’’ is in fact often not what the reader wants; it seems better to justbriefly 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 betweenscientific rigor and simplifying an idea enough to make it understandable, I gen-erally 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 technicaldetails they cannot follow My inspiration here were the words of the great Danishphysicist Niels Bohr, who noted that Wahrheit (truth) and Klarheit (clarity) arecomplementary: the more precisely you enforce one, the less precise the otherbecomes

Finally, it is my pleasure to express sincere thanks to all who have helped mewith this endeavor Obvious support came from my colleagues here in Bielefeld, inBrookhaven, at CERN, in Dubna and elsewhere They have been of crucialimportance in forming my view of things And last, but far from least, profoundthanks go to my wife, who has patiently borne with me during all these years

Bielefeld, May 2013 Helmut Satz

1 Horizons 1

1.1 The Horizon of Accessibility 3

1.2 Forbidden Rooms in the Universe 6

1.3 Ultimate Constituents 8

1.4 The End of the Earth 9

1.5 The Roof of Heaven 12

2 The Vanishing Stars 19

2.1 The Speed of Light 19

2.2 Why Is the Sky Dark at Night? 29

2.3 The Big Bang 33

2.4 Cosmic Inflation 37

2.5 The Absolute Elsewhere 38

3 The Secret Glow of Black Holes 43

3.1 The Escape Velocity 43

3.2 Tidal Effects 48

3.3 The Sea of Unborn Particles 51

3.4 Invisible Light on the Horizon 54

4 The Visions of an Accelerating Observer 59

4.1 Gravity and Acceleration 61

4.2 A Total End of Communication 63

4.3 The Temperature of the Vacuum 64

4.4 Lightning in Empty Space 66

4.5 Quantum Entanglement 67

5 The Smallest Possible Thing 71

5.1 Why Does the Sun Shine? 77

5.2 The Strong Nuclear Interaction 78

5.3 The Weak Nuclear Interaction 84

5.4 The Quarks 88

5.5 The Standard Model 95

5.6 The Confinement Horizon 98

xi

6 Quark Matter 103

6.1 Quarks Become Deconfined 105

6.2 Collective Behavior 108

6.3 The Ultimate Temperature of Matter 112

6.4 The Little Bang 114

6.5 Universal Hadrosynthesis 118

6.6 How Hot is the Quark–Gluon Plasma? 121

7 Hidden Symmetries 125

7.1 The Ising Model 130

7.2 Shadow Particles 136

7.3 Local Symmetries 138

7.4 Primordial Equality 140

8 The Last Veil 147

8.1 Ultimate Horizons in Time 148

8.2 Ultimate Horizons in Space 151

8.3 The End of Determinacy 152

8.4 Hyperspace 154

8.5 Cosmic Connections 156

Notes on Notation 161

Further Reading 165

Author Index 167

Index 171

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, ourview always ends at a horizon, beyond which we cannot see Moreover, horizons areelusive We see them, we’re surrounded by them, we try to reach them, and when weget “there”, they have moved to somewhere else Yet they always confront us withthe 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 skytouch in that sharp horizontal line Already more than three thousand years ago, onthe eastern shores of the Mediterranean Sea, the Phoenicians built navigable sailingvessels (Fig.1.1), and they were familiar with astronomical orientation Their shipsexplored the entire Mediterranean and passed beyond the limits of their world, thepillars of Hercules, today’s Strait of Gibraltar A thousand years ago, the ships ofthe 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 thehorizons began when the Portuguese sailors of Henry the Navigator dared to findout if the Earth ended somewhere The inquisitive curiosity to discover if and howthe known world continues—this was surely one of the driving forces that mademankind conquer the whole Earth and go on beyond Once all earthly horizons weresurpassed, the sky became the limit, receding back further and further At first, mancould 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 moredistant stellar regions Are there still regions in the universe which will remain foreverbeyond our reach?

Each horizon forms a boundary not only in space, but also in time If in ancienttimes a traveller saw a distant mountain range at the horizon, he knew that it wouldtake 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, 1 DOI: 10.1007/978-3-642-41657-6_1, © Springer-Verlag Berlin Heidelberg 2013

2 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 themountains, and for ages that was the solution Stage coaches defined the travel timeand comfort Postal relay stations were established, where tired riders and exhaustedhorses could be replaced, and in this way, news was distributed with remarkablespeed (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

China three thousand years ago, and in the Roman empire, post rider relays couldcover 300 km in a twenty-four hour period Post riders and post carriages determinedthe speed of communication until the nineteenth century, and it was the Pony Expressthat brought the American West within reach More than 400 horses and over tendays 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 aninteresting 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 daycoverage In that case, to send a message to some person in a place 900 km awaywould have taken at least three days For that length of time, the person to be reachedwas simply beyond our accessibility horizon Of course, the longer we are willing towait, the greater becomes the region with which we can communicate The resultingpartition of space and time into accessible and inaccessible regions is shown inFig.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 meantmany weeks on a steamboat; today it takes ten hours or less by plane In fact, if itcomes down to simply exchanging information with the “other side of the mountain”,

we don’t need a messenger; telephones can do that almost instantaneously, andsatellite stations connect us to all parts of the Earth For communication, our temporal

Fig 1.3 The accessibility

horizon in post rider times

3

2

1 accessibility horizonaccessible

inaccessible

500 distance [km]

1000

infor-of 20,000 km) takes about 1/15 infor-of a second, so for everyday purposes, it’s almostinstantaneous 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 distantstellar worlds only much later, and what we know of them is their remote past Thelight of the stars that we see now was emitted millions of years ago, and we don’tknow if these stars still exist today, and if they do, where they are So there arehorizons seemingly beyond our reach

Nevertheless, also that inaccessibility seems to be just a question of time If wewait long enough, even the light that distant starts emit now will eventually arrive onEarth 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 awaywas out of reach for us for three days; here and now, we have regions we cannotcommunicate with for some fractions of a second What is different, however, besidesthe sheer scale of things, is that by going from man to horse to train to plane, themessenger speed increased, and so did the range to the horizon at a given time; itssize 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 ultimatehorizon at any specific time, or in physics terminology, theevent horizon Whateverlies beyond this horizon is out of our reach—with that reach defined in terms of bothspace 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 lightyears away cannottodaysend 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; ourgreat-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 beamfrom 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 longerthe time needed to send it a signal, or to receive one it sent It is the event horizon thatforms 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 cosmologyprovide a more stringent form of limit: a truly final horizon, the absolute eventhorizon It defines thosehidden regionsof the world with which no communication

Fig 1.4 The event horizon

Was our universe always there? If not, how old is it? Modern cosmology tells us

of a beginning, aBig Bangabout 14 billion years ago, producing immensely hot anddense primordial matter, which has subsequently expanded to become our universe.Thetimeof the Big Bang is specified, but spatially it is not defined: 14 billion yearsago it began “everywhere”, the primordial world was not a hot little sphere, whichthen exploded That means that if there were, at that time, regions far away fromwhere 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 wesee 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 worldnow The longer we wait, the more

of the primordial world will become visible—or so it seems; the light from moredistant stars is “on its way to us” But while we are waiting, the universe does nothold still Recent astronomical observations have shown that it is in ever increasingexpansion If this expansion is rapid enough, there will be stars whose light cannever reach us, which will remain forever beyond our horizon And some of the starsfrom 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 begone for us

6 1 Horizons

But this cosmic event horizon is still “ours”; a distant galaxy we can see will haveits 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 ourhorizon, or at that of any other galaxy, absolutely nothing happens Its again thatelusive thing: the closer we get to it, the further away it moves

Besides these fleeting limitations to our outreach, there are, however, also moredefinite ones In many old fairy tales, there is a castle with many rooms You mayvisit them all, except one, which you should never ever enter: if you do, you willsuffer a horrible fate It turns out that this can also happen in outer space

If you enter a black hole, you will never come out again to tell what you saw andwhat happened to you At the horizon of the black hole, if you try to avoid fallinginto it, you will certainly experience some rather unpleasant effects And this willnot 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 becomecompact enough, the fusion of hydrogen to helium lets it shine, but eventually allthe fuel is burnt and gravity compresses the remaining ashes of the stellar mass to anever smaller sphere At the end we have an object of such a high gravity that it pullseverything in its vicinity into its range of attraction, even light Since no signal fromsuch a black hole can reach the outside, it appears to be completely decoupled fromour world We can never see what is inside, and for anything within its interior, weare behind an insurmountable event horizon

Thus, in the vast expanses of space, of the cosmos, there are indeed regionsremaining forever beyond our horizon But also at the other end of the scale, inthe microcosmos, in the very small, we find an ultimate limit Just as there is anend to our reach in the limit of large scales, there is one as we try to divide thingsinto ever smaller entities Since antiquity, man has tried to picture the complexworld we find around us as the result of a combined effort of many identical, simplebuilding 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 Themolecules themselves are made of atoms, which in turn consist of positively chargednuclei surrounded by negatively charged electrons, bound by electromagnetic forces

to form electrically neutral entities If we heat the system enough, or apply a verystrong electric field, such as a stroke of lightning, the atoms break up into theircharged constituents, forming a fourth state of matter, the plasma Our view of thestates of matter, with solids, liquids, gases and plasmas, thus agrees very well withthat of antiquity, having earth, water, air and fire (Fig.1.5) And already in antiquity

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, aquintessence, 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 ofhorizons For a trout, the surface of the water forms its horizon of existence, apartfrom short leaps up to catch flies; the shore as well is a definite limit to its livingspace In general, the boundary surfaces between the different states of matter (air–water, water–ice and so on)—in physics terminology: phase boundaries—separateworlds of different structure In ice, the molecules are arranged by firm bonds toform an orderly crystal pattern, a regular lattice with a periodic structure and of well-defined symmetry In water, that lattice is no longer present; the bonds soften andbecome flexible They now allow the molecules to move around in any direction, yetstill restrain them to a rather small spatial volume In the gaseous state, the bondsdissolve completely and we now have a system of balls colliding and scatteringoff each other, but otherwise free to move around in the entire container So thesame 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, thestates remain divisible into their constituents; we can isolate such a constituent andconsider it individually In fact, we can continue with the division, breaking up themolecule 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 evenheavier transuranium elements, artificially created by man For this binding, strongnuclear forces come into the game, overcoming the electric repulsion between thepositive protons Also these basic constituents of matter can in fact existin 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 experimentalstudy of the forces between individual nucleons has shown that they are not reallythe end of the line

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 allthe other newly created ones An understanding of such interactions ultimately led tofurther substructure: a nucleon is a bound state of three quarks, bound by an extremelystrong nuclear force—bound so strongly that an infinite amount of energy would beneeded to split a nucleon into quarks So we can never isolate a single quark TheRoman philosopher Lucretius had concluded over two thousand years ago that theultimate constituents of matter should not have an independent existence, that theycan only exist as parts of a larger whole And indeed this feature is today the basicproperty of quarks, whose bound states form our elementary particles (Fig.1.6) Thequarks are forever confined to their world, quite different from ours, a world thatdoes not have a vacuum, in which there is no empty space, in which they alwaysremain 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

Moreover, given the expansion of the universe, the strange world of the quarkswas 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 findgalaxies being compressed, less and less empty space existing, matter reaching evergreater 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 primordialtemporal horizons of the universe is thus the birth of the vacuum Human imaginationhas carried us back even further than that Electrons and quarks still have intrinsicmasses, and so, following again Lucretius, we can ask where they came from Wecan picture an even younger universe, in which such masses did not yet exist, onlyenergy The appearance of intrinsic masses thus defines yet another, even earlierhorizon 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 encounterhorizons, and beyond these, further horizons We have always been searching for thelast horizon, and the perseverance in keeping up this search is perhaps one of thefeatures that made mankind what it is today Is there an end to our search? Beforeturning to the stellar dimensions of the cosmos beyond what we can see, or to themicrocosmos at scales below what we can see, it seems natural to look at the worldaround us and remember how its limits were discovered

1.4 The End of the Earth

Around 1400A.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 tosea 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 andhusbands “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, seaserpents, giant octopuses and other monsters of the deep—more horrifying than allthese was the thought of falling over the edge of the Earth (Fig.1.7), of disappearinginto 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; butthen the world looked different In the year 1419, the Portuguese Prince Henrique,Infante of Portugal and “Henry the Navigator” for posterity, became governor of theAlgarve, and he dedicated his life to finding out what was beyond Bojador First, hehad collected all reports about the approach to the unknown regions, to establish atheoretical 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 tosee what, if anything, was to be found beyond Cape Bojador They either returnedwithout 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 GilEanes and his brave crew succeeded: they sailed around the end of the Earth andthereby showed that this it was not

The subsequent events are well-known: Following his course, Bartolomeu Diasreached the Cape of Good Hope in 1488, and noted that the coast of Africa thereturned north again Given this information, Vasco da Gama left Portugal in 1497 withthe 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, whoprovided 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 Gamaproceeded to present his credentials and royal Portuguese greetings to the Raja ofCalicut Some years earlier, in 1492, Christopher Columbus, in the service of theSpanish 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, Columbusinsisted all his life that it was India that he had found But when Fernando Magellan

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 viaIndia, it was clear to all: the Earth is a globe There is no mystical border, beyondwhich unknown forces operate

The Earth as a flat disk of finite size: even in the time of Henry the Navigator thatwas actually more of a maritime legend of old than accepted reality As early as fourcenturies before Christ, Aristotle had argued that the Earth must be a sphere, sinceviewed from the coast first the hull and only later the sails of departing ships woulddisappear 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 neververy 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 yearsbefore Christ, the Greek astronomer Eratosthenes had used solar measurements inEgypt to determine it He compared the positions of the Sun precisely at noon in thecity of Syene (today’s Assuan) with that in Alexandria The two cities lie on the samelongitude, so that they do not have a time shift He noted that when the Sun was at thezenith, directly overhead, in Syene (point a in Fig.1.8), in Alexandria (point b) it was

an angleα of 7.2◦off the zenith line (i.e., a line orthogonal to the surface of the Earth).

Simple geometry shows thatα 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

a sun

earth

12 1 Horizons

give the circumference of the Earth The separation distance had been determined byroyal step-markers of the Egyptian court, men who would walk from one city to theother in steps of as equal a length as possible They had found the distance betweenthe two cities to be 5,000 stadia, about 750 km The full circumference of the Earthmust 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 andGuido de Vivaldo from Genoa in Italy had left their city on board two well-armedships, theAllegranza and theSant’Antonio, along with a crew of 300 men, withthe aim of reaching India via the Atlantic So the idea of such a passage had alsobeen around for a while—theirs was the first known try The Genoese sailed southalong 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 earlyexplorers—Gil Eanes, Vasco da Gama, Christopher Columbus, Fernando Magel-lan—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 mankindever after, that was not theory, not thinking, not imagination, but reality

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 OmarKhayyam around 1100 after Christ Is the sky indeed something like a roof overthe 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 thebeginning, because up there everything is in motion So not only Sun and Moonwould 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 worldmeant 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 differentrotation speeds God lives behind the last and largest of the spheres and, as “primemover”, keeps them rotating

This is indeed a task for a god While it is quite easy to picture the Sun on onesphere around the Earth, and the Moon on another, to account for their positionsrelative to us, precision measurements of the relative Sun–Moon positions began topose 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 thesky… Nevertheless, astronomers of the time were up to the task The culminatinggeocentric scenario was developed by Claudius Ptolemy, a Roman citizen of Greekorigin living in Alexandria, Egypt, in the first century A.D.; his work is generallyknown by its Arab titleAlmagest, since it was preserved, as were many other Greekworks, 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 rotatingfirmament, on which the most distant “fixed stars” are attached The final patterntraced 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 involvedand apparently ad hoc: the epicycles of Ptolemy had to be determined specifically foreach planet, the center of the large circle was shifted from the Earth, and more Thecomplexity 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 compilationmade 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 wouldhave suggested something simpler” Hence it seemed not unreasonable to step backand 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 proposedthe Sun as the center of the observable stellar world He did acknowledge somehints from antiquity; the Greek astronomer Aristarchos of Samos had suggested aheliocentric universe already more than two centuries B.C Aristarchos had estimatedthe 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 amathematical model, in which the different planets circled around the Sun in differentdistances 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 sinceantiquity, 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 wasnow no longer the stationary center, the fixed point of the universe It rotates aboutits 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 wasapparently received favorably even by the Roman clergy This does not imply, how-ever, that it was accepted in the present sense It was rather considered an abstractconstruct, a mathematical scheme to calculate the motion and position of the heav-enly bodies, and even at that, it was not perfect It was left for Johannes Kepler toreplace the circular orbits by ellipses to obtain precise agreement And for much of thecommon world, a heliocentric universe with a rotating Earth was simply nonsense.Martin Luther is quoted as saying about Copernicus “that fool is turning astronomyupside 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 asolid empirical basis requiring a precise mathematical description Kepler, as well

as Galileo, considered the heliocentric universe as the true description of the mos, not just a model to compute the positions of planets As a result, strong protestcame from both the catholic and the protestant churches Moreover, his work wascarried out during the time of the 30 years’ War between the two christian fractions in

cos-Germany, and Kepler, refusing to take sides, had to flee several times from tion Nevertheless, he remained deeply religious.

persecu-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, knowntoday as Kepler’s laws of planetary motion These laws described with great precisionthe elliptical orbits of the planets around the Sun, without, however, explaining whythey 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 keepthe planets in orbit At his time, that was speculation—to be made into a physicaltheory almost 80 years later, by Isaac Newton, who wanted toexplainas well as

describe

The required abstraction was that the same forces that act on Earth also governthe 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 thenfell Correcting some Aristotelean misconceptions, Galileo Galilei had already estab-lished that the falling of all objects follows a universal law: the distance a body hasfallen grows with the square of the time and is the same no matter what the mass ofthe 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 samedistance as a heavier stone

The observations of Galileo soon led to what is today calledclassical mechanics—the beginning of physics as we now understand it Isaac Newton, in his celebrated

Philosophiae Naturalis Principia Mathematicaformulated the theory describing theeffect 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 onthe body, a cause for getting it to move Galileo, and following him more succintlyNewton, 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 toeach 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 someforce.That introduced the concept of force as the agent resulting in a change in the state ofbeing 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 betweencelestial 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, circumstanceand cause, whereas gravity was always there, everywhere and at all times A stonereleased would fall to the ground, in the same way, no matter who released it, whereand when There seemed to be a mysterious attraction of things to the Earth Itwas Newton’s great achievement to relate this everyday force to that determiningcelestial structure and motion Newton’s theory of gravitation states that a massiveobject 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,

F = G M1M2

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 overimmense distances without any apparent connection between the interacting objects,and, as it seemed, instantaneously It holds the Earth and the other planets in orbitsaround 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 isthis 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 mostuniversal force in the world, operative from our human scale to that of the entireuniverse

So, at this point, astronomers had a consistent theoretical explanation for thestructure and motion of the observable world: the Earth, the Moon, the Sun, theother 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 allobjects on Earth, making apples fall from trees and preventing stones from jumpinginto 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 lastsphere was brought up Instead, beyond the solar system, there could be an infinityfilled homogeneously with fixed stars; such a scenario had been considered by the

English astronomer Thomas Digges in 1576 Thoughts of this kind were always onthe verge of being heretic, in the eyes of the church The Italian philosopher GiordanoBruno 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 theminds of many scientists and philosophers Is there indeed some ultimate celestialsphere? And if so, what is in that forbidden “room” beyond it? The existence of afinal firmament, to which the fixed stars are attached, did in fact answer one rathercurious question Why is the sky dark at night? If there were no such sphere, ifinstead 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, somecloser, some further out, and further yet Copernicus insisted on a fixed outer spherewith a finite number of stars and thus avoided the problem Kepler had realized thedifficulty and therefore also ruled out the possibility of an infinite universe Still thequestion kept reappearing and is today known as Olbers’s paradox, after the Germanastronomer Heinrich Olbers, who formulated it most succinctly in 1823 It is anexcellent illustration of how a well-posed question can lead to progress in thinkingand understanding To answer it, however, we first have to address one of the basicissues of physics: what is light?

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

H Satz, Ultimate Horizons, The Frontiers Collection, 19 DOI: 10.1007/978-3-642-41657-6_2, © Springer-Verlag Berlin Heidelberg 2013

The question had been around for quite a while when Galilei, in his Renaissancetreatise on theTwo New Sciences, made hisalter egoSalviati ask it Already Aristotlehad complained more than 300 years before Christ that

Empedocles says that the light from the Sun arrives first in the intervening spacebefore it comes to the eye or reaches the Earth

He, Aristotle, was sure that this was completely wrong, that “light is not a ment”, and his belief dominated western thinking for almost 2,000 years The speed

move-of light is infinite—even great scientists and philosophers like Johannes Kepler andRené 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 ofphilosophy”

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 thetime 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, andthese 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 talented 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, afterthe island of Rømø, where they came from Ole had studied physics, mathematicsand astronomy in Copenhagen and eventually married the daughter of his professorthere In between, he had worked for King Louis XIV in Paris and took part in thedesign of the fountains of Versailles After this interlude, he returned to Denmarkfor an appointment as “royal mathematician”, where he introduced the first nationalsystem of weights and measures, as well as the Gregorian calender And besides allthis, he became Chief of the Copenhagen Police, responsible for the installation ofthe first street lights there In Paris, he had worked as an assistant for the astronomerGiovanni Domenico Cassini, and Cassini had made a remarkable observation Theplanet Jupiter, fifth around the Sun and largest of all, had a Moon, called Io (namedafter 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 ourearthly Moon takes for its orbit That meant that seen from the Earth, there would bemany “eclispses” of Io at any stage of the Earth ’s orbit around the Sun; the geometry

multi-is shown in Fig.2.1 One could thus measure the time at which Io disappears behindJupiter, and do this for a series of eclipses This provided a determination of the timebetween successive eclipses, giving a prediction for the next

And the striking observation first made by Cassini was that the onset of an eclipsefell more and morebehind schedulethe further away the Earth was from Jupiter.Cassini was not sure, but thought that perhapslight 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 21

Fig 2.1 Ole Rømer’s basis

for the determination of the

Io

Sun Earth

b

a

effects, and found that the delay in time of eclipse onsets seen from the point ofgreatest Earth—Jupiter separation (point a) compared to those seen from the smallestdistance (point b) was about 22 min From this he now concluded that the speed oflight 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 totravel 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 determinethe distance from Earth to Sun was actually available at that time, due mainly tothe studies of Cassini The first numerical value for the speed of light, however, wasapparently obtained by the Dutch physicist Christiaan Huygens in 1678, two yearsafter Ole Rømer had announced his conclusions

Kepler had, in this “third law” of celestial motion, concluded that the time for aplanet 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 theEarth and the Sun To arrive at an actual value for the Earth–Sun distance, some astro-nomical 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 measuredsimultaneously the position of Mars relative to the fixed star background, Cassini inParis and Richer in French Guiana This gave them an angle and a known terrestrialdistance, the 4,000 km between Paris and Guiana, and geometry then determinedthe distance between Mars and Earth They found it to be about 73 million km Atthe point of closest approach of Mars and Earth, that led to 146 million km for thedistance between Earth and Sun Since light travelled, according to Rømer, twice thatdistance in 22 min, Huygens noted that its speed must be about 220,000 km/s Thisresult, obtained over 300 years ago by a combination of logical thinking, abstractionand rudimentary measurements, is certainly one of the great achievements of the

stationary mirror

light source d

in a clever arrangement of mirrors Foucault, with his celebrated pendulum, had infact also provided for the first time direct proof of the rotation of the Earth aroundits axis But he now modified an older apparatus devised by Fizeau to measure onEarth 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 angleθ, 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

speedω 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 spacebefore

reaching the Earth, as Empedokles had supposed 2,500 years ago But what is thislight travelling through what we think is empty space? What is it that is moving at300,000 km/s?

This question led to another basic and universal phenomenon of the inanimateworld: electromagnetism Initially, electricity and magnetism entered as two quiteseparate and distinct features The first appearance of electricity in the life of humanswas lightning, for a long time thought to express the wrath of the gods in a frightfulway, and beyond human understanding A more mundane version was observed bythe ancient Egyptians, more than 3,000 years ago; they were familiar with electricfish which could produce remarkable bolts of electricity to stun their prey Thissource of electricity was supposedly used already in those days for the treatment ofneural illnesses In ancient Greece, it was noted that rubbing amber with a catskinmade it attract feathers and other light objects—and it was this feature that gave thename to the mysterious force, withelektronas 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 unrelatedphenomena began to be understood, and only in the last 100 years has electricitydramatically changed human life

Magnetism was more well-defined from the beginning Several millennia ago itwas noticed in China that a certain kind of stone attracts iron, and if suspended by astring, it would orient itself along a north–south axis Making use of this, the ancientChinese constructed the first magnetic compass for navigation In ancient Greece,Thales of Milos described the effect, and since the stones showing such behaviorthere came from a province called Magnesia, he called it magnetic In English, itbecame “leadstone” and finally “lodestone”, presumably because it could be used tolead 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 aspositiveandnegative; each form could be produced by rubbing,for example, and each kind can exist on its own If two metal balls were prepared tohave different “charges”, like and like repelled each other, while positive and negativeshowed 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 adistance provided by gravity (Fig.2.3) Coulomb’s law gives for the electric force

F = K q1q2

r2 ,

Moon Earth

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

where q1and q2measure 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, whilenorth/north or south/south meant repulsion But there was no way to get just onepole Cut a magnet in two in the middle, and you had two new magnets, each withits north and its south pole And until today, physicists are still wondering if thereisn’t some way to create a monopole The magnetic force was not quite of the inversesquare form encountered in Coulomb’s law of electric interaction or Newton’s law ofgravity, since each pair of magnets experienced both attraction, between the oppositepoles, and repulsion, between the equal poles Nevertheless, the interaction betweentwo magnets, as well as that between metals and magnets, was again by some invisiblemeans over the distance of separation

So both electric and magnetic interactions showed a mysterious feature alreadyencountered in the case of gravitation: an interaction over a distance, without anyapparent connection between the interacting objects How such an interaction couldarise was something that had puzzled people at all times Was there some invisiblemedium filling all of space to provide a connection? The beginning of an answer wasprovided by the British physicist Michael Faraday, who proposed that each chargewould be surrounded by an electric field, radiating out starlikelines of forceemergingfrom 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 towardsthe 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 discoveredthat there was a strange connection between electricity and magnetism It was knownthat certain materials—today’s conductors—allow a rapid spreading of electriccharge: they result in the flow of an electric current between opposite charges, form-ing an electric circuit Now Oersted observed that a magnet would align itself in adirection orthogonal to the line of current flow, as if the current had created magneticlines of force around its flow axis So one could imagine unending lines of forcecorresponding to magnetic fields, closed loops having neither beginning nor end.This would explain why cutting a magnet in two simply produced two magnets, anddid not yield an isolated pole

In the course of the nineteenth century, extensive studies showed that electric andmagnetic forces are indeed closely intertwined: electric currents produced magneticfields and moving magnetsinduceelectric currents This suggested a unified theory

of electromagnetic fields, and it was the great British physicist James Clerk Maxwellwho created it, with his famous equations Through Maxwell, electricity and mag-netism were unified to electromagnetism And in addition, he provided the basis for

an understanding of how the interaction of electromagnetic sources could occur overdistances Maxwell showed that a changing electric field generates a magnetic field,just as a changing magnetic field would through induction create an electric field Sothe combination of the two, electromagnetic fields, now gained an independent exis-tence, without the need of currents or magnets And one simple solution of Maxwell’sequations 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 occurthrough the exchange of electromagnetic signals in the form of such waves Theypropagate through space at a fixed speed, which can be measured and was found to

be the familiar speed of light The fundamental questionwhat is light?was thereforenow answered: it is an electromagnetic wave travelling through space, and the differ-ent colors of light simply correspond to different possible wavelengths Beyond therange of visible light, we recognize today electromagnetic radiation on both sides,with radio waves of longer wavelength (beyond the infrared) and X-rays of shorterwavelength (beyond the ultraviolet) And in a way, it also answered the question

of how distant charges could interact: through the exchange of an electromagneticsignal

But the answer was not really complete If distant charges communicated byelectromagnetic waves travelling between them:what was being excited to formsuch waves? In our everyday world, it can be a string, the surface of water, thedensity of air But what is it in empty space that is vibrating? And so theether

entered the world of physics, an invisible medium filling all of the so-called emptyspace This satisfied those who thought that truly empty space was “unnatural”,such as the French philosopher Blaise Pascal, who believed that “nature abhors avacuum” When Evangelista Torricelli in Italy succeeded in removing all the airfrom 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 lightlike 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

ocean and then travels towards some shore, so a change in the electromagnetic statesomewhere 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 ofthe most-travelled dead-end roads of physics From the time of Hooke to the time ofEinstein, a great number of well-known physicists tried their hand at it, and alwayswith rather limited success Is the ether stationary, or is it comoving with stars? Isthere 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 Michelsonand Edward Morley If light was travelling through the ether everywhere at its fixedspeed, then it would have to be slower if measured in the direction of the Earth’smotion 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 theEarth, travel the same distance and by means of a mirror arrangement meet again at agiven point (see Fig.2.5) The slowing effect of the Earth’s motion would throw themout of phase, so that a valley in the wave of one would hit a peak in that of the otherbeam, causing interference Much to their frustration, Michelson and Morley found

no effect whatsoever; all waves arrived completely in phase No matter how theypositioned 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 yearslater It is now definitely ruled out, at least as far as electromagnetism is concerned

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 ordark energy is; we shall return to these somewhat ether-like ideas later on

Maxwell’s equations implied a unique speed for electromagnetic waves travellingthrough empty space, the universal speed of light This is in fact much more dramaticthan it seems at first sight: such a behavior is simply not in accord with our everydayexperience A car moving at 100 km/h, as seen by a stationary observer, has a relativespeed of only 70 km/h for someone moving in the same direction at 30 km/h And twocars, 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 fallsstraight 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 bothfind the same value for its speed No matter how fast you move, the speed of lightyou 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 whichNewton formulated his laws, this was simply impossible In a fixed space with auniversal time, the speed of light would change for observers moving at differentspeeds 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 distanceand 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 standardmeter and his standard second, as seen by me here on Earth, must have taken ondifferent values than mine—and they do The resulting milestone in physics wasAlbert Einstein’s theory of relativity, more exactly, thespecialtheory of relativity.The “special” is ana posteriorimodification, indicating that it holds in a restrictedspatial region of the universe only The extension to the entire cosmos, including therole of gravity, followed 10 years later with thegeneraltheory of relativity, and again

it was Einstein who did it

To formulate his special theory of relativity, Einstein combined a principle posed by Galileo Galilei 400 years earlier with the recently discovered universalspeed of light Galileo had insisted that the laws of physics be the same for allobservers in uniform motion relative to each other In other words, if I measure thetime it takes a stone to fall to the ground from a height of one meter, once in thelaboratory and once on a high speed train, the results should be identical Einsteinrealized that if this was to hold and at the same time a universal speed of light was

pro-to be maintained for all observers in uniform relative motion, our ideas of spaceand time would have to be modified, space and time would have to be related, andtheir scales have to depend on the speed of the observer (see Box 1) In Newton’sworld, there was a unique time, the same everywhere, and one could talk about twoevents occurring at the same time In a relativistic world, synchronization over largedistances is not possible, and what is first for one observer, may be later for another

Another striking result of relativity theory was the conclusion that no materialbody 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 eventuallybring its speed to arbitrarily high values, faster than the speed of light Einsteinshowed 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 goesinto 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 thespeed corrections and work with a speed-independent inertial mass But in modernhigh-energy particle accelerators, such as the Large Hadron Collider at the EuropeanLaboratory 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 ismore 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 mustmean that from our point of view his length measure is shorter than ours, or hisclock 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 inhis moving space ship But his moving meter stick, as seen by us, becomes

shortened to the length d,

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 m0and Newton’s force law F = m0a.

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 Einsteinformulated 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 ifthey had a mass determined by their energy So we can weigh the photonstrapped 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 travellingfor many years, waves of electromagnetic energy moving through an empty spacecontaining 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 WithEdgar Allan Poe, the problem entered the literary world, leading to pictures that acentury later became science, such as an expanding universe starting from a BigBang 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, didstate 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, theyhave existed forever, and they have a finite size and brightness

Given these conditions, the whole sky should be as bright as a typical star; itshould never get dark at night So something must be wrong somewhere, and thatsomething leads us directly to the forefront of modern cosmology and its view of theorigin of the universe

If the age of the universe is finite, if there was a Big Bang starting everything acertain number of years ago, then the universe we can see today will also be of finitesize, because light has only had those years to travel To be sure, the numbers arehuge, but they are not infinite Moreover, the stars had to form sometime after theBig Bang, so their number is also finite In other words, a finite age of the universeallows us to see only a finite spatial part of it, and in that part only a finite number ofstars can have appeared since the Big Bang That is why the sky is dark at night—a

late answer to Heinrich Olbers, requiring both a finite speed of light and a Big Bangorigin 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 reasonswhy today most scientists tend to believe in the Big Bang theory—but let us approachthem slowly and step by step

A well-known effect in the physics of everyday phenomena is that the pitch of asound you hear is modified if the source of the sound is moving The sound of a racecar 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 intone of the whistle of a passing railroad engine was the typically cited example Thephenomenon is known as the Doppler effect, after the Austrian physicist ChristianDoppler The tone you hear is caused by sound waves of a certain wavelength, andwhen the source of the sound approaches you, the distance between wave peaks, thewavelength, is shortened, giving a higher sound, and when it moves away, it becomeslonger and hence results in a lower sound The same “Doppler effect” also occursfor light waves, so that one can in fact check if a given far-away star is stationary ormoving 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 wavelengthλ0, as measured by

an observer stationed on that star For an observer moving away from the star with aspeedv, 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 aredshift.The American astronomer Edwin Hubble, working in the 1920s at the MountWilson Observatory in California, had studied the light from very distant stars Frommeasurements of redshifts it was already known that they all seem to be moving awayfrom us at different speeds Hubble made the striking observation that the furtheraway they are, the faster they recede The Doppler shift, and hence the speed of thestars’ motion, was rather well measurable—the crucial factor for reaching Hubble’sconclusion was the determination of the distance of the stars in question To measurethe distance of fairly nearby objects in the sky, such as planets, one could use theparallax method employed by Cassini and Richer to determine the distance betweenMars and the Earth Howevever, for the very remote stars Hubble was after, theparallax angle became for too minute to be measurable The solution came throughthe extension of a very simple phenomenon The brightness of a given light sourcedecreases the further one is away from it Since light is emitted spherically from itssource, the light incident on a given surface becomes less and less with distance

The size of the spheres grows as d2, with d denoting that distance, and therefore the

light per area decreases as 1/d2 So if we know the original brightness of the sourceand 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 beendetermined; they were what astronomers today call standard candles Measuringtheir apparent luminosity as observed at Mount Wilson, Hubble had at least a good