From Eternity to Here

Table of Contents

Title PageCopyright PageDedication

PART ONE - TIME,EXPERIENCE, AND THEUNIVERSE

Chapter 1 - THE PAST IS PRESENT MEMORYChapter 2 - THE HEAVY HAND OF ENTROPYChapter 3 - THE BEGINNING AND END OF TIME

PART TWO - TIME INEINSTEINS UNIVERSE

Chapter 4 - TIME IS PERSONALChapter 5 - TIME IS FLEXIBLEChapter 6 - LOOPING THROUGH TIME

PART THREE - ENTROPY ANDTIMES ARROW

Chapter 7 - RUNNING TIME BACKWARDChapter 8 - ENTROPY AND DISORDERChapter 9 - INFORMATION AND LIFEChapter 10 - RECURRENT NIGHTMARESChapter 11 - QUANTUM TIME

PART FOUR - FROM THEKITCHEN TO THE MULTIVERSE

Chapter 12 - BLACK HOLES: THE ENDS OF TIMEChapter 13 - THE LIFE OF THE UNIVERSEChapter 14 - INFLATION AND THE MULTIVERSEChapter 15 - THE PAST THROUGH TOMORROWChapter 16 - EPILOGUEAPPENDIX: MATHNOTESBIBLIOGRAPHYAcknowledgementsINDEX

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Copyright 2010 by Sean CarrollAll rights reserved

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From eternity to here : the quest for the ultimate theory of time / SeanCarroll. p. cm.

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To Jennifer For all time

PROLOGUE

Does anybody really know what time it is?Chicago, Does Anybody Really Know What Time It

Is?

This book is about the nature of time, the beginning of theuniverse, and the underlying structure of physical reality.Were not thinking small here. The questions were tacklingare ancient and honorable ones: Where did time andspace come from? Is the universe we see all there is, orare there other universes beyond what we can observe?How is the future different from the past?

According to researchers at the Oxford EnglishDictionary, time is the most used noun in the Englishlanguage. We live through time, keep track of itobsessively, and race against it every dayyet,surprisingly, few people would be able to give a simpleexplanation of what time actually is.

In the age of the Internet, we might turn to Wikipedia forguidance. As of this writing, the entry on Time begins asfollows:

Time is a component of a measuring systemused to sequence events, to compare thedurations of events and the intervals betweenthem, and to quantify the motions of objects.Time has been a major subject of religion,philosophy, and science, but defining time ina non-controversial manner applicable to allfields of study has consistently eluded thegreatest scholars.1

Oh, its on. By the end of this book, we will have definedtime very precisely, in ways applicable to all fields. Lessclear, unfortunately, will be why time has the properties thatit doesalthough well examine some intriguing ideas.

Cosmology, the study of the whole universe, has madeextraordinary strides over the past hundred years. Fourteenbillion years ago, our universe (or at least the part of it wecan observe) was in an unimaginably hot, dense state thatwe call the Big Bang. Ever since, it has been expandingand cooling, and it looks like thats going to continue for theforeseeable future, and possibly forever.

A century ago, we didnt know any of thatscientistsunderstood basically nothing about the structure of theuniverse beyond the Milky Way galaxy. Now we have takenthe measure of the observable universe and are able todescribe in detail its size and shape, as well as itsconstituents and the outline of its history. But there areimportant questions we cannot answer, especiallyconcerning the early moments of the Big Bang. As we willsee, those questions play a crucial role in ourunderstanding of timenot just in the far-flung reaches ofthe cosmos, but in our laboratories on Earth and even inour everyday lives.

TIME SINCE THE BIG BANG

Its clear that the universe evolves as time passestheearly universe was hot and dense; the current universe iscold and dilute. But I am going to be drawing a muchdeeper connection. The most mysterious thing about timeis that it has a direction: the past is different from the future.Thats the arrow of timeunlike directions in space, all ofwhich are created pretty much equal, the universeindisputably has a preferred orientation in time. A majortheme of this book is that the arrow of time exists becausethe universe evolves in a certain way.

The reason why time has a direction is because theuniverse is full of irreversible processesthings thathappen in one direction of time, but never the other. Youcan turn an egg into an omelet, as the classic examplegoes, but you cant turn an omelet into an egg. Milkdisperses into coffee; fuels undergo combustion and turninto exhaust; people are born, grow older, and die.Everywhere in Nature we find sequences of events whereone kind of event always happens before, and another kindafter; together, these define the arrow of time.

Remarkably, a single concept underlies ourunderstanding of irreversible processes: something calledentropy, which measures the disorderliness of an objector conglomeration of objects. Entropy has a stubborntendency to increase, or at least stay constant, as timepassesthats the famous Second Law ofThermodynamics. 2 And the reason why entropy wants toincrease is deceptively simple: There are more ways to bedisorderly than to be orderly, so (all else being equal) anorderly arrangement will naturally tend toward increasingdisorder. Its not that hard to scramble the egg moleculesinto the form of an omelet, but delicately putting them backinto the arrangement of an egg is beyond our capabilities.

The traditional story that physicists tell themselves usuallystops there. But there is one absolutely crucial ingredientthat hasnt received enough attention: If everything in theuniverse evolves toward increasing disorder, it must havestarted out in an exquisitely ordered arrangement. Thiswhole chain of logic, purporting to explain why you cant turnan omelet into an egg, apparently rests on a deepassumption about the very beginning of the universe: It wasin a state of very low entropy, very high order.

The arrow of time connects the early universe tosomething we experience literally every moment of ourlives. Its not just breaking eggs, or other irreversibleprocesses like mixing milk into coffee or how an untendedroom tends to get messier over time. The arrow of time isthe reason why time seems to flow around us, or why (if youprefer) we seem to move through time. Its why weremember the past, but not the future. Its why we evolveand metabolize and eventually die. Its why we believe incause and effect, and is crucial to our notions of free will.

And its all because of the Big Bang.

WHAT WE SEE ISNT ALLTHERE IS

The mystery of the arrow of time comes down to this: Whywere conditions in the early universe set up in a veryparticular way, in a configuration of low entropy thatenabled all of the interesting and irreversible processes tocome? Thats the question this book sets out to address.Unfortunately, no one yet knows the right answer. But weve

reached a point in the development of modern sciencewhere we have the tools to tackle the question in a seriousway.

Scientists and prescientific thinkers have always tried tounderstand time. In ancient Greece, the pre-Socraticphilosophers Heraclitus and Parmenides staked outdifferent positions on the nature of time: Heraclitus stressedthe primacy of change, while Parmenides denied the realityof change altogether. The nineteenth century was the heroicera of statistical mechanicsderiving the behavior ofmacroscopic objects from their microscopic constituentsin which figures like Ludwig Boltzmann, James ClerkMaxwell, and Josiah Willard Gibbs worked out the meaningof entropy and its role in irreversible processes. But theydidnt know about Einsteins general relativity, or aboutquantum mechanics, and certainly not about moderncosmology. For the first time in the history of science, we atleast have a chance of putting together a sensible theory oftime and the evolution of the universe.

Im going to suggest the following way out: The Big Bangw a s not the beginning of the universe. Cosmologistssometimes say that the Big Bang represents a trueboundary to space and time, before which nothing existedindeed, time itself did not exist, so the concept ofbefore isnt strictly applicable. But we dont know enoughabout the ultimate laws of physics to make a statement likethat with confidence. Increasingly, scientists are takingseriously the possibility that the Big Bang is not really abeginningits just a phase through which the universegoes, or at least our part of the universe. If thats true, thequestion of our low-entropy beginnings takes on a differentcast: not Why did the universe start out with such a lowentropy? but rather Why did our part of the universe passthrough a period of such low entropy?

That might not sound like an easier question, but its adifferent one, and it opens up a new set of possibleanswers. Perhaps the universe we see is only part of amuch larger multiverse, which doesnt start in a low-entropyconfiguration at all. Ill argue that the most sensible modelfor the multiverse is one in which entropy increasesbecause entropy can always increasethere is no state ofmaximum entropy. As a bonus, the multiverse can becompletely symmetric in time: From some moment in themiddle where entropy is high, it evolves in the past andfuture to states where the entropy is even higher. Theuniverse we see is a tiny sliver of an enormously largerensemble, and our particular journey from a dense BigBang to an everlasting emptiness is all part of the widermultiverses quest to increase its entropy.

Thats one possibility, anyway. Im putting it out there asan example of the kind of scenarios cosmologists need tobe contemplating, if they want to take seriously theproblems raised by the arrow of time. But whether or notthis particular idea is on the right track, the problemsthemselves are fascinating and real. Through most of thisbook, well be examining the problems of time from avariety of anglestime travel, information, quantummechanics, the nature of eternity. When we arent sure ofthe final answer, it behooves us to ask the question in asmany ways as possible.

THERE WILL ALWAYS BESKEPTICS

Not everyone agrees that cosmology should play aprominent role in our understanding of the arrow of time. Ionce gave a colloquium on the subject to a large audienceat a major physics department. One of the older professorsin the department didnt find my talk very convincing andmade sure that everyone in the room knew of hisunhappiness. The next day he sent an e-mail around to thedepartment faculty, which he was considerate enough tocopy to me:

Finally, the magnitude of the entropy of theuniverse as a function of time is a veryinteresting problem for cosmology, but tosuggest that a law of physics depends on it issheer nonsense. Carrolls statement that thesecond law owes its existence to cosmologyis one of the dum mest [sic] remarks I heardin any of our physics colloquia, apart from[redacted]s earlier remarks aboutconsciousness in quantum mechanics. I amastounded that physicists in the audiencealways listen politely to such nonsense.Afterwards, I had dinner with some graduatestudents who readily understood myobjections, but Carroll remained adamant.

I hope he reads this book. Many dramatic-soundingstatements are contained herein, but Im going to be ascareful as possible to distinguish among three differenttypes: (1) remarkable features of modern physics thatsound astonishing but are nevertheless universallyaccepted as true; (2) sweeping claims that are notnecessarily accepted by many working physicists but thatshould be, as there is no question they are correct; and (3)speculative ideas beyond the comfort zone ofcontemporary scientific state of the art. We certainly wontshy away from speculation, but it will always be clearlylabeled. When all is said and done, youll be equipped tojudge for yourself which parts of the story make sense.

The subject of time involves a large number of ideas,from the everyday to the mind-blowing. Well be looking atthermodynamics, quantum mechanics, special and generalrelativity, information theory, cosmology, particle physics,and quantum gravity. Part One of the book can be thoughtof as a lightning tour of the terrainentropy and the arrowof time, the evolution of the universe, and differentconceptions of the idea of time itself. Then we will get abit more systematic; in Part Two we will think deeply aboutspacetime and relativity, including the possibility of travelbackward in time. In Part Three we will think deeply aboutentropy, exploring its role in multiple contexts, from theevolution of life to the mysteries of quantum mechanics.

In Part Four we will put it all together to confront head-onthe mysteries that entropy presents to the moderncosmologist: What should the universe look like, and howdoes that compare to what it actually does look like? Illargue that the universe doesnt look anything like it should,after being careful about what that is supposed to meanat least, not if the universe we see is all there is. If ouruniverse began at the Big Bang, it is burdened with a finelytuned boundary condition for which we have no goodexplanation. But if the observed universe is part of a biggerensemblethe multiversethen we might be able toexplain why a tiny part of that ensemble witnesses such adramatic change in entropy from one end of time to theother.

All of which is unapologetically speculative but worth

taking seriously. The stakes are bigtime, space, theuniverseand the mistakes we are likely to make along theway will doubtless be pretty big as well. Its sometimeshelpful to let our imaginations roam, even if our ultimategoal is to come back down to Earth and explain whatsgoing on in the kitchen.

PART ONE

TIME, EXPERIENCE, AND THEUNIVERSE

1THE PAST IS PRESENTMEMORY

What is time? If no one asks me, Iknow. If I wish to explain it to onethat asketh, I know not.

St. Augustine, Confessions

The next time you find yourself in a bar, or on an airplane, orstanding in line at the Department of Motor Vehicles, youcan pass the time by asking the strangers around you howthey would define the word time. Thats what I starteddoing, anyway, as part of my research for this book. Youllprobably hear interesting answers: Time is what moves usalong through life, Time is what separates the past fromthe future, Time is part of the universe, and more alongthose lines. My favorite was Time is how we know whenthings happen.

All of these concepts capture some part of the truth. Wemight struggle to put the meaning of time into words, butlike St. Augustine we nevertheless manage to deal withtime pretty effectively in our everyday lives. Most peopleknow how to read a clock, how to estimate the time it willtake to drive to work or make a cup of coffee, and how tomanage to meet their friends for dinner at roughly theappointed hour. Even if we cant easily articulate whatexactly it is we mean by time, its basic workings makesense at an intuitive level.

Like a Supreme Court justice confronted with obscenity,we know time when we see it, and for most purposes thatsgood enough. But certain aspects of time remain deeplymysterious. Do we really know what the word means?

WHAT WE MEAN BY TIME

The world does not present us with abstract conceptswrapped up with pretty bows, which we then must work tounderstand and reconcile with other concepts. Rather, theworld presents us with phenomena, things that we observeand make note of, from which we must then work to deriveconcepts that help us understand how those phenomenarelate to the rest of our experience. For subtle conceptssuch as entropy, this is pretty clear. You dont walk downthe street and bump into some entropy; you have toobserve a variety of phenomena in nature and discern apattern that is best thought of in terms of a new concept youlabel entropy. Armed with this helpful new concept, youobserve even more phenomena, and you are inspired torefine and improve upon your original notion of whatentropy really is.

For an idea as primitive and indispensable as time, thefact that we invent the concept rather than having it handedto us by the universe is less obvioustime is something we

literally dont know how to live without. Nevertheless, part ofthe task of science (and philosophy) is to take our intuitivenotion of a basic concept such as time and turn it intosomething rigorous. What we find along the way is that wehavent been using this word in a single unambiguousfashion; it has a few different meanings, each of whichmerits its own careful elucidation.

Time comes in three different aspects, all of which aregoing to be important to us.

1. Time labels moments in the universe.Time is a coordinate; it helps us locatethings.

2 . Time measures the duration elapsedbetween events.

Time is what clocks measure.3. Time is a medium through which we move.

Time is the agent of change. We movethrough it, orequivalentlytime flows pastus, from the past, through the present, towardthe future.

At first glance, these all sound somewhat similar. Timelabels moments, it measures duration, and it moves frompast to futuresure, nothing controversial about any of that.But as we dig more deeply, well see how these ideas dontneed to be related to one anotherthey represent logicallyindependent concepts that happen to be tightly intertwinedin our actual world. Why that is so? The answer mattersmore than scientists have tended to think.

1. Time labels moments in the universe

John Archibald Wheeler, an influential American physicistwho coined the term black hole, was once asked how hewould define time. After thinking for a while, he came upwith this: Time is Natures way of keeping everything fromhappening at once.

There is a lot of truth there, and more than a little wisdom.When we ordinarily think about the world, not as scientistsor philosophers but as people getting through life, we tendto identify the world as a collection of things, located invarious places. Physicists combine all of the placestogether and label the whole collection space, and theyhave different ways of thinking about the kinds of things thatexist in spaceatoms, elementary particles, quantumfields, depending on the context. But the underlying idea isthe same. Youre sitting in a room, there are various piecesof furniture, some books, perhaps food or other people,certainly some air moleculesthe collection of all thosethings, everywhere from nearby to the far reaches ofintergalactic space, is the world.

And the world changes. We find objects in someparticular arrangement, and we also find them in someother arrangement. (Its very hard to craft a sensiblesentence along those lines without referring to the conceptof time.) But we dont see the different configurationssimultaneously, or at once. We see one configurationhere you are on the sofa, and the cat is in your lapandthen we see another configurationthe cat has jumped offyour lap, annoyed at the lack of attention while you areengrossed in your book. So the world appears to us againand again, in various configurations, but theseconfigurations are somehow distinct. Happily, we can labelthe various configurations to keep straight which is whichMiss Kitty is walking away now; she was on your lapthen. That label is time.

So the world exists, and what is more, the worldhappens, again and again. In that sense, the world is likethe different frames of a film reela film whose cameraview includes the entire universe. (There are also, as far aswe can tell, an infinite number of frames, infinitesimallyseparated.) But of course, a film is much more than a pileof individual frames. Those frames better be in the rightorder, which is crucial for making sense of the movie. Timeis the same way. We can say much more than thathappened, and that also happened, and that happened,too. We can say that this happened before that happened,and the other thing is going to happen after. Time isnt justa label on each instance of the world; it provides asequence that puts the different instances in order.

A real film, of course, doesnt include the entire universewithin its field of view. Because of that, movie editingtypically involves cutsabrupt jumps from one scene orcamera angle to another. Imagine a movie in which everysingle transition between two frames was a cut to acompletely different scene. When shown through aprojector, it would be incomprehensibleon the screen itwould look like random static. Presumably there is someFrench avant-garde film that has already used thistechnique.

The real universe is not an avant-garde film. Weexperience a degree of continuity through timeif the cat ison your lap now, there might be some danger that she willstalk off, but there is little worry that she will simplydematerialize into nothingness one moment later. Thiscontinuity is not absolute, at the microscopic level; particlescan appear and disappear, or at least transform under theright conditions into different kinds of particles. But there isnot a wholesale rearrangement of reality from moment tomoment.

This phenomenon of persistence allows us to think aboutthe world in a different way. Instead of a collection ofthings distributed through space that keep changing intodifferent configurations, we can think of the entire history ofthe world, or any particular thing in it, in one fell swoop.Rather than thinking of Miss Kitty as a particulararrangement of cells and fluids, we can think of her entirelife stretching through time, from birth to death. The historyof an object (a cat, a planet, an electron) through timedefines its world linethe trajectory the object takesthrough space as time passes.3 The world line of an objectis just the complete set of positions the object has in theworld, labeled by the particular time it was in each position.

Figure 1: The world, ordered into different moments oftime. Objects (including people and cats) persist frommoment to moment, defining world lines that stretch throughtime.

Finding ourselvesThinking of the entire history of the universe all at once,rather than thinking of the universe as a set of things thatare constantly moving around, is the first step towardthinking of time as kind of like space, which we willexamine further in the chapters to come. We use both timeand space to help us pinpoint things that happen in theuniverse. When you want to meet someone for coffee, orsee a certain showing of a movie, or show up for workalong with everyone else, you need to specify a time: Letsmeet at the coffee shop at 6:00 P.M. this Thursday.

If you want to meet someone, of course, its not sufficientjust to specify a time; you also need to specify a place.(Which coffee shop are we talking about here?) Physicistssay that space is three-dimensional. What that means isthat we require three numbers to uniquely pick out aparticular location. If the location is near the Earth, aphysicist might give the latitude, longitude, and heightabove ground. If the location is somewhere far away,astronomically speaking, we might give its direction in thesky (two numbers, analogous to latitude and longitude),plus the distance from Earth. It doesnt matter how wechoose to specify those three numbers; the crucial point isthat you will always need exactly three. Those threenumbers are the coordinates of that location in space. Wecan think of a little label attached to each point, telling usprecisely what the coordinates of that point are.

Figure 2: Coordinates attached to each point in space.In everyday life, we can often shortcut the need to specify

all three coordinates of space. If you say the coffee shop atEighth and Main Street, youre implicitly giving twocoordinatesEighth and Main Streetand youreassuming that we all agree the coffee shop is likely to be atground level, rather than in the air or underground. Thats aconvenience granted to us by the fact that much of thespace we use to locate things in our daily lives is effectivelytwo-dimensional, confined near the surface of the Earth.But in principle, all three coordinates are needed to specifya point in space.

Each point in space occurs once at each moment oftime. If we specify a certain location in space at onedefinite moment in time, physicists call that an event. (Thisis not meant to imply that its an especially exciting event;any random point in empty space at any particular momentof time would qualify, so long as its uniquely specified.)What we call the universe is just the set of all eventsevery point in space, at every moment of time. So we needfour numbersthree coordinates of space, and one of timeto uniquely pick out an event. Thats why we say that theuniverse is four-dimensional. This is such a useful conceptthat we will often treat the whole collection, every point inspace at every moment of time, as a single entity calledspacetime.

This is a big conceptual leap, so its worth pausing totake it in. Its natural to think of the world as a three-dimensional conglomeration that keeps changing(happening over and over again, slightly differently eachtime). Were now suggesting that we can think of the wholeshebang, the entire history of the world, as a single four-dimensional thing, where the additional dimension is time.In this sense, time serves to slice up the four-dimensionaluniverse into copies of space at each moment in timethewhole universe at 10:00 A.M. on January 20, 2010; thewhole universe at 10:01 A.M. on January 20, 2010; and soon. There are an infinite number of such slices, togethermaking up the universe.

2. Time measures the duration elapsed betweenevents

The second aspect of time is that it measures the durationelapsed between events. That sounds pretty similar to thelabels moments in the universe aspect already discussed,but there is a difference. Time not only labels and ordersdifferent moments; it also measures the distance betweenthem.

When taking up the mantle of philosopher or scientist

and trying to make sense of a subtle concept, its helpful tolook at things operationallyhow do we actually use thisidea in our experience? When we use time, we refer to themeasurements that we get by reading clocks. If you watch aTV show that is supposed to last one hour, the reading onyour clock at the end of the show will be one hour later thanwhat it read when the show began. Thats what it means tosay that one hour elapsed during the broadcast of thatshow: Your clock read an hour later when it ended thanwhen it began.

But what makes a good clock? The primary criterion isthat it should be consistentit wouldnt do any good tohave a clock that ticked really fast sometimes and reallyslowly at others. Fast or slow compared to what? Theanswer is: other clocks. As a matter of empirical fact (ratherthan logical necessity), there are some objects in theuniverse that are consistently periodicthey do the samething over and over again, and when we put them next toone another we find them repeating in predictable patterns.

Think of planets in the Solar System. The Earth orbitsaround the Sun, returning to the same position relative tothe distant stars once every year. By itself, thats not someaningfulits just the definition of a year. But Mars, asit turns out, returns to the same position once every 1.88years. That kind of statement is extremely meaningfulwithout recourse to the concept of a year, we can say thatEarth moves around the Sun 1.88 times every time Marsorbits just once.4 Likewise, Venus moves around the Sun1.63 times every time Earth orbits just once.

The key to measuring time is synchronized repetitionawide variety of processes occur over and over again, andthe number of times that one process repeats itself whileanother process returns to its original state is reliablypredictable. The Earth spins on its axis, and its going to doso 365.25 times every time the Earth moves around theSun. The tiny crystal in a quartz watch vibrates2,831,155,200 times every time the Earth spins on its axis.(Thats 32,768 vibrations per second, 3,600 seconds in anhour, 24 hours in a day.5) The reason why quartz watchesare reliable is that quartz crystal has extremely regularvibrations; even as the temperature or pressure changes,the crystal will vibrate the same number of times for everyone rotation of the Earth.

So when we say that something is a good clock, wemean that it repeats itself in a predictable way relative toother good clocks. It is a fact about the universe that suchclocks exist, and thank goodness. In particular, at themicroscopic level where all that matters are the rules ofquantum mechanics and the properties (masses, electriccharges) of individual elementary particles, we find atomsand molecules that vibrate with absolutely predictablefrequencies, forming a widespread array of excellent clocksmarching in cheerful synchrony. A universe without goodclocksin which no processes repeated themselves apredictable number of times relative to other repeatingprocesseswould be a scary universe indeed.6

Still, good clocks are not easy to come by. Traditionalmethods of timekeeping often referred to celestial objectsthe positions of the Sun or stars in the skybecausethings down here on Earth tend to be messy andunpredictable. In 1581, a young Galileo Galilei reportedlymade a breakthrough discovery while he sat bored during achurch service in Pisa. The chandelier overhead wouldswing gently back and forth, but it seemed to move morequickly when it was swinging widely (after a gust of wind, forexample) and more slowly when wasnt moving as far.

Intrigued, Galileo decided to measure how much time ittook for each swing, using the only approximately periodicevent to which he had ready access: the beating of his ownpulse. He found something interesting: The number ofheartbeats between swings of the chandelier was roughlythe same, regardless of whether the swings were wide ornarrow. The size of the oscillationshow far the pendulumswung back and forthdidnt affect the frequency of thoseoscillations. Thats not unique to chandeliers in Pisanchurches; its a robust property of the kind of pendulumphysicists call a simple harmonic oscillator. And thatswhy pendulums form the centerpiece of grandfather clocksand other timekeeping devices: Their oscillations areextremely reliable. The craft of clock making involves thesearch for ever-more-reliable forms of oscillations, fromvibrations in quartz to atomic resonances.

Our interest here is not really in the intricacies of clockconstruction, but in the meaning of time. We live in a worldthat contains all sorts of periodic processes, which repeat apredictable number of times in comparison to certain otherperiodic processes. And thats how we measure duration:by the number of repetitions of such a process. When wesay that our TV program lasts one hour, we mean that thequartz crystal in our watch will oscillate 117,964,800 timesbetween the start and end of the show (32,768 oscillationsper second, 3,600 seconds in an hour).

Notice that, by being careful about defining time, weseem to have eradicated the concept entirely. Thats justwhat any decent definition should doyou dont want todefine something in terms of itself. The passage of timecan be completely recast in terms of certain thingshappening together, in synchrony. The program lasts onehour is equivalent to there will be 117,964,800 oscillationsof the quartz crystal in my watch between the beginning andend of the program (give or take a few commercials). If youreally wanted to, you could reinvent the entire superstructureof physics in a way that completely eliminated the conceptof time, by replacing it with elaborate specifications ofhow certain things happen in coincidence with certain otherthings.7 But why would we want to? Thinking in terms oftime is convenient, and more than that, it reflects a simpleunderlying order in the way the universe works.

Figure 3: Good clocks exhibit synchronized repetition.Every time one day passes, the Earth rotates once about

its axis, a pendulum with a period of 1 second oscillates86,400 times, and a quartz watch crystal vibrates2,831,155,200 times.

Slowing, stopping, bendingtimeArmed with this finely honed understanding of what wemean by the passage of time, at least one big question canbe answered: What would happen if time were to slowdown throughout the universe? The answer is: Thats not asensible question to ask. Slow down relative to what? Iftime is what clocks measure, and every clock were to slowdown by the same amount, it would have absolutely noeffect at all. Telling time is about synchronized repetition,and as long as the rate of one oscillation is the samerelative to some other oscillation, all is well.

As human beings we feel the passage of time. Thatsbecause there are periodic processes occurring within ourown metabolismbreaths, heartbeats, electrical pulses,digestion, rhythms of the central nervous system. We are acomplicated, interconnected collection of clocks. Ourinternal rhythms are not as reliable as a pendulum or aquartz crystal; they can be affected by external conditions orour emotional states, leading to the impression that time ispassing more quickly or more slowly. But the truly reliableclocks ticking away inside our bodiesvibratingmolecules, individual chemical reactionsarent movingany faster or slower than usual.8

What could happen, on the other hand, is that certainphysical processes that we thought were good clockswould somehow go out of synchronizationone clockslows down, or speeds up, compared to all the rest. Asensible response in that case would be to blame thatparticular clock, rather than casting aspersions on timeitself. But if we stretch a bit, we can imagine a particularcollection of clocks (including molecular vibrations andother periodic processes) that all change in concert withone another, but apart from the rest of the world. Then wewould have to ask whether it was appropriate to say thatthe rate at which time passes had really changed within thatcollection.

Consider an extreme example. Nicholson Bakers novelThe Fermata tells the story of a man, Arno Strine, with theability to stop time. (Mostly he uses this miraculous powerto go around undressing women.) It wouldnt mean anythingif time stopped everywhere; the point is that Arno keepsmoving through time, while everything around him stops.We all know this is unrealistic, but its instructive to reflectupon the way in which it flouts the laws of physics. What thisapproach to stopping time entails is that every kind ofmotion and rhythm in Arnos body continues as usual, whileevery kind of motion and rhythm in the outside worldfreezes absolutely still. Of course we have to imagine thattime continues for all of the air and fluids within Arno,otherwise he would instantly die. But if the air in the rest ofthe room has truly stopped experiencing time, eachmolecule must remain suspended precisely in its location;consequently, Arno would be unable to move, trapped in aprison of rigidly stationary air molecules. Okay, lets begenerous and assume that time would proceed normally forany air molecules that came sufficiently close to Arnosskin. (The book alludes to something of the sort.) But

everything outside, by assumption, is not changing in anyway. In particular, no sound or light would be able to travelto him from the outside world; Arno would be completelydeaf and blind. It turns out not to be a promisingenvironment for a Peeping Tom.9

What if, despite all the physical and narrative obstacles,something like this really could happen? Even if we cantstop time around us, presumably we can imagine speedingup the motion of some local clocks. If we truly measure timeby synchronized repetition, and we arranged an ensembleof clocks that were all running fast compared to the outsideworld while they remained in synchrony with one another,wouldnt that be something like time running faster withinthat arrangement?

It depends. Weve wandered far afield from what mightactually happen in the real world, so lets establish somerules. Were fortunate enough to live in a universe thatfeatures very reliable clocks. Without such clocks, we cantuse time to measure the duration between events. In theworld of The Fermata, we could say that time slowed downfor the universe outside Arno Strineor, equivalently andperhaps more usefully, that time for him sped up, while therest of the world went on as usual. But just as well, we couldsay that time was completely unaffected, and whatchanged were the laws of particle physics (masses,charges on different particles) within Arnos sphere ofinfluence. Concepts like time are not handed to usunambiguously by the outside world but are invented byhuman beings trying to make sense of the universe. If theuniverse were very different, we might have to make senseof it in a different way.

Meanwhile, there is a very real way for one collection ofclocks to measure time differently than another: have themmove along different paths through spacetime. Thatscompletely compatible with our claim that good clocksshould measure time in the same way, because we cantreadily compare clocks unless theyre next to one anotherin space. The total amount of time elapsed on two differenttrajectories can be different without leading to anyinconsistencies. But it does lead to something importantthe theory of relativity.

Twisty paths throughspacetimeThrough the miracle of synchronized repetition, timedoesnt simply put different moments in the history of theuniverse into order; it also tells us how far apart they are(in time). We can say more than 1776 happened before2010; we can say 1776 happened 234 years before2010.

I should emphasize a crucial distinction betweendividing the universe into different moments andmeasuring the elapsed time between events, a distinctionthat will become enormously important when we get torelativity. Lets imagine you are an ambitious temporal10engineer, and youre not satisfied to just have yourwristwatch keep accurate time; you want to be able to knowwhat time it is at every other event in spacetime as well.You might be tempted to wonder: Couldnt we(hypothetically) construct a time coordinate all throughoutthe universe, just by building an infinite number of clocks,synchronizing them to the same time, and scattering them

throughout space? Then, wherever we went in spacetime,there would be a clock sitting at each point telling us whattime it was, once and for all.

The real world, as we will see, doesnt let us construct anabsolute universal time coordinate. For a long time peoplethought it did, under no less an authority than that of SirIsaac Newton. In Newtons view of the universe, there wasone particular right way to slice up the universe into slicesof space at a particular moment of time. And we couldindeed, at least in a thought-experiment kind of way, sendclocks all throughout the universe to set up a timecoordinate that would uniquely specify when a certain eventwas taking place.

But in 1905, along comes Einstein with his special theoryof relativity.11 The central conceptual breakthrough ofspecial relativity is that our two aspects of time, time labelsdifferent moments and time is what clocks measure, arenot equivalent, or even interchangeable. In particular, thescheme of setting up a time coordinate by sending clocksthroughout the universe would not work: two clocks, leavingthe same event and arriving at the same event but takingdifferent paths to get there, will generally experiencedifferent durations along the journey, slipping out ofsynchronization. Thats not because we havent beencareful enough to pick good clocks, as defined above. Itsbecause the duration elapsed along two trajectoriesconnecting two events in spacetime need not be thesame.

This idea isnt surprising, once we start thinking thattime is kind of like space. Consider an analogousstatement, but for space instead of time: The distancetraveled along two paths connecting two points in spaceneed not be the same. Doesnt sound so surprising at all,does it? Of course we can connect two points in space bypaths with different lengths; one could be straight and onecould be curved, and we would always find that the distancealong the curved path was greater. But the difference incoordinates between the same two points is always thesame, regardless of how we get from one point to another.Thats because, to drive home the obvious, the distanceyou travel is not the same as your change in coordinates.Consider a running back in football who zips back and forthwhile evading tacklers, and ends up advancing from the 30-yard line to the 80-yard line. (It should really be theopponents 20-yard line, but the point is clearer this way.)The change in coordinates is 50 yards, no matter how longor short was the total distance he ran.

Figure 4: Yard lines serve as coordinates on a footballfield. A running back who advances the ball from the 30-yard line to the 80-yard line has changed coordinates by 50yards, even though the distance traveled may have beenmuch greater.

The centerpiece of special relativity is the realization thattime is like that. Our second definition, time is duration asmeasured by clocks, is analogous to the total length of apath through space; the clock itself is analogous to an

odometer or some other instrument that measures the totaldistance traveled. This definition is simply not the same asthe concept of a coordinate labeling different slices ofspacetime (analogous to the yard lines on a football field).And this is not some kind of technical problem that we canfix by building better clocks or making better choicesabout how we travel through spacetime; its a feature ofhow the universe works, and we need to learn to live with it.

As fascinating and profound as it is that time works inmany ways similar to space, it will come as no surprise thatthere are crucial differences as well. Two of them arecentral elements of the theory of relativity. First, while thereare three dimensions of space, there is only one of time;that brute fact has important consequences for physics, asyou might guess. And second, while a straight line betweentwo points in space describes the shortest distance, astraight trajectory between two events in spacetimedescribes the longest elapsed duration.

But the most obvious, blatant, unmistakable differencebetween time and space is that time has a direction, andspace doesnt. Time points from the past toward the future,while (out there in space, far away from local disturbanceslike the Earth) all directions of space are created equal. Wecan invert directions in space without doing damage to howphysics works, but all sorts of real processes can happen inone direction of time but not the other. Its to this crucialdifference that we now turn.

3. Time is a medium through which we move

The sociology experiment suggested at the beginning ofthis chapter, in which you ask strangers how they woulddefine time, also serves as a useful tool for distinguishingphysicists from non-physicists. Nine times out of ten, aphysicist will say something related to one of the first twonotions abovetime is a coordinate, or time is a measureof duration. Equally often, a non-physicist will saysomething related to the third aspect we mentionedtimeis something that flows from past to future. Time whooshesalong, from back then to now and on toward later.

Or, conversely, someone might say that we move throughtime, as if time were a substance through which we couldpass. In the Afterword to his classic Zen and the Art ofMotorcycle Maintenance, Robert Pirsig relates a particulartwist on this metaphor. The ancient Greeks, according toPirsig, saw the future as something that came upon themfrom behind their backs, with the past receding awaybefore their eyes.12 When you think about it, that seems abit more honest than the conventional view that we marchtoward the future and away from the past. We knowsomething about the past, from experience, while the futureis more conjectural.

Common to these perspectives is the idea that time is athing, and its a thing that can changeflow around us, orpass by as we move through it. But conceptualizing time assome sort of substance with its own dynamics, perhapseven the ability to change at different rates depending oncircumstances, raises one crucially important question.

What in the world is that supposed to mean?Consider something that actually does flow, such as a

river. We can think about the river from a passive or anactive perspective: Either we are standing still as the waterrushes by, or perhaps we are on a boat moving along withthe river as the banks on either side move relative to us.

The river flows, no doubt about that. And what that means

is that the location of some particular drop of river waterchanges with timehere it is at some moment, there it isjust a bit later. And we can talk sensibly about the rate atwhich the river flows, which is just the velocity of the waterin other words, the distance that the water travels in a givenamount of time. We could measure it in miles per hour, ormeters per second, or whatever units of distance traveledper interval of time you prefer. The velocity may very wellchange from place to place or moment to momentsometimes the river flows faster; sometimes it flows moreslowly. When we are talking about the real flow of actualrivers, all this language makes perfect sense.

But when we examine carefully the notion that time itselfsomehow flows, we hit a snag. The flow of the river was achange with timebut what is it supposed to mean to saythat time changes with time? A literal flow is a change oflocation over timebut time doesnt have a location. Sowhat is it supposed to be changing with respect to?

Think of it this way: If time does flow, how would wedescribe its speed? It would have to be something like xhours per houran interval of time per unit time. And I cantell you what x is going to beits 1, all the time. The speedof time is 1 hour per hour, no matter what else might begoing on in the universe.

The lesson to draw from all this is that its not quite rightto think of time as something that flows. Its a seductivemetaphor, but not one that holds up under closer scrutiny.To extract ourselves from that way of thinking, its helpful tostop picturing ourselves as positioned within the universe,with time flowing around us. Instead, lets think of theuniverseall of the four-dimensional spacetime around usas somehow a distinct entity, as if we were observing itfrom an external perspective. Only then can we appreciatetime for what it truly is, rather than privileging our positionright here in the middle of it.

The view from nowhenWe cant literally stand outside the universe. The universe isnot some object that sits embedded in a larger space (asfar as we know); its the collection of everything that exists,space and time included. So were not wondering what theuniverse would really look like from the point of view ofsomeone outside it; no such being could possibly exist.Rather, were trying to grasp the entirety of space and timeas a single entity. Philosopher Huw Price calls this theview from nowhen, a perspective separate from anyparticular moment in time.13 We are all overly familiar withtime, having dealt with it every day of our lives. But we canthelp but situate ourselves within time, and its useful tocontemplate all of space and time in a single picture.

And what do we see, when looking down from nowhen?We dont see anything changing with time, because we areoutside of time ourselves. Instead, we see all of history atoncepast, present, and future. Its like thinking of spaceand time as a book, which we could in principle open to anypassage, or even cut apart and spread out all the pagesbefore us, rather than as a movie, where we are forced towatch events in sequence at specific times. We could alsocall this the Tralfamadorian perspective, after the aliens inKurt Vonneguts Slaughterhouse-Five. According toprotagonist Billy Pilgrim,

The Tralfamadorians can look at all thedifferent moments just the way we can look at

different moments just the way we can look ata stretch of the Rocky Mountains, forinstance. They can see how permanent all themoments are, and they can look at anymoment that interests them. It is just anillusion we have here on earth that onemoment follows another like beads on astring, and that once a moment is gone it isgone forever.14

How do we reconstruct our conventional understanding offlowing time from this lofty timeless Tralfamadorian perch?What we see are correlated events, arranged in asequence. There is a clock reading 6:45, and a personstanding in their kitchen with a glass of water in one handand an ice cube in the other. In another scene, the clockreads 6:46 and the person is again holding the glass ofwater, now with the ice cube inside. In yet another one, theclock reads 6:50 and the person is holding a slightly colderglass of water, now with the ice cube somewhat melted.

In the philosophical literature, this is sometimes calledthe block time or block universe perspective, thinking ofall space and time as a single existing block of spacetime.For our present purposes, the important point is that wecan think about time in this way. Rather than carrying apicture in the back of our minds in which time is asubstance that flows around us or through which we move,we can think of an ordered sequence of correlated events,together constituting the entire universe. Time is thensomething we reconstruct from the correlations in theseevents. This ice cube melted over the course of tenminutes is equivalent to the clock reads ten minutes laterwhen the ice cube has melted than it does when the icecube is put into the glass. Were not committing ourselvesto some dramatic conceptual stance to the effect that itswrong to think of ourselves as embedded within time; it justturns out to be more useful, when we get around to askingwhy time and the universe are the way they are, to be ableto step outside and view the whole ball of wax from theperspective of nowhen.

Opinions differ, of course. The struggle to understandtime is a puzzle of long standing, and what is real andwhat is useful have been very much up for debate. One ofthe most influential thinkers on the nature of time was St.Augustine, the fifth-century North African theologian andFather of the Church. Augustine is perhaps best known fordeveloping the doctrine of original sin, but he wasinterdisciplinary enough to occasionally turn his hand tometaphysical issues. In Book XI of his Confessions, hediscusses the nature of time.

What is by now evident and clear is thatneither future nor past exists, and it is inexactlanguage to speak of three timespast,present, and future. Perhaps it would beexact to say: there are three times, a presentof things past, a present of things present, apresent of things to come. In the soul thereare these three aspects of time, and I do notsee them anywhere else. The presentconsidering the past is memory, the presentconsidering the present is immediateawareness, the present considering thefuture is expectation.15

Augustine doesnt like this block-universe business. He iswhat is known as a presentist, someone who thinks thatonly the present moment is realthe past and future arethings that we here in the present simply try to reconstruct,

things that we here in the present simply try to reconstruct,given the data and knowledge available to us. Theviewpoint weve been describing, on the other hand, is(sensibly enough) known as eternalism, which holds thatpast, present, and future are all equally real.16

Concerning the debate between eternalism andpresentism, a typical physicist would say: Who cares?Perhaps surprisingly, physicists are not overly concernedwith adjudicating which particular concepts are real or not.They care very much about how the real world works, but tothem its a matter of constructing comprehensivetheoretical models and comparing them with empiricaldata. Its not the individual concepts characteristic of eachmodel (past, future, time) that matter; its the structureas a whole. Indeed, it often turns out to be the case that onespecific model can be described in two completely differentways, using an entirely different set of concepts. 17

So, as scientists, our goal is to construct a model ofreality that successfully accounts for all of these differentnotions of timetime is measured by clocks, time is acoordinate on spacetime, and our subjective feeling thattime flows. The first two are actually very well understood interms of Einsteins theory of relativity, as we will cover inPart Two of the book. But the third remains a bitmysterious. The reason why I am belaboring the notion ofstanding outside time to behold the entire universe as asingle entity is because we need to distinguish the notion oftime in and of itself from the perception of time asexperienced from our parochial view within the presentmoment. The challenge before us is to reconcile these twoperspectives.

2THE HEAVY HAND OFENTROPY

Eating is unattractive too . . . Variousitems get gulped into my mouth, andafter skillful massage with tongueand teeth I transfer them to the platefor additional sculpture with knife andfork and spoon. That bits quitetherapeutic at least, unless yourehaving soup or something, which canbe a real sentence. Next you facethe laborious business of cooling, ofreassembly, of storage, before thereturn of these foodstuffs to theSuper ette, where, admittedly, I ampromptly and generously reimbursedfor my pains. Then you tool down theaisles, with trolley or basket,returning each can and packet to itsrightful place.

Martin Amis, Times Arrow18

Forget about spaceships, rocket guns, clashes withextraterrestrial civilizations. If you want to tell a story thatpowerfully evokes the feeling of being in an alienenvironment, you have to reverse the direction of time.

You could, of course, simply take an ordinary story andtell it backward, from the conclusion to the beginning. Thisis a literary device known as reverse chronology andappears at least as early as Virgils Aeneid. But to really jarreaders out of their temporal complacency, you want tohave some of your characters experience time backward.The reason its jarring, of course, is that all of us nonfictionalcharacters experience time in the same way; thats due tothe consistent increase of entropy throughout the universe,which defines the arrow of time.

THROUGH THE LOOKINGGLASS

F. Scott Fitzgeralds short story The Curious Case ofBenjamin Buttonmore recently made into a film starringBrad Pittfeatures a protagonist who is born as an oldman and gradually grows younger as time passes. Thenurses of the hospital at which Benjamin is born are,understandably, somewhat at a loss.

Wrapped in a voluminous white blanket, andpartly crammed into one of the cribs, theresat an old man apparently about seventyyears of age. His sparse hair was almostwhite, and from his chin dripped a longsmoke-coloured beard, which waved

absurdly back and forth, fanned by the breezecoming in at the window. He looked up at Mr.Button with dim, faded eyes in which lurked apuzzled question.

Am I mad? thundered Mr. Button, histerror resolving into rage. Is this someghastly hospital joke?

It doesnt seem like a joke to us, repliedthe nurse severely. And I dont know whetheryoure mad or notbut that is most certainlyyour child.

The cool perspiration redoubled on Mr.Buttons forehead. He closed his eyes, andthen, opening them, looked again. There wasno mistakehe was gazing at a man ofthreescore and tena baby of threescoreand ten, a baby whose feet hung over thesides of the crib in which it was reposing.19

No mention is made in the story of what poor Mrs. Buttonmust have been feeling around this time. (In the movieversion, at least the newborn Benjamin is baby-sized, albeitold and wrinkled.)

Because it is so bizarre, having time run backward forsome characters in a story is often played for comic effect.In Lewis Carrolls Through the Looking-Glass, Alice isastonished upon first meeting the White Queen, who livesin both directions of time. The Queen is shouting andshaking her finger in pain:

What IS the matter? [Alice] said, as soon asthere was a chance of making herself heard.Have you pricked your finger?

I havent pricked it YET, the Queen said,but I soon shalloh, oh, oh!

When do you expect to do it? Aliceasked, feeling very much inclined to laugh.

When I fasten my shawl again, the poorQueen groaned out: the brooch will comeundone directly. Oh, oh! As she said thewords the brooch flew open, and the Queenclutched wildly at it, and tried to clasp itagain.

Take care! cried Alice. Youre holding itall crooked! And she caught at the brooch;but it was too late: the pin had slipped, andthe Queen had pricked her finger.20

Carroll (no relation21) is playing with a deep feature of thenature of timethe fact that causes precede effects. Thescene makes us smile, while serving as a reminder of howcentral the arrow of time is to the way we experience theworld.

Time can be reversed in the service of tragedy, as wellas comedy. Martin Amiss novel Times Arrow is a classicof the reversing-time genre, even accounting for the factthat its a pretty small genre.22 Its narrator is a disembodiedconsciousness who lives inside another person, OdiloUnverdorben. The host lives life in the ordinary sense,forward in time, but the homunculus narrator experienceseverything backwardhis first memory is Unverdorbensdeath. He has no control over Unverdorbens actions, noraccess to his memories, but passively travels through life inreverse order. At first Unverdorben appears to us as adoctor, which strikes the narrator as quite a morbidoccupationpatients shuffle into the emergency room,where staff suck medicines out of their bodies and rip off

their bandages, sending them out into the night bleedingand screaming. But near the end of the book, we learn thatUnverdorben was an assistant at Auschwitz, where hecreated life where none had been beforeturningchemicals and electricity and corpses into living persons.Only now, thinks the narrator, does the world finally makesense.

THE ARROW OF TIME

There is a good reason why reversing the relative directionof time is an effective tool of the imagination: In the actual,non-imaginary world, it never happens. Time has adirection, and it has the same direction for everybody.None of us has met a character like the White Queen, whoremembers what we think of as the future rather than (or inaddition to) the past.

What does it mean to say that time has a direction, anarrow pointing from the past to the future? Think aboutwatching a movie played in reverse. Generally, its prettyclear if we are seeing something running the wrong way intime. A classic example is a diver and a pool. If the diverdives, and then there is a big splash, followed by wavesbouncing around in the water, all is normal. But if we see apool that starts with waves, which collect into a big splash,in the process lifting a diver up onto the board andbecoming perfectly calm, we know something is up: Themovie is being played backward.

Certain events in the real world always happen in thesame order. Its dive, splash, waves; never waves, splash,spit out a diver. Take milk and mix it into a cup of blackcoffee; never take coffee with milk and separate the twoliquids. Sequences of this sort are called irreversibleprocesses. We are free to imagine that kind of sequenceplaying out in reverse, but if we actually see it happen, wesuspect cine matic trickery rather than a faithfulreproduction of reality.

Irreversible processes are at the heart of the arrow oftime. Events happen in some sequences, and not in others.Furthermore, this ordering is perfectly consistent, as far aswe know, throughout the observable universe. Someday wemight find a planet in a distant solar system that containsintelligent life, but nobody suspects that we will find a planeton which the aliens regularly separate (the indigenousequivalents of) milk and coffee with a few casual swirls of aspoon. Why isnt that surprising? Its a big universe outthere; things might very well happen in all sorts ofsequences. But they dont. For certain kinds of processesroughly speaking, complicated actions with lots ofindividual moving partsthere seems to be an allowedorder that is somehow built into the very fabric of the world.

Tom Stoppards play Arcadia uses the arrow of time asa central organizing metaphor. Heres how Thomasina, ayoung prodigy who was well ahead of her time, explains theconcept to her tutor:

THOMASINA: When you stir your ricepudding, Septimus, the spoonful of jamspreads itself round making red trails like thepicture of a meteor in my astronomical atlas.But if you need stir backward, the jam will notcome together again. Indeed, the puddingdoes not notice and continues to turn pink justas before. Do you think this odd?

SEPTIMUS: No.THOMASINA: Well, I do. You cannot stirthings apart.SEPTIMUS: No more you can, time mustneeds run backward, and since it will not, wemust stir our way onward mixing as we go,disorder out of disorder into disorder untilpink is complete, unchanging andunchangeable, and we are done with it forever. This is known as free will or self-determination.23

The arrow of time, then, is a brute fact about our universe.Arguably the brute fact about our universe; the fact thatthings happen in one order and not in the reverse order isdeeply ingrained in how we live in the world. Why is it likethat? Why do we live in a universe where X is often followedby Y, but Y is never followed by X?

The answer lies in the concept of entropy that Imentioned above. Like energy or temperature, entropy tellsus something about the particular state of a physicalsystem; specifically, it measures how disorderly the systemis. A collection of papers stacked neatly on top of oneanother has a low entropy; the same collection, scatteredhaphazardly on a desktop, has a high entropy. The entropyof a cup of coffee along with a separate teaspoon of milk islow, because there is a particular orderly segregation of themolecules into milk and coffee, while the entropy of thetwo mixed together is comparatively large. All of theirreversible processes that reflect times arrowwe canturn eggs into omelets but not omelets into eggs, perfumedisperses through a room but never collects back into thebottle, ice cubes in water melt but glasses of warm waterdont spontaneously form ice cubesshare a commonfeature: Entropy increases throughout, as the systemprogresses from order to disorder. Whenever we disturbthe universe, we tend to increase its entropy.

A big part of our task in this book will be to explain howthe single idea of entropy ties together such a disparate setof phenomena, and then to dig more deeply into whatexactly this stuff called entropy really is, and why it tends toincrease. The final taskstill a profound open question incontemporary physicsis to ask why the entropy was solow in the past, so that it could be increasing ever since.

FUTURE AND PAST VS. UPAND DOWN

But first, we need to contemplate a prior question: Shouldwe really be surprised that certain things happen in onedirection of time, but not in the other? Who ever said thateverything should be reversible, anyway?

Think of time as a label on events as they happen. Thatsone of the ways in which time is like spacethey both helpus locate things in the universe. But from that point of view,there is also a crucial difference between time and spacedirections in space are created equal, while directions intime (namely, the past and the future) are very different.Here on Earth, directions in space are easily distinguisheda compass tells us whether we are moving north, south,east, or west, and nobody is in any danger of confusing upwith down. But thats not a reflection of deep underlyinglaws of natureits just because we live on a giant planet,

with respect to which we can define different directions. Ifyou were floating in a space suit far away from any planets,all directions in space would truly be indistinguishablethere would be no preferred notion of up or down.

The technical way to say this is that there is a symmetryin the laws of natureevery direction in space is as goodas every other. Its easy enough to reverse the direction ofspacetake a photograph and print it backward, or forthat matter just look in a mirror. For the most part, the viewin a mirror appears pretty unremarkable. The obviouscounterexample is writing, for which its easy to tell that weare looking at a reversed image; thats because writing,like the Earth, does pick out a preferred direction (yourereading this book from left to right). But the images of mostscenes not full of human creations look equally natural tous whether we see them directly or we see them through amirror.

Contrast that with time. The equivalent of looking at animage through a mirror (reversing the direction of space)is simply playing a movie backward (reversing thedirection of time). And in that case, its easy to tell whentime has been invertedthe irreversible processes thatdefine the arrow of time are suddenly occurring in thewrong order. What is the origin of this profound differencebetween space and time?

While its true that the presence of the Earth beneath ourfeet picks out an arrow of space by distinguishing up fromdown, its pretty clear that this is a local, parochialphenomenon, rather than a reflection of the underlying lawsof nature. We can easily imagine ourselves out in spacewhere there is no preferred direction. But the underlyinglaws of nature do not pick out a preferred direction of time,any more than they pick out a preferred direction in space.If we confine our attention to very simple systems with just afew moving parts, whose motion reflects the basic laws ofphysics rather than our messy local conditions, there is noarrow of timewe cant tell when a movie is being runbackward. Think about Galileos chandelier, rockingpeacefully back and forth. If someone showed you a movieof the chandelier, you wouldnt be able to tell whether it wasbeing shown forward or backwardits motion is sufficientlysimple that it works equally well in either direction of time.

Figure 5: The Earth defines a preferred direction in space,while the Big Bang defines a preferred direction in time.

The arrow of time, therefore, is not a feature of theunderlying laws of physics, at least as far as we know.Rather, like the up/down orientation space picked out bythe Earth, the preferred direction of time is also aconsequence of features of our environment. In the case oftime, its not that we live in the spatial vicinity of aninfluential object; its that we live in the temporal vicinity ofan influential event: the birth of the universe. The beginningof our observable universe, the hot dense state known as

the Big Bang, had a very low entropy. The influence of thatevent orients us in time, just as the presence of the Earthorients us in space.

NATURES MOST RELIABLELAW

The principle underlying irreversible processes is summedup in the Second Law of Thermodynamics:

The entropy of an isolated system eitherremains constant or increases with time.

(The First Law states that energy is conserved.24) TheSecond Law is arguably the most dependable law in all ofphysics. If you were asked to predict what currentlyaccepted principles of physics would still be consideredinviolate a thousand years from now, the Second Lawwould be a good bet. Sir Arthur Eddington, a leadingastrophysicist of the early twentieth century, put itemphatically:

If someone points out to you that your pettheory of the universe is in disagreement withMaxwells equations [the laws of electricityand magnetism]then so much the worse forMaxwells equations. If it is found to becontradicted by observationwell, theseexperimentalists do bungle thingssometimes. But if your theory is found to beagainst the Second Law of ThermodynamicsI can give you no hope; there is nothing for itbut to collapse in deepest humiliation.25

C. P. SnowBritish intellectual, physicist, and novelistisperhaps best known for his insistence that the TwoCultures of the sciences and the humanities had grownapart and should both be a part of our common civilization.When he came to suggest the most basic item of scientificknowledge that every educated person should understand,he chose the Second Law:

A good many times I have been present atgatherings of people who, by the standardsof the traditional culture, are thought highlyeducated and who have with considerablegusto been expressing their incredulity at theilliteracy of scientists. Once or twice I havebeen provoked and have asked the companyhow many of them could describe theSecond Law of Thermodynamics, the law ofentropy. The response was cold: it was alsonegative. Yet I was asking something whichis about the scientific equivalent of: Haveyou read a work of Shakespeares?26

Im sure Baron Snow was quite the hit at Cambridgecocktail parties. (To be fair, he did later admit that evenphysicists didnt really understand the Second Law.)

Our modern definition of entropy was proposed byAustrian physicist Ludwig Boltzmann in 1877. But theconcept of entropy, and its use in the Second Law ofThermodynamics, dates back to German physicist RudolfClausius in 1865. And the Second Law itself goes backeven earlierto French military engineer Nicolas LonardSadi Carnot in 1824. How in the world did Clausius use

entropy in the Second Law without knowing its definition,and how did Carnot manage to formulate the Second Lawwithout even using the concept of entropy at all?

The nineteenth century was the heroic age ofthermodynamicsthe study of heat and its properties. Thepioneers of thermodynamics studied the interplay betweentemperature, pressure, volume, and energy. Their interestwas by no means abstractthis was the dawn of theindustrial age, and much of their work was motivated by thedesire to build better steam engines.

Today physicists understand that heat is a form of energyand that the temperature of an object is simply a measureof the average kinetic energy (energy of motion) of theatoms in the object. But in 1800, scientists didnt believe inatoms, and they didnt understand energy very well. Carnot,whose pride was wounded by the fact that the English wereahead of the French in steam engine technology, sethimself the task of understanding how efficient such anengine could possibly behow much useful work could youdo by burning a certain amount of fuel? He showed thatthere is a fundamental limit to such extraction. By taking anintellectual leap from real machines to idealized heatengines, Carnot demonstrated there was a best possibleengine, which got the most work out of a given amount offuel operating at a given temperature. The trick,unsurprisingly, was to minimize the production of wasteheat. We might think of heat as useful in warming ourhouses during the winter, but it doesnt help in doing whatphysicists think of as workgetting something like apiston or a flywheel to move from place to place. WhatCarnot realized was that even the most efficient enginepossible is not perfect; some energy is lost along the way.In other words, the operation of a steam engine is anirreversible process.

So Carnot appreciated that engines did something thatcould not be undone. It was Clausius, in 1850, whounderstood that this reflected a law of nature. He formulatedhis law as heat does not spontaneously flow from coldbodies to warm ones. Fill a balloon with hot water andimmerse it in cold water. Everyone knows that thetemperatures will tend to average out: The water in theballoon will cool down as the surrounding liquid warms up.The opposite never happens. Physical systems evolvetoward a state of equilibriuma quiescent configurationthat is as uniform as possible, with equal temperatures inall components. From this insight, Clausius was able to re-derive Carnots results concerning steam engines.

So what does Clausius law (heat never flowsspontaneously from colder bodies to hotter ones) have todo with the Second Law (entropy never spontaneouslydecreases)? The answer is, they are the same law. In 1865Clausius managed to reformulate his original maxim interms of a new quantity, which he called the entropy. Takean object that is gradually cooling downemitting heat intoits surroundings. As this process happens, consider atevery moment the amount of heat being lost, and divide itby the temperature of the object. The entropy is then theaccumulated amount of this quantity (the heat lost dividedby the temperature) over the course of the entire process.Clausius showed that the tendency of heat to flow from hotobjects to cold ones was precisely equivalent to the claimthat the entropy of a closed system would only ever go up,never go down. An equilibrium configuration is simply onein which the entropy has reached its maximum value, andhas nowhere else to go; all the objects in contact are at thesame temperature.

If that seems a bit abstract, there is a simple way ofsumming up this view of entropy: It measures theuselessness of a certain amount of energy.27 There isenergy in a gallon of gasoline, and its usefulwe can put itto work. The process of burning that gasoline to run anengine doesnt change the total amount of energy; as longas we keep careful track of what happens, energy is alwaysconserved.28 But along the way, that energy becomesincreasingly useless. It turns into heat and noise, as well asthe motion of the vehicle powered by that engine, but eventhat motion eventually slows down due to friction. And asenergy transforms from useful to useless, its entropyincreases all the while.

The Second Law doesnt imply that the entropy of asystem can never decrease. We could invent a machinethat separated out the milk from a cup of coffee, forexample. The trick, though, is that we can only decrease theentropy of one thing by creating more entropy elsewhere.We human beings, and the machines that we might use torearrange the milk and coffee, and the food and fuel eachconsumeall of these also have entropy, which willinevitably increase along the way. Physicists draw adistinction between open systemsobjects that interactsignificantly with the outside world, exchanging entropy andenergyand closed systemsobjects that are essentiallyisolated from external influences. In an open system, likethe coffee and milk we put into our machine, entropy cancertainly decrease. But in a closed systemsay, the totalsystem of coffee plus milk plus machine plus humanoperators plus fuel and so onthe entropy will alwaysincrease, or at best stay constant.

THE RISE OF ATOMS

The great insights into thermodynamics of Carnot,Clausius, and their colleagues all took place within aphenomenological framework. They knew the big picturebut not the underlying mechanisms. In particular, they didntknow about atoms, so they didnt think of temperature andenergy and entropy as properties of some microscopicsubstrate; they thought of each of them as real things, inand of themselves. It was common in those days to think ofenergy in particular as a form of fluid, which could flow fromone body to another. The energy-fluid even had a name:caloric. And this level of understanding was perfectlyadequate to formulating the laws of thermodynamics.

But over the course of the nineteenth century, physicistsgradually became convinced that the many substances wefind in the world can all be understood as differentarrangements of a fixed number of elementary constituents,known as atoms. (The physicists actually lagged behindthe chemists in their acceptance of atomic theory.) Its anold idea, dating back to Democritus and other ancientGreeks, but it began to catch on in the nineteenth centuryfor a simple reason: The existence of atoms could explainmany observed properties of chemical reactions, whichotherwise were simply asserted. Scientists like it when asingle simple idea can explain a wide variety of observedphenomena.

These days it is elementary particles such as quarks andleptons that play the role of Democrituss atoms, but theidea is the same. What a modern scientist calls an atomis the smallest possible unit of matter that still counts as a

distinct chemical element, such as carbon or nitrogen. Butwe now understand that such atoms are not indivisible; theyconsist of electrons orbiting the atomic nucleus, and thenucleus is made of protons and neutrons, which in turn aremade of different combinations of quarks. The search forrules obeyed by these elementary building blocks of matteris often called fundamental physics, although elementaryphysics would be more accurate (and arguably less self-aggrandizing). Henceforth, Ill use atoms in the establishednineteenth-century sense of chemical elements, not theancient Greek sense of elementary particles.

The fundamental laws of physics have a fascinatingfeature: Despite the fact that they govern the behavior of allthe matter in the universe, you dont need to know them toget through your everyday life. Indeed, you would be hard-pressed to discover them, merely on the basis of yourimmediate experiences. Thats because very largecollections of particles obey distinct, autonomous rules ofbehavior, which dont really depend on the smallerstructures underneath. The underlying rules are referred toas microscopic or simply fundamental, while theseparate rules that apply only to large systems are referredto as macroscopic or emergent. The behavior oftemperature and heat and so forth can certainly beunderstood in terms of atoms: Thats the subject known asstatistical mechanics. But it can equally well beunderstood without knowing anything whatsoever aboutatoms: Thats the phenomenological approach weve beendiscussing, known as thermodynamics. It is a commonoccurrence in physics that in complex, macroscopicsystems, regular patterns emerge dynamically fromunderlying microscopic rules. Despite the way it issometimes portrayed, there is no competition betweenfundamental physics and the study of emergentphenomena; both are fascinating and crucially important toour understanding of nature.

One of the first physicists to advocate atomic theory wasa Scotsman, James Clerk Maxwell, who was alsoresponsible for the final formulation of the modern theory ofelectricity and magnetism. Maxwell, along with Boltzmann inAustria (and following in the footsteps of numerous others),used the idea of atoms to explain the behavior of gases,according to what was known as kinetic theory. Maxwelland Boltzmann were able to figure out that the atoms in agas in a container, fixed at some temperature, should havea certain distribution of velocitiesthis many would bemoving fast, that many would be moving slowly, and so on.These atoms would naturally keep banging against thewalls of the container, exerting a tiny force each time theydid so. And the accumulated impact of those tiny forceshas a name: It is simply the pressure of the gas. In this way,kinetic theory explained features of gases in terms ofsimpler rules.

ENTROPY AND DISORDER

But the great triumph of kinetic theory was its use byBoltzmann in formulating a microscopic understanding ofentropy. Boltzmann realized that when we look at somemacroscopic system, we certainly dont keep track of theexact properties of every single atom. If we have a glass ofwater in front of us, and someone sneaks in and (say)switches some of the water molecules around without

changing the overall temperature and density and so on, wewould never notice. There are many different arrangementsof particular atoms that are indistinguishable from ourmacroscopic perspective. And then he noticed that low-entropy objects are more delicate with respect to suchrearrangements. If you have an egg, and start exchangingbits of the yolk with bits of the egg white, pretty soon you willnotice. The situations that we characterize as low-entropyseem to be easily disturbed by rearranging the atomswithin them, while high-entropy ones are more robust.

Figure 6: Ludwig Boltzmanns grave in the Zentralfriedhof,Vienna. The inscribed equation, S = k log W, is his formulafor entropy in terms of the number of ways you canrearrange microscopic components of a system withoutchanging its macroscopic appearance. (See Chapter Eightfor details.)

So Boltzmann took the concept of entropy, which hadbeen defined by Clausius and others as a measure of theuselessness of energy, and redefined it in terms of atoms:

Entropy is a measure of the number ofparticular microscopic arrangements ofatoms that appear indistinguishable from amacroscopic perspective.29

It would be difficult to overemphasize the importance of thisinsight. Before Boltzmann, entropy was aphenomenological thermodynamic concept, which followedits own rules (such as the Second Law). After Boltzmann,the behavior of entropy could be derived from deeperunderlying principles. In particular, it suddenly makesperfect sense why entropy tends to increase:

In an isolated system entropy tends toincrease, because there are more ways to behigh entropy than to be low entropy.

At least, that formulation sounds like it makes perfectsense. In fact, it sneaks in a crucial assumption: that westart with a system that has a low entropy. If we start with asystem that has a high entropy, well be in equilibriumnothing will happen at all. That word start sneaks in anasymmetry in time, by privileging earlier times over laterones. And this line of reasoning takes us all the way back tothe low entropy of the Big Bang. For whatever reason, ofthe many ways we could arrange the constituents of theuniverse, at early times they were in a very special, lo w-entropy configuration.

This caveat aside, there is no question that Boltzmannsformulation of the concept of entropy represented a greatleap forward in our understanding of the arrow of time. Thisincrease in understanding, however, came at a cost.

Before Boltzmann, the Second Law was absoluteanironclad law of nature. But the definition of entropy in termsof atoms comes with a stark implication: entropy doesntnecessarily increase, even in a closed system; it is simplylikely to increase. (Overwhelmingly likely, as we shall see,but still.) Given a box of gas evenly distributed in a high-entropy state, if we wait long enough, the random motion ofthe atoms will eventually lead them all to be on one side ofthe box, just for a momenta statistical fluctuation. Whenyou run the numbers, it turns out that the time you wouldhave to wait before expecting to see such a fluctuation ismuch larger than the age of the universe. Its not somethingwe have to worry about, as a practical matter. But its there.

Some people didnt like that. They wanted the SecondLaw of Thermodynamics, of all things, to be utterly inviolate,not just something that holds true most of the time.Boltzmanns suggestion met with a great deal ofcontroversy, but these days it is universally accepted.

ENTROPY AND LIFE

This is all fascinating stuff, at least to physicists. But theramifications of these ideas go far beyond steam enginesand cups of coffee. The arrow of time manifests itself inmany different waysour bodies change as we get older,we remember the past but not the future, effects alwaysfollow causes. It turns out that all of these phenomena canbe traced back to the Second Law. Entropy, quite literally,makes life possible.

The major source of energy for life on Earth is light fromthe Sun. As Clausius taught us, heat naturally flows from ahot object (the Sun) to a cooler object (the Earth). But if thatwere the end of the story, before too long the two objectswould come into equilibrium with each otherthey wouldattain the same temperature. In fact, that is just what wouldhappen if the Sun filled our entire sky, rather thandescribing a disk about one degree across. The resultwould be an unhappy world indeed. It would be completelyinhospitable to the existence of lifenot simply becausethe temperature was high, but because it would be static.Nothing would ever change in such an equilibrium world.

In the real universe, the reason why our planet doesntheat up until it reaches the temperature of the Sun isbecause the Earth loses heat by radiating it out into space.And the only reason it can do that, Clausius would proudlynote, is because space is much colder than Earth.30 It isbecause the Sun is a hot spot in a mostly cold sky that theEarth doesnt just heat up, but rather can absorb the Sunsenergy, process it, and radiate it into space. Along the way,of course, entropy increases; a fixed amount of energy inthe form of solar radiation has a much lower entropy thanthe same amount of energy in the form of the Earthsradiation into space.

This process, in turn, explains why the biosphere of theEarth is not a static place.31 We receive energy from theSun, but it doesnt just heat us up until we reach equilibrium;its very low-entropy radiation, so we can make use of itand then release it as high-entropy radiation. All of which ispossible only because the universe as a whole, and theSolar System in particular, have a relatively low entropy atthe present time (and an even lower entropy in the past). Ifthe universe were anywhere near thermal equilibrium,nothing would ever happen.

Nothing good lasts forever. Our universe is a lively placebecause there is plenty of room for entropy to increasebefore we hit equilibrium and everything grinds to a halt. Itsnot a foregone conclusionentropy might be able to simplygrow forever. Alternatively, entropy may reach a maximum