One can answer this on the basis of the weak anthropicprinciple. Conditions in the contracting phase would not besuitable for the existence of intelligent beings who could ask thequestion: why is disorder increasing in the same direction oftime as that in which the universe is expanding? The inflationin the early stages of the universe, which the no boundaryproposal predicts, means that the universe must be expandingat very close to the critical rate at which it would just avoidrecollapse, and so will not recollapse for a very long time. Bythen all the stars will have burned out and the protons andneutrons in them will probably have decayed into light particlesand radiation. The universe would be in a state of almostcomplete disorder. There would be no strong thermodynamicarrow of time. Disorder couldn’t increase much because theuniverse would be in a state of almost complete disorderalready. However, a strong thermodynamic arrow is necessaryfor intelligent life to operate. In order to survive, human beingshave to consume food, which is an ordered form of energy,and convert it into heat, which is a disordered form of energy.
Thus intelligent life could not exist in the contracting phase ofthe universe. This is the explanation of why we observe thatthe thermodynamic and cosmological arrows of time point inthe same direction. It is not that the expansion of the universecauses disorder to increase. Rather, it is that the no boundarycondition causes disorder to increase and the conditions to besuitable for intelligent life only in the expanding phase.
To summarize, the laws of science do not distinguishbetween the forward and backward directions of time. However,there are at least three arrows of time that do distinguish thepast from the future. They are the thermodynamic arrow, thedirection of time in which disorder increases; the psychologicalarrow, the direction of time in which we remember the pastand not the future; and the cosmological arrow, the direction oftime in which the universe expands rather than contracts. Ihave shown that the psychological arrow is essentially the sameas the thermodynamic arrow, so that the two would alwayspoint in the same direction. The no boundary proposal for theuniverse predicts the existence of a well-defined thermodynamicarrow of time because the universe must start off in a smoothand ordered state. And the reason we observe thisthermodynamic arrow to agree with the cosmological arrow isthat intelligent beings can exist only in the expanding phase.
The contracting phase will be unsuitable because it has nostrong thermodynamic arrow of time.
The progress of the human race in understanding theuniverse has established a small corner of order in anincreasingly disordered universe. If you remember every wordin this book, your memory will have recorded about two millionpieces of information: the order in your brain will haveincreased by about two million units. However, while you havebeen reading the book, you will have converted at least athousand calories of ordered energy, in the form of food, intodisordered energy, in the form of heat that you lose to the airaround you by convection and sweat. This will increase thedisorder of the universe by about twenty million million millionmillion units – or about ten million million million times theincrease in order in your brain – and that’s if you remembereverything in this book. In the next chapter but one I will tryto increase the order in our neck of the woods a little furtherby explaining how people are trying to fit together the partialtheories I have described to form a complete unified theorythat would cover everything in the universe.
CHAPTER 10 WORMHOLES AND TIME TRAVEL
The last chapter discussed why we see time go forward:
why disorder increases and why we remember the past butnot the future. Time was treated as if it were a straight railwayline on which one could only go one way or the other.
But what if the railway line had loops and branches so thata train could keep going forward but come back to a station ithad already passed? In other words, might it be possible forsomeone to travel into the future or the past?
H. G. Wells in The Time Machine explored these possibilitiesas have countless other writers of science fiction. Yet many ofthe ideas of science fiction, like submarines and travel to themoon, have become matters of science fact. So what are theprospects for time travel?
The first indication that the laws of physics might really allowpeople to travel in time came in 1949 when Kurt Godeldiscovered a new space-time allowed by general relativity. Godelwas a mathematician who was famous for proving that it isimpossible to prove all true statements, even if you limityourself to trying to prove all the true statements in a subjectas apparently cut and dried as arithmetic. Like the uncertaintyprinciple, Godel’s incompleteness theorem may be a fundamentallimitation on our ability to understand and predict the universe,but so far at least it hasn’t seemed to be an obstacle in oursearch for a complete unified theory.
Godel got to know about general relativity when he andEinstein spent their later years at the Institute for AdvancedStudy in Princeton. His space-time had the curious propertythat the whole universe was rotating. One might ask: “Rotatingwith respect to what?” The answer is that distant matter wouldbe rotating with respect to directions that little tops orgyroscopes point in.
This had the side effect that it would be possible forsomeone to go off in a rocket ship and return to earth beforehe set out. This property really upset Einstein, who hadthought that general relativity wouldn’t allow time travel.
However, given Einstein’s record of ill-founded opposition togravitational collapse and the uncertainty principle, maybe thiswas an encouraging sign. The solution Godel found doesn’tcorrespond to the universe we live in because we can showthat the universe is not rotating. It also had a non-zero valueof the cosmological constant that Einstein introduced when hethought the universe was unchanging. After Hubble discoveredthe expansion of the universe, there was no need for acosmological constant and it is now generally believed to bezero. However, other more reasonable space-times that areallowed by general relativity and which permit travel into thepast have since been found. One is in the interior of a rotatingblack hole. Another is a space-time that contains two cosmicstrings moving past each other at high speed. As their namesuggests, cosmic strings are objects that are like string in thatthey have length but a tiny cross section. Actually, they aremore like rubber bands because they are under enormoustension, something like a million million million million tons. Acosmic string attached to the earth could accelerate it from 0to 60 mph in 1/30th of a second. Cosmic strings may soundlike pure science fiction but there are reasons to believe theycould have formed in the early universe as a result ofsymmetry-breaking of the kind discussed in Chapter 5. Becausethey would be under enormous tension and could start in anyconfiguration, they might accelerate to very high speeds whenthey straighten out.
The Godel solution and the cosmic string space-time start outso distorted that travel into the past was always possible. Godmight have created such a warped universe but we have noreason to believe he did. Observations of the microwavebackground and of the abundances of the light elementsindicate that the early universe did not have the kind ofcurvature required to allow time travel. The same conclusionfollows on theoretical grounds if the no boundary proposal iscorrect. So the question is: if the universe starts out withoutthe kind of curvature required for time travel, can wesubsequently warp local regions of space-time sufficiently toallow it?
A closely related problem that is also of concern to writersof science fiction is rapid interstellar or intergalactic travel.
According to relativity, nothing can travel faster than light. If wetherefore sent a spaceship to our nearest neighboring star,Alpha Centauri, which is about four light-years away, it wouldtake 佛山夜生活论坛邀请码 at least eight years before we could expect the travelers toreturn and tell us what they had found. If the expedition wereto the center of our galaxy, it would be at least a hundredthousand years before it came back. The theory of relativitydoes allow one consolation. This is the so-called twins paradoxmentioned in Chapter 2.
Because there is no unique standard of time, but ratherobservers each have their own time as measured by clocksthat they carry with them, it is possible for the journey toseem to be much shorter for the space travelers than forthose who remain on earth. But there would not be much joyin returning from a space voyage a few years older to findthat everyone you had left behind was dead and gonethousands of years ago. So in order to have any humaninterest in their stories, science fiction writers had to 佛山夜生活网 supposethat we would one day discover how to travel faster than light.
What most of thee authors don’t seem to have realized is thatif you can travel faster than light, the theory of relativity impliesyou can also travel back in the, as the following limerick says:
There was a young lady of WightWho traveled much faster than light.
She departed one day,In a relative way,And arrived on the previous nightThe point is that the theory of relativity says hat there is nounique measure of time that all observers will agree on Rather,each observer has his or her own measure of time. If it ispossible for a rocket traveling below the speed of light to getfrom event A (say, the final of the 100-meter race of theOlympic Games in 202) to event B (say, the opening of the100,004th meeting of the Congress of Alpha Centauri), then allobservers 佛山桑拿飞机论坛网 will agree that event A happened before event Baccording to their times. Suppose, however, that the spaceshipwould have to travel faster than light to carry the news of therace to the Congress. Then observers moving at differentspeeds can disagree about whether event A occurred before Bor vice versa. According to the time of an observer who is atrest with respect to the earth, it may be that the Congressopened after the race. Thus this observer would think that aspaceship could get from A to B in time if only it could ignorethe speed-of-light speed limit. However, to an observer at AlphaCentauri moving away from the earth at nearly the speed oflight, it would appear that event B, the opening of theCongress, would occur before event A, the 100-meter race. Thetheory of relativity says that the laws of physics appear thesame to 佛山桑拿会所 observers moving at different speeds.
This has been well tested by experiment and is likely toremain a feature even if we find a more advanced theory toreplace relativity Thus the moving observer would say that iffaster-than-light travel is possible, it should be possible to getfrom event B, the opening of the Congress, to event A, the100-meter race. If one went
slightly faster, one could even getback before the race and place a bet on it in the sureknowledge that one would win.
There is a problem with breaking the speed-of-light barrier.
The theory of relativity says that the rocket power needed toaccelerate a spaceship gets greater and greater the nearer itgets to the speed of light. We have experimental evidence forthis, not with spaceships but with elementary particles in particleaccelerators like those at Fermilab or CERN (European Centrefor Nuclear Research). We can 佛山桑拿去哪里好 accelerate particles to 99.99percent of the speed of light, but however much power wefeed in, we can’t get them beyond the speed-of-light barrier.
Similarly with spaceships: no matter how much rocket powerthey have, they can’t accelerate beyond the speed of light.
That might seem to rule out both rapid space travel andtravel back in time. However, there is a
possible way out. Itmight be that one could warp space-time so that there was ashortcut between A and B One way of doing this would be tocreate a wormhole between A and B. As its name suggests, awormhole is a thin tube of space-time which can connect twonearly flat regions far apart.
There need be no relation between the distance through thewormhole and the separation of its ends in the nearly Hatbackground. Thus one could imagine that one could create orfind a wormhole that world lead from the vicinity of the SolarSystem to 佛山桑拿按摩上门 Alpha Centauri. The distance through the wormholemight be only a few million miles even though earth and AlphaCentauri are twenty million million miles apart in ordinary space.
This would allow news of the 100-meter race to reach theopening of the Congress. But then an observer moving toward6e earth should also be able to find another wormhole thatwould enable him to get from the opening of the Congress onAlpha Centauri back to earth before the start of the race. Sowormholes, like any other possible form of travel faster thanlight, would allow one to travel into the past.
The idea of wormholes between different regions ofspace-time was not an invention of science fiction writers butcame from a very respectable source.
In 1935, Einstein and Nathan Rosen wrote a paper in whichthey showed that general relativity allowed what they called“bridges,” but which are now known as wormholes. TheEinstein-Rosen bridges didn’t last long enough for a spaceshipto get through: the ship would run into a singularity as thewormhole pinched off. However, it has been suggested that itmight be possible for an advanced civilization to keep awormhole open. To do this, or to warp space-time in anyother way so as to permit time travel, one can show that oneneeds a region of space-time with negative curvature, like thesurface of a saddle. Ordi-nary matter, which has a positiveenergy density, gives space-time a positive curvature, like thesurface of a sphere. So what one needs, in order to warpspace-time in a way that will allow travel into the past, ismatter with negative energy density.
Energy is a bit like money: if you have a positive balance,you can distribute it in various ways, but according to theclassical laws that were believed at the beginning of the century,you weren’t allowed to be overdrawn. So these classical lawswould have ruled out any possibility of time travel. However, ashas been described in earlier chapters, the classical laws weresuperseded by quantum laws based on the uncertaintyprinciple. The quantum laws are more liberal and allow you tobe overdrawn on one or two accounts provided the totalbalance is positive. In other words, quantum theory allows theenergy density to be negative in some places, provided that thisis made up for by positive energy densities in other places, sothat the total energy re-mains positive. An example of howquantum theory can allow negative energy densities is providedby what is called the Casimir effect. As we saw in Chapter 7,even what we think of as “empty” space is filled with pairs ofvirtual particles and antiparticles that appear together, moveapart, and come back together and annihilate each other. Now,suppose one has two parallel metal plates a short distanceapart. The plates will act like mirrors for the virtual photons orparticles of light. In fact they will form a cavity between them,a bit like an organ pipe that will resonate only at certain notes.
This means that virtual photons can occur in the spacebetween the plates only if their wavelengths (the distancebetween the crest of one wave and the next) fit a wholenumber of times into the gap between the plates. If the widthof a cavity is a whole number of wavelengths plus a fraction ofa wave-length, then after some reflections backward andforward between the plates, the crests of one wave will coincidewith the troughs of another and the waves will cancel out.
Because the virtual photons between the plates can haveonly the resonant wavelengths, there will be slightly fewer ofthem than in the region outside the plates where virtualphotons can have any wavelength. Thus there will be slightlyfewer virtual photons hitting the inside surfaces of the platesthan the outside surfaces. One would therefore expect a forceon the plates, pushing them toward each other. This force hasactually been detected and has the predicted value. Thus wehave experimental evidence that virtual particles exist and havereal effects.
The fact that there are fewer virtual photons between theplates means that their energy density will be less thanelsewhere. But the total energy density in “empty” space faraway from the plates must be zero, because otherwise theenergy density would warp the space and it would not bealmost flat. So, if the energy density between the plates is lessthan the energy density far away, it must be negative.
We thus have experimental evidence both that space-timecan be warped (from the bending of light during eclipses) andthat it can be curved in the way necessary to allow time travel(from the Casimir effect). One might hope therefore that as weadvance in science and technology, we would eventually manageto build a time machine. But if so, why hasn’t anyone comeback from the future and told us how to do it? There mightbe good reasons why it would be unwise to give us the secretof time travel at our present primitive state of development, butunless human nature changes radically, it is difficult to believethat some visitor from the future wouldn’t spill the beans. Ofcourse, some people would claim that sightings of UFOs areevidence that we are being visited either by aliens or by peoplefrom the future. (If the aliens were to get here in reasonabletime, they would need faster-than-light travel, so the twopossibilities may be equivalent.)However, I think that any visit by aliens or people from thefuture would be much more obvious and, probably, much moreunpleasant. If they are going to reveal themselves at all, whydo so only to those who are not regarded as reliablewitnesses? If they are trying to warn us of some great danger,they are not being very effective.
A possible way to explain the absence of visitors from thefuture would be to say that the past is fixed because we haveobserved it and seen that it does not have the kind of warpingneeded to allow travel back from the future. On the otherhand, the future is unknown and open, so it might well havethe curvature required. This would mean that any time travelwould be confined to the future. There would be no chance ofCaptain Kirk and the Starship Enterprise turning up at thepresent time.
This might explain why we have not yet been overrun bytourists from the future, but it would not avoid the problemsthat would arise if one were able to go back and changehistory. Suppose, for example, you went back and killed yourgreat-great-grandfather while he was still a child. There aremany versions of this paradox but they are essentiallyequivalent: one would get contradictions if one were free tochange the past.
There seem to be two possible resolutions to the paradoxesposed by time travel. One I shall call the consistent historiesapproach. It says that even if space-time is warped so that itwould be possible to travel into the past, what happens inspace-time must be a consistent solution of the laws of physics.
According to this viewpoint, you could not go back in timeunless history showed that you had already arrived in the pastand, while there, had not killed your great-great-grandfather orcommitted any other acts that would conflict with your currentsituation in the present. Moreover, when you did go back, youwouldn’t be able to change recorded history. That means youwouldn’t have free will to do what you wanted. Of course, onecould say that free will is an illusion anyway. If there really is acomplete unified theory that governs everything, it presumablyalso determines your actions. But it does so in a way that isimpossible to calculate for an organism that is as complicatedas a human being. The reason we say that humans have freewill is because we can’t predict what they will do. However, ifthe human then goes off in a rocket ship and comes backbefore he or she set off, we will be able to predict what he orshe will do because it will be part of recorded history. Thus, inthat situation, the time traveler would have no free will.
The other possible way to resolve the paradoxes of timetravel might be called the alternative histories hypothesis. Theidea here is that when time travelers go back to the past, theyenter alternative histories which differ from recorded history.
Thus they can act freely, without the constraint of consistencywith their previous history. Steven Spiel-berg had fun with thisnotion in the Back to the Future films: Marty McFly was ableto go back and change his parents’ courtship to a moresatisfactory history.
The alternative histories hypothesis sounds rather like RichardFeynman’s way of expressing quantum theory as a sum overhistories, which was described in Chapters 4 and 8. This saidthat the universe didn’t just have a single history: rather it hadevery possible history, each with its own probability. However,there seems to be an important difference between Feynman’sproposal and alternative histories. In Feynman’s sum, eachhistory comprises a complete space-time and everything in it.
The space-time may be so warped that it is possible to travelin a rocket into the past. But the rocket would remain in thesame space-time and therefore the same history, which wouldhave to be consistent. Thus Feynman’s sum over historiesproposal seems to support the consistent histories hypothesisrather than the alternative histories.
The Feynman sum over histories does allow travel into thepast on a microscopic scale. In Chapter 9 we saw that thelaws of science are unchanged by combinations of theoperations C, P, and T. This means that an antiparticle spinningin the anticlockwise direction and moving from A to B can alsobe viewed as an ordinary particle spinning clockwise andmoving backward in time from B to A. Similarly, an ordinaryparticle moving forward in time is equivalent to an antiparticlemoving backward in time. As has been discussed in thischapter and Chapter 7, “empty” space is filled with pairs ofvirtual particles and antiparticles that appear together, moveapart, and then come back together and annihilate each other.
So, one can regard the pair of particles as a single particlemoving on a closed loop in space-time. When the pair ismoving forward in time (from the event at which it appears tothat at which it annihilates), it is called a particle. But when theparticle is traveling back in time (from the event at which thepair annihilates to that at which it appears), it is said to be anantiparticle traveling forward in time.
The explanation of how black holes can emit particles andradiation (given in Chapter 7) was that one member of avirtual particle/ antiparticle pair (say, the antiparticle) might fallinto the black hole, leaving the other member without a partnerwith which to annihilate. The forsaken particle might fall intothe hole as well, but it might also escape from the vicinity ofthe black hole. If so, to an observer at a distance it wouldappear to be a particle emitted by the black hole.
One can, however, have a different but equivalent intuitivepicture of the mechanism for emission from black holes. Onecan regard the member of the virtual pair that fell into theblack hole (say, the antiparticle) as a particle traveling backwardin time out of the hole. When it gets to the point at which thevirtual particle/antiparticle pair appeared together, it is scatteredby the gravitational field into a particle traveling forward in timeand escaping from the black hole. If, instead, it were theparticle member of the virtual pair that fell into the hole, onecould regard it as an antiparticle traveling back in time andcoming out of the black hole. Thus the radiation by black holesshows that quantum theory allows travel back in time on amicroscopic scale and that such time travel can produceobservable effects.
One can therefore ask: does quantum theory allow timetravel on a macroscopic scale, which people could use? At firstsight, it seems it should. The Feynman sum over historiesproposal is supposed to be over all histories. Thus it shouldinclude histories in which space-time is so warped that it ispossible to travel into the past. Why then aren’t we in troublewith history? Suppose, for example, someone had gone backand given the Nazis the secret of the atom bomb?
One would avoid these problems if what I call thechronology protection conjecture holds. This says that the lawsof physics conspire to prevent macroscopic bodies from carryinginformation into the past. Like the cosmic censorship conjecture,it has not been proved but there are reasons to believe it istrue.
The reason to believe that chronology protection operates isthat when space-time is warped enough to make travel into thepast possible, virtual particles moving on closed loops inspace-time can become real particles traveling forward in timeat or below the speed of light. As these particles can go roundthe loop any number of times, they pass each point on theirroute many times. Thus their energy is counted over and overagain and the energy density will become very large. This couldgive space-time a positive curvature that would not allow travelinto the past. It is not yet clear whether these particles wouldcause positive or negative curvature or whether the curvatureproduced by some kinds of virtual particles might cancel thatproduced by other kinds. Thus the possibility of time travelremains open. But I’m not going to bet on it. My opponentmight have the unfair advantage of knowing the future.
CHAPTER 11 THE UNIFICATION OF PHYSICS
As was explained in the first chapter, it would be verydifficult to construct a complete unified theory of everything inthe universe all at one go. So instead we have made progressby finding partial theories that describe a limited range ofhappenings and by neglecting other effects or approximatingthem by certain numbers. (Chemistry, for example, allows us tocalculate the interactions of atoms, without knowing the internalstructure of an atom’s nucleus.) Ultimately, however, one wouldhope to find a complete, consistent, unified theory that wouldinclude all these partial theories as approximations, and that didnot need to be adjusted to fit the facts by picking the valuesof certain arbitrary numbers in the theory. The quest for sucha theory is known as “the unification of physics.” Einstein spentmost of his later years unsuccessfully searching for a unifiedtheory, but the time was not ripe: there were partial theoriesfor gravity and the electromagnetic force, but very little wasknown about the nuclear forces. Moreover, Einstein refused tobelieve in the reality of quantum mechanics, despite theimportant role he had played in its development. Yet it seemsthat the uncertainty principle is a fundamental feature of theuniverse we live in. A successful unified theory must, therefore,necessarily incorporate this principle.
As I shall describe, the prospects for finding such a theoryseem to be much better now because we know so much moreabout the universe. But we must beware of overconfidence -we have had false dawns before! At the beginning of thiscentury, for example, it was thought that everything could beexplained in terms of the properties of continuous matter, suchas elasticity and heat conduction. The discovery of atomicstructure and the uncertainty principle put an emphatic end tothat. Then again, in 1928, physicist and Nobel Prize winnerMax Born told a group of visitors to Gottingen University,“Physics, as we know it, will be over in six months.” Hisconfidence was based on the recent discovery by Dirac of theequation that governed the electron. It was thought that asimilar equation would govern the proton, which was the onlyother particle known at the time, and that would be the end oftheoretical physics. However, the discovery of the neutron andof nuclear forces knocked that one on the head too. Havingsaid this, I still believe there are grounds for cautious optimismthat we may now be near the end of the search for theultimate laws of nature.
In previous chapters I have described general relativity, thepartial theory of gravity, and the partial theories that governthe weak, the strong, and the electromagnetic forces. The lastthree may be combined in so-called grand unified theories, orGUTs, which are not very satisfactory because they do notinclude gravity and because they contain a number ofquantities, like the relative masses of different particles, thatcannot be predicted from the theory but have to be chosen tofit observations. The main difficulty in finding a theory thatunifies gravity with the other forces is that general relativity is a“classical” theory; that is, it does not incorporate the uncertaintyprinciple of quantum mechanics. On the other hand, the otherpartial theories depend on quantum mechanics in an essentialway. A necessary first step, therefore, is to combine generalrelativity with the uncertainty principle. As we have seen, thiscan produce some remark-able consequences, such as blackholes not being black, and the universe not having anysingularities but being completely self-contained and without aboundary. The trouble is, as explained in Chapter 7, that theuncertainty principle means that even “empty” space is filledwith pairs of virtual particles and antiparticles. These pairswould have an infinite amount of energy and, therefore, byEinstein’s famous equation E = mc2, they would have aninfinite amount of mass. Their gravitational attraction would thuscurve up the universe to infinitely small size.
Rather similar, seemingly absurd infinities occur in the otherpartial theories, but in all these cases the infinities can becanceled out by a process called renormalization. This involvescanceling the infinities by introducing other infinities. Althoughthis technique is rather dubious mathematically, it does seem towork in practice, and has been used with these theories tomake predictions that agree with observations to anextraordinary degree of accuracy. Renormalization, however,does have a serious drawback from the point of view of tryingto find a complete theory, because it means that the actualvalues of the masses and the strengths of the forces cannot bepredicted from the theory, but have to be chosen to fit theobservations.
In attempting to incorporate the uncertainty principle intogeneral relativity, one has only two quantities that can beadjusted: the strength of gravity and the value of thecosmological constant. But adjusting these is not sufficient toremove all the infinities. One therefore has a theory that seemsto predict that certain quantities, such as the curvature ofspace-time, are really infinite, yet these quantities can beobserved and measured to be perfectly finite! This problem incombining general relativity and the uncertainty principle hadbeen suspected for some time, but was finally confirmed bydetailed calculations in 1972. Four years later, a possiblesolution, called “supergravity,” was suggested. The idea was tocombine the spin-2 particle called the graviton, which carriesthe gravitational force, with certain other particles of spin 3/2,1, ?, and 0. In a sense, all these particles could then beregarded as different aspects of the same “superparticle,” thusunifying the matter particles with spin ? and 3/2 with theforce-carrying particles of spin 0, 1, and 2. The virtualparticle/antiparticle pairs of spin ? and 3/2 would havenegative energy, and so would tend to cancel out the positiveenergy of the spin 2, 1, and 0 virtual pairs. This would causemany of the possible infinities to cancel out, but it wassuspected that some infinities might still remain. However, thecalculations required to find out whether or not there were anyinfinities left uncanceled were so long and difficult that no onewas prepared to undertake them. Even with a computer it wasreckoned it would take at least four years, and the chanceswere very high that one would make at least one mistake,probably more. So one would know one had the right answeronly if someone else repeated the calculation and got the sameanswer, and that did not seem very likely!
Despite these problems, and the fact that the particles in thesuper-gravity theories did not seem to match the observedparticles, most scientists believed that supergravity was probablythe right answer to the problem of the unification of physics. Itseemed the best way of unifying gravity with the other forces.
However, in 1984 there was a remarkable change of opinion infavor of what are called string theories. In these theories thebasic objects are not particles, which occupy a single point ofspace, but things that have a length but no other dimension,like an infinitely thin piece of string. These strings may haveends (the so-called open strings) or they may be joined upwith themselves in closed loops (closed strings) (Fig. 11.1 andFig. 11.2). A particle occupies one point of space at each instantof time. Thus its history can be represented by a line inspace-time (the “world-line”). A string, on the other hand,occupies a line in space at each moment of time. So its historyin space-time is a two-dimensional surface called theworld-sheet. (Any point on such a world-sheet can bedescribed by two numbers, one specifying the time and theother the position of the point on the string.) The world-sheetof an open string is a strip: its edges represent the pathsthrough space-time of the ends of the string (Fig. 11.1). Theworld-sheet of a closed string is a cylinder or tube (Fig. 11.2):
a slice through the tube is a circle, which represents theposition of the string at one particular time.
Two pieces of string can join together to form a singlestring; in the case of open strings they simply join at the ends(Fig. 11.3), while in the case of closed strings it is like the twolegs joining on a pair of trousers (Fig. 11.4). Similarly, a singlepiece of string can divide into two strings. In string theories,what were previously thought of as particles are now picturedas waves traveling down the string, like waves on a vibratingkite string. The emission or absorption of one particle byanother corresponds to the dividing or joining together ofstrings. For example, the gravitational force of the sun on theearth was pictured in particle theories as being caused by theemission of a graviton by a particle in the sun and itsabsorption by a particle in the earth (Fig. 11.5). In stringtheory, this process corresponds to an H-shaped tube or pipe(Fig. 11.6) (string theory is rather like plumbing, in a way). Thetwo vertical sides of the H correspond to the particles in thesun and the earth, and the horizontal crossbar corresponds tothe graviton that travels between them.
String theory has a curious history. It was originally inventedin the late 1960s in an attempt to find a theory to describe thestrong force. The idea was that particles like the proton andthe neutron could be regarded as waves on a string. Thestrong forces between the particles would correspond to piecesof string that went between other bits of string, as in a spider’sweb. For this theory to give the observed value of the strongforce between particles, the strings had to be like rubber bandswith a pull of about ten tons.
In 1974 Joel Scherk from Paris and John Schwarz from theCalifornia Institute of Technology published a paper in whichthey showed that string theory could describe the gravitationalforce, but only if the tension in the string were very muchhigher, about a thousand million million million million millionmillion tons (1 with thirty-nine zeros after it). The predictions ofthe string theory would be just the same as those of generalrelativity on normal length scales, but they would differ at verysmall distances, less than a thousand million million millionmillion millionth of a centimeter (a centimeter divided by 1 withthirty-three zeros after it). Their work did not receive muchattention, however, because at just about that time most peopleabandoned the original string theory of the strong force infavor of the theory based on quarks and gluons, which seemedto fit much better with observations. Scherk died in tragiccircumstances (he suffered from diabetes and went into a comawhen no one was around to give him an injection of insulin).
So Schwarz was left alone as almost the only supporter ofstring theory, but now with the much higher pro-posed valueof the string tension.
In 1984 interest in strings suddenly revived, apparently fortwo reasons. One was that people were not really makingmuch progress toward showing that supergravity was finite orthat it could explain the kinds of particles that we observe. Theother was the publication of a paper by John Schwarz andMike Green of Queen Mary College, London, that showed thatstring theory might be able to explain the existence of particlesthat have a built-in left-handedness, like some of the particlesthat we observe. Whatever the reasons, a large number ofpeople soon began to work on string theory and a newversion was developed, the so-called heterotic string, whichseemed as if it might be able to explain the types of particlesthat we observe.
String theories also lead to infinities, but it is thought theywill all cancel out in versions like the heterotic string (thoughthis is not yet known for certain). String theories, however,have a bigger problem: they seem to be consistent only ifspace-time has either ten or twenty-six dimensions, instead ofthe usual four! Of course, extra space-time dimensions are acommonplace of science fiction indeed, they provide an idealway of overcoming the normal restriction of general relativitythat one cannot travel faster than light or back in time (seeChapter 10). The idea is to take a shortcut through the extradimensions. One can picture this in the following way. Imaginethat the space we live in has only two dimensions and iscurved like the surface of an anchor ring or torus (Fig. 11.7). Ifyou were on one side of the inside edge of the ring and youwanted to get to a point on the other side, you would have togo round the inner edge of the ring. However, if you wereable to travel in the third dimension, you could cut straightacross.
Why don’t we notice all these extra dimensions, if they arereally there? Why do we see only three space dimensions andone time dimension? The suggestion is that the otherdimensions are curved up into a space of very small size,something like a million million million million millionth of aninch. This is so small that we just don’t notice it: we see onlyone time dimension and three space dimensions, in whichspace-time is fairly flat. It is like the surface of a straw. If youlook at it closely, you see it is two-dimensional (the position ofa point on the straw is described by two numbers, the lengthalong the straw and the distance round the circular direction).
But if you look at it from a distance, you don’t see thethickness of the straw and it looks one-dimensional (theposition of a point is specified only by the length along thestraw). So it is with space-time: on a very small scale it isten-dimensional and highly curved, but on bigger scales youdon’t see the curvature or the extra dimensions. If this pictureis correct, it spells bad news for would-be space travelers: theextra dimensions would be far too small to allow a spaceshipthrough. However, it raises another major problem. Why shouldsome, but not all, of the dimensions be curled up into a smallball? Presumably, in the very early universe all the dimensionswould have been very curved. Why did one time dimensionand three space dimensions flatten out, while the otherdimensions remain tightly curled up?
One possible answer is the anthropic principle. Two spacedimensions do not seem to be enough to allow for thedevelopment of complicated beings like us. For example,two-dimensional animals living on a one-dimensional earth wouldhave to climb over each other in order to get past each other.
If a two-dimensional creature ate something it could not digestcompletely, it would have to bring up the remains the sameway it swallowed them, because if there were a passage rightthrough its body, it would divide the creature into two separatehalves: our two-dimensional being would fall apart (Fig. 11.8).
Similarly, it is difficult to see how there could be any circulationof the blood in a two-dimensional creature.
There would also be problems with more than three spacedimensions. The gravitational force between two bodies woulddecrease more rapidly with distance than it does in threedimensions. (In three dimensions, the gravitational force dropsto 1/4 if one doubles the distance. In four dimensions it woulddrop to 1/5, in five dimensions to 1/6, and so on.) Thesignificance of this is that the orbits of planets, like the earth,around the sun would be unstable: the least disturbance froma circular orbit (such as would be caused by the gravitationalattraction of other planets) would result in the earth spiralingaway from or into the sun. We would either freeze or beburned up. In fact, the same behavior of gravity with distancein more than three space dimensions means that the sunwould not be able to exist in a stable state with pressurebalancing gravity. It would either fall apart or it would collapseto form a black hole. In either case, it would not be of muchuse as a source of heat and light for life on earth. On asmaller scale, the electrical forces that cause the electrons toorbit round the nucleus in an atom would behave in the sameway as gravitational forces. Thus the electrons would eitherescape from the atom altogether or would spiral into thenucleus. In either case, one could not have atoms as we knowthem.
It seems clear then that life, at least as we know it, canexist only in regions of space-time in which one time dimensionand three space dimensions are not curled up small. Thiswould mean that one could appeal to the weak anthropicprinciple, provided one could show that string theory does atleast allow there to be such regions of the universe – and itseems that indeed string theory does. There may well be otherregions of the universe, or other universes (whatever that maymean), in which all the dimensions are curled up small or inwhich more than four dimensions are nearly flat, but therewould be no intelligent beings in such regions to observe thedifferent number of effective dimensions.
Another problem is that there are at least four differentstring theories (open strings and three different closed stringtheories) and millions of ways in which the extra dimensionspredicted by string theory could be curled up. Why should justone string theory and one kind of curling up be picked out?
For a time there seemed no answer, and progress got boggeddown. Then, from about 1994, people started discovering whatare called dualities: different string theories and different waysof curling up the extra dimensions could lead to the sameresults in four dimensions. Moreover, as well as particles, whichoccupy a single point of space, and strings, which are lines,there were found to be other objects called p-branes, whichoccupied two-dimensional or higher-dimensional volumes inspace. (A particle can be regarded as a 0-brane and a stringas a 1-brane but there were also p-branes for p=2 to p=9.)What this seems to indicate is that there is a sort ofdemocracy among supergravity, string, and p-brane theories:
they seem to fit together but none can be said to be morefundamental than the others. They appear to be differentapproximations to some fundamental theory that are valid indifferent situations.
People have searched for this underlying theory, but withoutany success so far. However, I believe there may not be anysingle formulation of the fundamental theory any more than, asGodel showed, one could formulate arithmetic in terms of asingle set of axioms. Instead it may be like maps – you can’tuse a single map to describe the surface of the earth or ananchor ring: you need at least two maps in the case of theearth and four for the anchor ring to cover every point. Eachmap is valid only in a limited region, but different maps willhave a region of overlap. The collection of maps provides acomplete description of the surface. Similarly, in physics it maybe necessary to use different formulations in different situations,but two different formulations would agree in situations wherethey can both be applied. The whole collection of differentformulations could be regarded as a complete unified theory,though one that could not be expressed in terms of a singleset of postulates.
But can there really be such a unified theory? Or are weperhaps just chasing a mirage? There seem to be threepossibilities:
1. There really is a complete unified theory (or a collection ofoverlapping formulations), which we will someday discover if weare smart enough.