For SOLUTION PROVIDERS ONLY

Chapter 1 - Our Picture of the Universe Chapter 2 - Space and Time Chapter 3 - The Expanding Universe Chapter 4 - The Uncertainty Principle Chapter 5 - Elementary Particles and the Forces of Nature Chapter 6 - Black Holes Chapter 7 - Black Holes Ain't So Black Chapter 8 - The Origin and Fate of the Universe Chapter 9 - The Arrow of Time Chapter 10 - Wormholes and Time Travel Chapter 11 - The Unification of Physics Chapter 12 - Conclusion Glossary Acknowledgments & About The Author     FOREWARD I didn’t write a foreword to the original edition of A Brief History of Time. That was done by Carl Sagan. Instead, I wrote a short piece titled “Acknowledgments” in which I was advi\ sed to thank everyone. Some of the foundations that had given me support weren’t too pleased to have bee\ n mentioned, however, because it led to a great increase in applications.

I don’t think anyone, my publishers, my agent, or myself, expected th\ e book to do anything like as well as it did.

It was in the London Sunday Times best-seller list for 237 weeks, longer than any other book (apparently,\ the Bible and Shakespeare aren’t counted). It has been translated into s\ omething like forty languages and has sold about one copy for every 750 men, women, and children in the world. As N\ athan Myhrvold of Microsoft (a former post-doc of mine) remarked: I have sold more books on physics th\ an Madonna has on sex.

The success of A Brief History indicates that there is widespread interest in the big questions like: W\ here did we come from? And why is the universe the way it is?

I have taken the opportunity to update the book and include new theoreti\ cal and observational results obtained since the book was first published (on April Fools’ Day, 1988). I h\ ave included a new chapter on wormholes and time travel. Einstein’s General Theory of Relativity seems to off\ er the possibility that we could create and maintain wormholes, little tubes that connect different regions of space\ -time. If so, we might be able to use them for rapid travel around the galaxy or travel back in time. Of cours\ e, we have not seen anyone from the A Brief History of Time - Stephen Hawking file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/A Brief History in Time.html (1 of 2) [2/20/2001 3:13:58 AM] future (or have we?) but I discuss a possible explanation for this.

I also describe the progress that has been made recently in finding “\ dualities” or correspondences between apparently different theories of physics. These correspondences are a st\ rong indication that there is a complete unified theory of physics, but they also suggest that it may not be poss\ ible to express this theory in a single fundamental formulation. Instead, we may have to use different reflectio\ ns of the underlying theory in different situations. It might be like our being unable to represent the surface o\ f the earth on a single map and having to use different maps in different regions. This would be a revolution in o\ ur view of the unification of the laws of science but it would not change the most important point: that the unive\ rse is governed by a set of rational laws that we can discover and understand.

On the observational side, by far the most important development has bee\ n the measurement of fluctuations in the cosmic microwave background radiation by COBE (the Cosmic Backgroun\ d Explorer satellite) and other collaborations. These fluctuations are the finger-prints of creation, ti\ ny initial irregularities in the otherwise smooth and uniform early universe that later grew into galaxies, stars, \ and all the structures we see around us.

Their form agrees with the predictions of the proposal that the universe\ has no boundaries or edges in the imaginary time direction; but further observations will be necessary to \ distinguish this proposal from other possible explanations for the fluctuations in the background. However, w\ ithin a few years we should know whether we can believe that we live in a universe that is completely sel\ f-contained and without beginning or end.

Stephen Hawking   A Brief History of Time - Stephen Hawking file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/A Brief History in Time.html (2 of 2) [2/20/2001 3:13:58 AM] CHAPTER 1 OUR PICTURE OF THE UNIVERSE   A well-known scientist (some say it was Bertrand Russell) once gave a \ public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, \ orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a litt\ le old lady at the back of the room got up and said: “What you have told us is rubbish. The world is really a flat p\ late supported on the back of a giant tortoise.” The scientist gave a superior smile before replying, “W\ hat is the tortoise standing on.” “You’re very clever, young man, very clever,” said the old lady. “But it’s t\ urtles all the way down!” Most people would find the picture of our universe as an infinite tower \ of tortoises rather ridiculous, but why do we think we know better? What do we know about the universe, and how do \ we know it? Where did the universe come from, and where is it going? Did the universe have a begin\ ning, and if so, what happened before then? What is the nature of time? Will it ever come to an end? Can we go\ back in time? Recent breakthroughs in physics, made possible in part by fantastic new technologies, suggest\ answers to some of these longstanding questions. Someday these answers may seem as obvious to us \ as the earth orbiting the sun – or perhaps as ridiculous as a tower of tortoises. Only time (whatever that\ may be) will tell.

As long ago as 340 BC the Greek philosopher Aristotle, in his book On the Heavens, was able to put forward two good arguments for believing that the earth was a round sphere rathe\ r than a Hat plate. First, he realized that eclipses of the moon were caused by the earth coming between the su\ n and the moon. The earth’s shadow on the moon was always round, which would be true only if the ear\ th was spherical. If the earth had been a flat disk, the shadow would have been elongated and elliptical, u\ nless the eclipse always occurred at a time when the sun was directly under the center of the disk. Second, the\ Greeks knew from their travels that the North Star appeared lower in the sky when viewed in the south than i\ t did in more northerly regions. (Since the North Star lies over the North Pole, it appears to be directly above\ an observer at the North Pole, but to someone looking from the equator, it appears to lie just at the horizon.\ From the difference in the apparent position of the North Star in Egypt and Greece, Aristotle even quoted an\ estimate that the distance around the earth was 400,000 stadia. It is not known exactly what length a stadium \ was, but it may have been about 200 yards, which would make Aristotle’s estimate about twice the currentl\ y accepted figure. The Greeks even had a third argument that the earth must be round, for why else does one first\ see the sails of a ship coming over the horizon, and only later see the hull?

Aristotle thought the earth was stationary and that the sun, the moon, t\ he planets, and the stars moved in circular orbits about the earth. He believed this because he felt, for m\ ystical reasons, that the earth was the center of the universe, and that circular motion was the most perfect. T\ his idea was elaborated by Ptolemy in the second century AD into a complete cosmological model. The earth stoo\ d at the center, surrounded by eight spheres that carried the moon, the sun, the stars, and the five planets \ known at the time, Mercury, Venus, Mars, Jupiter, and Saturn.

A Brief History of Time - Stephen Hawking... Chapter 1 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/n.html (1 of 7) [2/20/2001 3:14:06 AM] Figure 1:1 The planets themselves moved on smaller circles attached to their respec\ tive spheres in order to account for their rather complicated observed paths in the sky. The outermost sphere\ carried the so-called fixed stars, which always stay in the same positions relative to each other but which\ rotate together across the sky. What lay beyond the last sphere was never made very clear, but it certainly w\ as not part of mankind’s observable universe.

Ptolemy’s model provided a reasonably accurate system for predicting \ the positions of heavenly bodies in the sky. But in order to predict these positions correctly, Ptolemy had to m\ ake an assumption that the moon followed a path that sometimes brought it twice as close to the earth as\ at other times. And that meant that the moon ought sometimes to appear twice as big as at other times! Ptolemy r\ ecognized this flaw, but nevertheless his model was generally, although not universally, accepted. It was adop\ ted by the Christian church as the picture of the universe that was in accordance with Scripture, for it ha\ d the great advantage that it left lots of room outside the sphere of fixed stars for heaven and hell.

A simpler model, however, was proposed in 1514 by a Polish priest, Nicho\ las Copernicus. (At first, perhaps for fear of being branded a heretic by his church, Copernicus circulated his\ model anonymously.) His idea was that the sun was stationary at the center and that the earth and the planets \ moved in circular orbits around the sun.

Nearly a century passed before this idea was taken seriously. Then two a\ stronomers – the German, Johannes A Brief History of Time - Stephen Hawking... Chapter 1 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/n.html (2 of 7) [2/20/2001 3:14:06 AM] Kepler, and the Italian, Galileo Galilei – started publicly to suppor\ t the Copernican theory, despite the fact that the orbits it predicted did not quite match the ones observed. The death\ blow to the Aristotelian/Ptolemaic theory came in 1609. In that year, Galileo started observing the night s\ ky with a telescope, which had just been invented. When he looked at the planet Jupiter, Galileo found that it wa\ s accompanied by several small satellites or moons that orbited around it. This implied that everything\ did not have to orbit directly around the earth, as Aristotle and Ptolemy had thought. (It was, of course, still \ possible to believe that the earth was stationary at the center of the universe and that the moons of Jupiter m\ oved on extremely complicated paths around the earth, giving the appearance that they orbited Jupiter. Howev\ er, Copernicus’s theory was much simpler.) At the same time, Johannes Kepler had modified Copernicus’\ s theory, suggesting that the planets moved not in circles but in ellipses (an ellipse is an elongated circle\ ). The predictions now finally matched the observations.

As far as Kepler was concerned, elliptical orbits were merely an ad hoc \ hypothesis, and a rather repugnant one at that, because ellipses were clearly less perfect than circles. Having\ discovered almost by accident that elliptical orbits fit the observations well, he could not reconcile them\ with his idea that the planets were made to orbit the sun by magnetic forces. An explanation was provided only much \ later, in 1687, when Sir Isaac Newton published his Philosophiae Naturalis Principia Mathematica, probably the most important single work ever published in the physical sciences. In it Newton not only put forward a \ theory of how bodies move in space and time, but he also developed the complicated mathematics needed to analyz\ e those motions. In addition, Newton postulated a law of universal gravitation according to which each\ body in the universe was attracted toward every other body by a force that was stronger the more massive th\ e bodies and the closer they were to each other. It was this same force that caused objects to fall to the gr\ ound. (The story that Newton was inspired by an apple hitting his head is almost certainly apocryphal. All Newton \ himself ever said was that the idea of gravity came to him as he sat “in a contemplative mood” and “wa\ s occasioned by the fall of an apple.”) Newton went on to show that, according to his law, gravity causes the moon to m\ ove in an elliptical orbit around the earth and causes the earth and the planets to follow elliptical paths ar\ ound the sun.

The Copernican model got rid of Ptolemy’s celestial spheres, and with\ them, the idea that the universe had a natural boundary. Since “fixed stars” did not appear to change the\ ir positions apart from a rotation across the sky caused by the earth spinning on its axis, it became natural to suppo\ se that the fixed stars were objects like our sun but very much farther away.

Newton realized that, according to his theory of gravity, the stars shou\ ld attract each other, so it seemed they could not remain essentially motionless. Would they not all fall togethe\ r at some point? In a letter in 1691 to Richard Bentley, another leading thinker of his day, Newton argued that \ this would indeed happen if there were only a finite number of stars distributed over a finite region of space.\ But he reasoned that if, on the other hand, there were an infinite number of stars, distributed more or less uniform\ ly over infinite space, this would not happen, because there would not be any central point for them to fall to\ .

This argument is an instance of the pitfalls that you can encounter in t\ alking about infinity. In an infinite universe, every point can be regarded as the center, because every point\ has an infinite number of stars on each side of it. The correct approach, it was realized only much later, \ is to consider the finite situation, in which the stars all fall in on each other, and then to ask how things change i\ f one adds more stars roughly uniformly distributed outside this region. According to Newton’s law, the extra\ stars would make no difference at all to the original ones on average, so the stars would fall in just as fast. We ca\ n add as many stars as we like, but they will still always collapse in on themselves. We now know it is impossibl\ e to have an infinite static model of the universe in which gravity is always attractive.

It is an interesting reflection on the general climate of thought before\ the twentieth century that no one had suggested that the universe was expanding or contracting. It was general\ ly accepted that either the universe had existed forever in an unchanging state, or that it had been created \ at a finite time in the past more or less as we observe it today. In part this may have been due to people’s te\ ndency to believe in eternal truths, as well as the comfort they found in the thought that even though they may grow \ old and die, the universe is eternal and unchanging.

Even those who realized that Newton’s theory of gravity showed that t\ he universe could not be static did not think to suggest that it might be expanding. Instead, they attempted to \ modify the theory by making the A Brief History of Time - Stephen Hawking... Chapter 1 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/n.html (3 of 7) [2/20/2001 3:14:06 AM] gravitational force repulsive at very large distances. This did not sign\ ificantly affect their predictions of the motions of the planets, but it allowed an infinite distribution of stars\ to remain in equilibrium – with the attractive forces between nearby stars balanced by the repulsive forces from those \ that were farther away. However, we now believe such an equilibrium would be unstable: if the stars in some \ region got only slightly nearer each other, the attractive forces between them would become stronger and domi\ nate over the repulsive forces so that the stars would continue to fall toward each other. On the other ha\ nd, if the stars got a bit farther away from each other, the repulsive forces would dominate and drive them fart\ her apart.

Another objection to an infinite static universe is normally ascribed to\ the German philosopher Heinrich Olbers, who wrote about this theory in 1823. In fact, various contemporaries of \ Newton had raised the problem, and the Olbers article was not even the first to contain plausible arguments aga\ inst it. It was, however, the first to be widely noted. The difficulty is that in an infinite static universe near\ ly every line of sight would end on the surface of a star. Thus one would expect that the whole sky would be as \ bright as the sun, even at night.

Olbers’ counter-argument was that the light from distant stars would \ be dimmed by absorption by intervening matter. However, if that happened the intervening matter would eventuall\ y heat up until it glowed as brightly as the stars. The only way of avoiding the conclusion that the whole of the\ night sky should be as bright as the surface of the sun would be to assume that the stars had not been shinin\ g forever but had turned on at some finite time in the past. In that case the absorbing matter might not hav\ e heated up yet or the light from distant stars might not yet have reached us. And that brings us to the question \ of what could have caused the stars to have turned on in the first place.

The beginning of the universe had, of course, been discussed long before\ this. According to a number of early cosmologies and the Jewish/Christian/Muslim tradition, the universe star\ ted at a finite, and not very distant, time in the past. One argument for such a beginning was the feeling that\ it was necessary to have “First Cause” to explain the existence of the universe. (Within the universe, you alw\ ays explained one event as being caused by some earlier event, but the existence of the universe itself could be\ explained in this way only if it had some beginning.) Another argument was put forward by St. Augustine in his bo\ ok The City of God. He pointed out that civilization is progressing and we remember who performed this deed\ or developed that technique. Thus man, and so also perhaps the universe, could not have been around all th\ at long. St. Augustine accepted a date of about 5000 BC for the Creation of the universe according to the \ book of Genesis. (It is interesting that this is not so far from the end of the last Ice Age, about 10,000 BC, wh\ ich is when archaeologists tell us that civilization really began.) Aristotle, and most of the other Greek philosophers, on the other hand, \ did not like the idea of a creation because it smacked too much of divine intervention. They believed, there\ fore, that the human race and the world around it had existed, and would exist, forever. The ancients had \ already considered the argument about progress described above, and answered it by saying that there had been \ periodic floods or other disasters that repeatedly set the human race right back to the beginning of civilizatio\ n.

The questions of whether the universe had a beginning in time and whethe\ r it is limited in space were later extensively examined by the philosopher Immanuel Kant in his monumental \ (and very obscure) work Critique of Pure Reason, published in 1781. He called these questions antinomies (that is, contr\ adictions) of pure reason because he felt that there were equally compelling arguments for believi\ ng the thesis, that the universe had a beginning, and the antithesis, that it had existed forever. His argument\ for the thesis was that if the universe did not have a beginning, there would be an infinite period of time before a\ ny event, which he considered absurd.

The argument for the antithesis was that if the universe had a beginning\ , there would be an infinite period of time before it, so why should the universe begin at any one particular t\ ime? In fact, his cases for both the thesis and the antithesis are really the same argument. They are both based on \ his unspoken assumption that time continues back forever, whether or not the universe had existed forever.\ As we shall see, the concept of time has no meaning before the beginning of the universe. This was first poin\ ted out by St. Augustine. When asked:

“What did God do before he created the universe?” Augustine didn’\ t reply: “He was preparing Hell for people who asked such questions.” Instead, he said that time was a property \ of the universe that God created, and that time did not exist before the beginning of the universe.

When most people believed in an essentially static and unchanging univer\ se, the question of whether or not it had a beginning was really one of metaphysics or theology. One could acc\ ount for what was observed equally well on the theory that the universe had existed forever or on the theor\ y that it was set in motion at some finite A Brief History of Time - Stephen Hawking... Chapter 1 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/n.html (4 of 7) [2/20/2001 3:14:06 AM] time in such a manner as to look as though it had existed forever. But i\ n 1929, Edwin Hubble made the landmark observation that wherever you look, distant galaxies are moving\ rapidly away from us. In other words, the universe is expanding. This means that at earlier times objects woul\ d have been closer together. In fact, it seemed that there was a time, about ten or twenty thousand million years\ ago, when they were all at exactly the same place and when, therefore, the density of the universe was infi\ nite. This discovery finally brought the question of the beginning of the universe into the realm of science.

Hubble’s observations suggested that there was a time, called the big\ bang, when the universe was infinitesimally small and infinitely dense. Under such conditions all th\ e laws of science, and therefore all ability to predict the future, would break down. If there were events earlier th\ an this time, then they could not affect what happens at the present time. Their existence can be ignored because\ it would have no observational consequences. One may say that time had a beginning at the big bang, in \ the sense that earlier times simply would not be defined. It should be emphasized that this beginning in tim\ e is very different from those that had been considered previously. In an unchanging universe a beginning in tim\ e is something that has to be imposed by some being outside the universe; there is no physical necessi\ ty for a beginning. One can imagine that God created the universe at literally any time in the past. On the \ other hand, if the universe is expanding, there may be physical reasons why there had to be a beginning. One could\ still imagine that God created the universe at the instant of the big bang, or even afterwards in just such\ a way as to make it look as though there had been a big bang, but it would be meaningless to suppose that it was \ created before the big bang. An expanding universe does not preclude a creator, but it does place limits\ on when he might have carried out his job!

In order to talk about the nature of the universe and to discuss questio\ ns such as whether it has a beginning or an end, you have to be clear about what a scientific theory is. I shall \ take the simpleminded view that a theory is just a model of the universe, or a restricted part of it, and a set o\ f rules that relate quantities in the model to observations that we make. It exists only in our minds and does not have\ any other reality (whatever that might mean). A theory is a good theory if it satisfies two requirements. It m\ ust accurately describe a large class of observations on the basis of a model that contains only a few arbitrary \ elements, and it must make definite predictions about the results of future observations. For example, Arist\ otle believed Empedocles’s theory that everything was made out of four elements, earth, air, fire, and water. T\ his was simple enough, but did not make any definite predictions. On the other hand, Newton’s theory of gravi\ ty was based on an even simpler model, in which bodies attracted each other with a force that was proportional to \ a quantity called their mass and inversely proportional to the square of the distance between them. Yet i\ t predicts the motions of the sun, the moon, and the planets to a high degree of accuracy.

Any physical theory is always provisional, in the sense that it is only \ a hypothesis: you can never prove it. No matter how many times the results of experiments agree with some theory,\ you can never be sure that the next time the result will not contradict the theory. On the other hand, you c\ an disprove a theory by finding even a single observation that disagrees with the predictions of the theory. As\ philosopher of science Karl Popper has emphasized, a good theory is characterized by the fact that it makes a n\ umber of predictions that could in principle be disproved or falsified by observation. Each time new experi\ ments are observed to agree with the predictions the theory survives, and our confidence in it is increased; \ but if ever a new observation is found to disagree, we have to abandon or modify the theory.

At least that is what is supposed to happen, but you can always question\ the competence of the person who carried out the observation.

In practice, what often happens is that a new theory is devised that is \ really an extension of the previous theory.

For example, very accurate observations of the planet Mercury revealed a\ small difference between its motion and the predictions of Newton’s theory of gravity. Einstein’s gene\ ral theory of relativity predicted a slightly different motion from Newton’s theory. The fact that Einstein’s pr\ edictions matched what was seen, while Newton’s did not, was one of the crucial confirmations of the new the\ ory. However, we still use Newton’s theory for all practical purposes because the difference between its prediction\ s and those of general relativity is very small in the situations that we normally deal with. (Newton’s theory\ also has the great advantage that it is much simpler to work with than Einstein’s!) The eventual goal of science is to provide a single theory that describe\ s the whole universe. However, the A Brief History of Time - Stephen Hawking... Chapter 1 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/n.html (5 of 7) [2/20/2001 3:14:06 AM] approach most scientists actually follow is to separate the problem into\ two parts. First, there are the laws that tell us how the universe changes with time. (If we know what the univer\ se is like at any one time, these physical laws tell us how it will look at any later time.) Second, there is the \ question of the initial state of the universe.

Some people feel that science should be concerned with only the first pa\ rt; they regard the question of the initial situation as a matter for metaphysics or religion. They would sa\ y that God, being omnipotent, could have started the universe off any way he wanted. That may be so, but in that \ case he also could have made it develop in a completely arbitrary way. Yet it appears that he chose to m\ ake it evolve in a very regular way according to certain laws. It therefore seems equally reasonable to supp\ ose that there are also laws governing the initial state.

It turns out to be very difficult to devise a theory to describe the uni\ verse all in one go. Instead, we break the problem up into bits and invent a number of partial theories. Each of th\ ese partial theories describes and predicts a certain limited class of observations, neglecting the effects\ of other quantities, or representing them by simple sets of numbers. It may be that this approach is completely wr\ ong. If everything in the universe depends on everything else in a fundamental way, it might be impossible \ to get close to a full solution by investigating parts of the problem in isolation. Nevertheless, it is cer\ tainly the way that we have made progress in the past. The classic example again is the Newtonian theory of gravit\ y, which tells us that the gravitational force between two bodies depends only on one number associated with each\ body, its mass, but is otherwise independent of what the bodies are made of. Thus one does not need to ha\ ve a theory of the structure and constitution of the sun and the planets in order to calculate their orbi\ ts.

Today scientists describe the universe in terms of two basic partial the\ ories – the general theory of relativity and quantum mechanics. They are the great intellectual achievements of t\ he first half of this century. The general theory of relativity describes the force of gravity and the larg\ e-scale structure of the universe, that is, the structure on scales from only a few miles to as large as a million m\ illion million million (1 with twenty-four zeros after it) miles, the size of the observable universe. Quantum mec\ hanics, on the other hand, deals with phenomena on extremely small scales, such as a millionth of a millionth \ of an inch. Unfortunately, however, these two theories are known to be inconsistent with each other – the\ y cannot both be correct. One of the major endeavors in physics today, and the major theme of this book, is t\ he search for a new theory that will incorporate them both – a quantum theory of gravity. We do not yet ha\ ve such a theory, and we may still be a long way from having one, but we do already know many of the properties \ that it must have. And we shall see, in later chapters, that we already know a fair amount about the predicat\ ions a quantum theory of gravity must make.

Now, if you believe that the universe is not arbitrary, but is governed \ by definite laws, you ultimately have to combine the partial theories into a complete unified theory that will de\ scribe everything in the universe. But there is a fundamental paradox in the search for such a complete unified\ theory. The ideas about scientific theories outlined above assume we are rational beings who are free to ob\ serve the universe as we want and to draw logical deductions from what we see.

In such a scheme it is reasonable to suppose that we might progress ever\ closer toward the laws that govern our universe. Yet if there really is a complete unified theory, it would\ also presumably determine our actions.

And so the theory itself would determine the outcome of our search for i\ t! And why should it determine that we come to the right conclusions from the evidence? Might it not equally we\ ll determine that we draw the wrong conclusion.? Or no conclusion at all?

The only answer that I can give to this problem is based on Darwin’s \ principle of natural selection. The idea is that in any population of self-reproducing organisms, there will be vari\ ations in the genetic material and upbringing that different individuals have. These differences will mean \ that some individuals are better able than others to draw the right conclusions about the world around them an\ d to act accordingly. These individuals will be more likely to survive and reproduce and so their pattern of beh\ avior and thought will come to dominate.

It has certainly been true in the past that what we call intelligence an\ d scientific discovery have conveyed a survival advantage. It is not so clear that this is still the case: our \ scientific discoveries may well destroy us all, and even if they don’t, a complete unified theory may not make much d\ ifference to our chances of survival.

However, provided the universe has evolved in a regular way, we might ex\ pect that the reasoning abilities that natural selection has given us would be valid also in our search for a c\ omplete unified theory, and so would not lead us to the wrong conclusions.

A Brief History of Time - Stephen Hawking... Chapter 1 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/n.html (6 of 7) [2/20/2001 3:14:06 AM] Because the partial theories that we already have are sufficient to make\ accurate predictions in all but the most extreme situations, the search for the ultimate theory of the universe s\ eems difficult to justify on practical grounds. (It is worth noting, though, that similar arguments could have\ been used against both relativity and quantum mechanics, and these theories have given us both nuclear energy \ and the microelectronics revolution!) The discovery of a complete unified theory, therefore, may\ not aid the survival of our species. It may not even affect our lifestyle. But ever since the dawn of civilizati\ on, people have not been content to see events as unconnected and inexplicable. They have craved an understandin\ g of the underlying order in the world. Today we still yearn to know why we are here and where we came fr\ om. Humanity’s deepest desire for knowledge is justification enough for our continuing quest. And our goal\ is nothing less than a complete description of the universe we live in.

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Chapter 1 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/n.html (7 of 7) [2/20/2001 3:14:06 AM] CHAPTER 2 SPACE AND TIME   Our present ideas about the motion of bodies date back to Galileo and Ne\ wton. Before them people believed Aristotle, who said that the natural state of a body was to be at rest a\ nd that it moved only if driven by a force or impulse. It followed that a heavy body should fall faster than a light o\ ne, because it would have a greater pull toward the earth.

The Aristotelian tradition also held that one could work out all the law\ s that govern the universe by pure thought: it was not necessary to check by observation. So no one until G\ alileo bothered to see whether bodies of different weight did in fact fall at different speeds. It is said tha\ t Galileo demonstrated that Aristotle’s belief was false by dropping weights from the leaning tower of Pisa. The story \ is almost certainly untrue, but Galileo did do something equivalent: he rolled balls of different weights down a\ smooth slope. The situation is similar to that of heavy bodies falling vertically, but it is easier to observe bec\ ause the Speeds are smaller. Galileo’s measurements indicated that each body increased its speed at the same ra\ te, no matter what its weight. For example, if you let go of a ball on a slope that drops by one meter for \ every ten meters you go along, the ball will be traveling down the slope at a speed of about one meter per secon\ d after one second, two meters per second after two seconds, and so on, however heavy the ball. Of course a\ lead weight would fall faster than a feather, but that is only because a feather is slowed down by air resist\ ance. If one drops two bodies that don’t have much air resistance, such as two different lead weights, they fall \ at the same rate. On the moon, where there is no air to slow things down, the astronaut David R. Scott perfor\ med the feather and lead weight experiment and found that indeed they did hit the ground at the same tim\ e.

Galileo’s measurements were used by Newton as the basis of his laws o\ f motion. In Galileo’s experiments, as a body rolled down the slope it was always acted on by the same force (it\ s weight), and the effect was to make it constantly speed up. This showed that the real effect of a force is alwa\ ys to change the speed of a body, rather than just to set it moving, as was previously thought. It also meant tha\ t whenever a body is not acted on by any force, it will keep on moving in a straight line at the same speed. This\ idea was first stated explicitly in Newton’s Principia Mathematica, published in 1687, and is known as Newton’s first law. What happens t\ o a body when a force does act on it is given by Newton’s second law. This states tha\ t the body will accelerate, or change its speed, at a rate that is proportional to the force. (For example, the a\ cceleration is twice as great if the force is twice as great.) The acceleration is also smaller the greater the mass \ (or quantity of matter) of the body. (The same force acting on a body of twice the mass will produce half the acce\ leration.) A familiar example is provided by a car: the more powerful the engine, the greater the acceler\ ation, but the heavier the car, the smaller the acceleration for the same engine. In addition to his laws of\ motion, Newton discovered a law to describe the force of gravity, which states that every body attracts eve\ ry other body with a force that is proportional to the mass of each body. Thus the force between two bodies\ would be twice as strong if one of the bodies (say, body A) had its mass doubled. This is what you might \ expect because one could think of the new body A as being made of two bodies with the original mass. Each would attract b\ ody B with the original force. Thus the total force between A and B would be twice the original force. And if, say, one of the bodies had twice the mass, and the other had three times the mass, then the force w\ ould be six times as strong. One can now see why all bodies fall at the same rate: a body of twice the weight\ will have twice the force of gravity pulling it down, but it will also have twice the mass. According to Newt\ on’s second law, these two effects will exactly cancel each other, so the acceleration will be the same in all c\ ases.

Newton’s law of gravity also tells us that the farther apart the bodi\ es, the smaller the force. Newton’s law of gravity says that the gravitational attraction of a star is exactly one \ quarter that of a similar star at half the distance. This law predicts the orbits of the earth, the moon, and the p\ lanets with great accuracy. If the law were that the gravitational attraction of a star went down faster or inc\ reased more rapidly with distance, the orbits of the planets would not be elliptical, they would either spiral \ in to the sun or escape from the sun.

The big difference between the ideas of Aristotle and those of Galileo a\ nd Newton is that Aristotle believed in a preferred state of rest, which any body would take up if it were not dri\ ven by some force Or impulse. In particular, he thought that the earth was at rest. But it follows from N\ ewton’s laws that there is no unique A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (1 of 12) [2/20/2001 3:14:15 AM] standard of rest. One could equally well say that body A was at rest and body B was moving at constant speed with respect to body A, or that body B was at rest and body A was moving. For example, if one sets aside for a moment the rotation of the earth and its orbit round the sun, one could \ say that the earth was at rest and that a train on it was traveling north at ninety miles per hour or that the tra\ in was at rest and the earth was moving south at ninety miles per hour. If one carried out experiments with movi\ ng bodies on the train, all Newton’s laws would still hold. For instance, playing Ping-Pong on the train, one woul\ d find that the ball obeyed Newton’s laws just like a ball on a table by the track. So there is no way to tell whe\ ther it is the train or the earth that is moving.

The lack of an absolute standard of rest meant that one could not determ\ ine whether two events that took place at different times occurred in the same position in space. For example, \ suppose our Ping-Pong ball on the train bounces straight up and down, hitting the table twice on the same spot o\ ne second apart. To someone on the track, the two bounces would seem to take place about forty meters apart\ , because the train would have traveled that far down the track between the bounces. The nonexistence o\ f absolute rest therefore meant that one could not give an event an absolute position in space, as Aristotle \ had believed. The positions of events and the distances between them would be different for a person on the tr\ ain and one on the track, and there would be no reason to prefer one person’s position to the other’s.\ Newton was very worried by this lack of absolute position, or absolute s\ pace, as it was called, because it did not accord with his idea of an absolute God. In fact, he refused to acce\ pt lack of absolute space, even though it was implied by his laws. He was severely criticized for this irrational \ belief by many people, most notably by Bishop Berkeley, a philosopher who believed that all material objects an\ d space and time are an illusion. When the famous Dr. Johnson was told of Berkeley’s opinion, he cried, “\ I refute it thus!” and stubbed his toe on a large stone.

Both Aristotle and Newton believed in absolute time. That is, they belie\ ved that one could unambiguously measure the interval of time between two events, and that this time woul\ d be the same whoever measured it, provided they used a good clock. Time was completely separate from and i\ ndependent of space. This is what most people would take to be the commonsense view. However, we have had \ to change our ideas about space and time. Although our apparently commonsense notions work well when dea\ ling with things like apples, or planets that travel comparatively slowly, they don’t work at all for \ things moving at or near the speed of light.

The fact that light travels at a finite, but very high, speed was first \ discovered in 1676 by the Danish astronomer Ole Christensen Roemer. He observed that the times at which the moons of\ Jupiter appeared to pass behind Jupiter were not evenly spaced, as one would expect if the moons went ro\ und Jupiter at a constant rate. As the earth and Jupiter orbit around the sun, the distance between them varies\ . Roemer noticed that eclipses of Jupiter’s moons appeared later the farther we were from Jupiter. He a\ rgued that this was because the light from the moons took longer to reach us when we were farther away. His measure\ ments of the variations in the distance of the earth from Jupiter were, however, not very accurate, and\ so his value for the speed of light was 140,000 miles per second, compared to the modern value of 186,000 miles \ per second. Nevertheless, Roemer’s achievement, in not only proving that light travels at a fin\ ite speed, but also in measuring that speed, was remarkable – coming as it did eleven years before Newton’s pub\ lication of Principia Mathematica. A proper theory of the propagation of light didn’t come until 1865, when the B\ ritish physicist James Clerk Maxwell succeeded in unifying the partial theories that up to then had been used\ to describe the forces of electricity and magnetism. Maxwell’s equations predicted that there could be wavelike\ disturbances in the combined electromagnetic field, and that these would travel at a fixed speed, lik\ e ripples on a pond. If the wavelength of these waves (the distance between one wave crest and the next) is a me\ ter or more, they are what we now call radio waves. Shorter wavelengths are known as microwaves (a few centime\ ters) or infrared (more than a ten-thousandth of a centimeter). Visible light has a wavelength of betw\ een only forty and eighty millionths of a centimeter. Even shorter wavelengths are known as ultraviolet, X rays, a\ nd gamma rays.

Maxwell’s theory predicted that radio or light waves should travel at\ a certain fixed speed. But Newton’s theory had got rid of the idea of absolute rest, so if light was supposed to tr\ avel at a fixed speed, one would have to say what that fixed speed was to be measured relative to.

It was therefore suggested that there was a substance called the "ether"\ that was present everywhere, even in "empty" space. Light waves should travel through the ether as sound wave\ s travel through air, and their speed should therefore be relative to the ether. Different observers, moving r\ elative to the ether, would see light A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (2 of 12) [2/20/2001 3:14:15 AM] coming toward them at different speeds, but light's speed relative to th\ e ether would remain fixed. In particular, as the earth was moving through the ether on its orbit round the sun, th\ e speed of light measured in the direction of the earth's motion through the ether (when we were moving \ toward the source of the light) should be higher than the speed of light at right angles to that motion (when \ we are not moving toward the source). In 1887Albert Michelson (who later became the first American to receive th\ e Nobel Prize for physics) and Edward Morley carried out a very careful experiment at the Case School of Appli\ ed Science in Cleveland. They compared the speed of light in the direction of the earth's motion with \ that at right angles to the earth's motion.

To their great surprise, they found they were exactly the same!

Between 1887 and 1905 there were several attempts, most notably by the D\ utch physicist Hendrik Lorentz, to explain the result of the Michelson-Morley experiment in terms of object\ s contracting and clocks slowing down when they moved through the ether. However, in a famous paper in 1905, a\ hitherto unknown clerk in the Swiss patent office, Albert Einstein, pointed out that the whole idea of\ an ether was unnecessary, providing one was willing to abandon the idea of absolute time. A similar point was ma\ de a few weeks later by a leading French mathematician, Henri Poincare. Einstein’s arguments were close\ r to physics than those of Poincare, who regarded this problem as mathematical. Einstein is usually given the\ credit for the new theory, but Poincare is remembered by having his name attached to an important part \ of it.

The fundamental postulate of the theory of relativity, as it was called,\ was that the laws of science should be the same for all freely moving observers, no matter what their speed. Th\ is was true for Newton’s laws of motion, but now the idea was extended to include Maxwell’s theory and\ the speed of light: all observers should measure the same speed of light, no matter how fast they are moving. Thi\ s simple idea has some remarkable consequences. Perhaps the best known are the equivalence of mass and ene\ rgy, summed up in Einstein’s famous equation E=mc 2 (where E is energy, m is mass, and c is the speed of light), and the law that nothing may travel faster than the speed of light. Because of the equivalence of\ energy and mass, the energy which an object has due to its motion will add to its mass. In other words, it wi\ ll make it harder to increase its speed. This effect is only really significant for objects moving at speeds close to \ the speed of light. For example, at 10 percent of the speed of light an object’s mass is only 0.5 percent mo\ re than normal, while at 90 percent of the speed of light it would be more than twice its normal mass. As an object\ approaches the speed of light, its mass rises ever more quickly, so it takes more and more energy to speed it up\ further. It can in fact never reach the speed of light, because by then its mass would have become infinite, and\ by the equivalence of mass and energy, it would have taken an infinite amount of energy to get it there\ . For this reason, any normal object is forever confined by relativity to move at speeds slower than the speed o\ f light. Only light, or other waves that have no intrinsic mass, can move at the speed of light.

An equally remarkable consequence of relativity is the way it has revolu\ tionized our ideas of space and time. In Newton’s theory, if a pulse of light is sent from one place to anothe\ r, different observers would agree on the time that the journey took (since time is absolute), but will not alwa\ ys agree on how far the light traveled (since space is not absolute). Since the speed of the light is just the distan\ ce it has traveled divided by the time it has taken, different observers would measure different speeds for the light.\ In relativity, on the other hand, all observers must agree on how fast light travels. They still, however, do not agree on th\ e distance the light has traveled, so they must therefore now also disagree over the time it has \ taken. (The time taken is the distance the light has traveled – which the observers do not agree on – div\ ided by the light’s speed – which they do agree on.) In other words, the theory of relativity put an end to the i\ dea of absolute time! It appeared that each observer must have his own measure of time, as recorded by a clock carri\ ed with him, and that identical clocks carried by different observers would not necessarily agree.

Each observer could use radar to say where and when an event took place \ by sending out a pulse of light or radio waves. Part of the pulse is reflected back at the event and the ob\ server measures the time at which he receives the echo. The time of the event is then said to be the time hal\ fway between when the pulse was sent and the time when the reflection was received back: the distance of the \ event is half the time taken for this round trip, multiplied by the speed of light. (An event, in this sense,\ is something that takes place at a single point in space, at a specified point in time.) This idea is shown here,\ which is an example of a space-time diagram... A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (3 of 12) [2/20/2001 3:14:16 AM] Figure 2:1 Using this procedure, observers who are moving relative to each other wi\ ll assign different times and positions to the same event. No particular observer’s measurements are any more\ correct than any other observer’s, but all the measurements are related. Any observer can work out precisely wh\ at time and position any other observer will assign to an event, provided he knows the other observer’\ s relative velocity.

Nowadays we use just this method to measure distances precisely, because\ we can measure time more accurately than length. In effect, the meter is defined to be the distan\ ce traveled by light in 0.000000003335640952 second, as measured by a cesium clock. (The reason\ for that particular number is that it corresponds to the historical definition of the meter – in terms o\ f two marks on a particular platinum bar kept in Paris.) Equally, we can use a more convenient, new unit of length ca\ lled a light-second. This is simply defined as the distance that light travels in one second. In the theory \ of relativity, we now define distance in A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (4 of 12) [2/20/2001 3:14:16 AM] terms of time and the speed of light, so it follows automatically that e\ very observer will measure light to have the same speed (by definition, 1 meter per 0.000000003335640952 second)\ . There is no need to introduce the idea of an ether, whose presence anyway cannot be detected, as the Miche\ lson-Morley experiment showed.

The theory of relativity does, however, force us to change fundamentally\ our ideas of space and time. We must accept that time is not completely separate from and independent of spac\ e, but is combined with it to form an object called space-time.

It is a matter of common experience that one can describe the position o\ f a point in space by three numbers, or coordinates. For instance, one can say that a point in a room is seven f\ eet from one wall, three feet from another, and five feet above the floor. Or one could specify that a poin\ t was at a certain latitude and longitude and a certain height above sea level. One is free to use any three suita\ ble coordinates, although they have only a limited range of validity. One would not specify the position of the m\ oon in terms of miles north and miles west of Piccadilly Circus and feet above sea level. Instead, one might d\ escribe it in terms of distance from the sun, distance from the plane of the orbits of the planets, and the angle\ between the line joining the moon to the sun and the line joining the sun to a nearby star such as Alpha Centauri\ . Even these coordinates would not be of much use in describing the position of the sun in our galaxy or the p\ osition of our galaxy in the local group of galaxies. In fact, one may describe the whole universe in terms of a col\ lection of overlapping patches. In each patch, one can use a different set of three coordinates to specify the p\ osition of a point.

An event is something that happens at a particular point in space and at\ a particular time. So one can specify it by four numbers or coordinates. Again, the choice of coordinates is arbi\ trary; one can use any three well-defined spatial coordinates and any measure of time. In relativity,\ there is no real distinction between the space and time coordinates, just as there is no real difference between \ any two space coordinates. One could choose a new set of coordinates in which, say, the first space coordinat\ e was a combination of the old first and second space coordinates. For instance, instead of measuring the positio\ n of a point on the earth in miles north of Piccadilly and miles west of Piccadilly, one could use miles northeas\ t of Piccadilly, and miles north-west of Piccadilly. Similarly, in relativity, one could use a new time coordinat\ e that was the old time (in seconds) plus the distance (in light-seconds) north of Piccadilly.

It is often helpful to think of the four coordinates of an event as spec\ ifying its position in a four-dimensional space called space-time. It is impossible to imagine a four-dimensional \ space. I personally find it hard enough to visualize three-dimensional space! However, it is easy to draw diagra\ ms of two-dimensional spaces, such as the surface of the earth. (The surface of the earth is two-dimensional \ because the position of a point can be specified by two coordinates, latitude and longitude.) I shall generall\ y use diagrams in which time increases upward and one of the spatial dimensions is shown horizontally. The othe\ r two spatial dimensions are ignored or, sometimes, one of them is indicated by perspective. (These are call\ ed space-time diagrams, like Figure 2:1 .) A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (5 of 12) [2/20/2001 3:14:16 AM] Figure 2:2 For example, in Figure 2:2 time is measured upward in years and the distance along the line from t\ he sun to Alpha Centauri is measured horizontally in miles. The paths of the sun a\ nd of Alpha Centauri through space-time are shown as the vertical lines on the left and right of the \ diagram. A ray of light from the sun follows the diagonal line, and takes four years to get from the sun to A\ lpha Centauri.

As we have seen, Maxwell’s equations predicted that the speed of ligh\ t should be the same whatever the speed of the source, and this has been confirmed by accurate measurement\ s. It follows from this that if a pulse of light is emitted at a particular time at a particular point in space,\ then as time goes on it will spread out as a sphere of light whose size and position are independent of the speed of \ the source. After one millionth of a second the light will have spread out to form a sphere with a radius of \ 300 meters; after two millionths of a second, the radius will be 600 meters; and so on. It will be like the ri\ pples that spread out on the surface of a pond when a stone is thrown in. The ripples spread out as a circle that \ gets bigger as time goes on. If one stacks snapshots of the ripples at different times one above the other, \ the expanding circle of ripples will mark out a cone whose tip is at the place and time at which the stone hit the\ water Figure 2:3. A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (6 of 12) [2/20/2001 3:14:16 AM] Figure 2:3 Similarly, the light spreading out from an event forms a (three-dimensi\ onal) cone in (the four-dimensional) space-time. This cone is called the future light cone of the event. In t\ he same way we can draw another cone, called the past light cone, which is the set of events from which a puls\ e of light is able to reach the given event Figure 2:4 . A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (7 of 12) [2/20/2001 3:14:16 AM] Figure 2:4 Given an event P, one can divide the other events in the universe into t\ hree classes. Those events that can be reached from the event P by a particle or wave traveling at or below the\ speed of light are said to be in the future of P. They will lie within or on the expanding sphere of light em\ itted from the event P. Thus they will lie within or on the future light cone of P in the space-time diagram. Only \ events in the future of P can be affected by what happens at P because nothing can travel faster than light.

Similarly, the past of P can be defined as the set of all events from wh\ ich it is possible to reach the event P traveling at or below the speed of light. It is thus the set of events t\ hat can affect what happens at P. The events that do not lie in the future or past of P are said to lie in the\ elsewhere of P Figure 2:5. A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (8 of 12) [2/20/2001 3:14:16 AM] Figure 2:5 What happens at such events can neither affect nor be affected by what h\ appens at P. For example, if the sun were to cease to shine at this very moment, it would not affect things o\ n earth at the present time because they would be in the elsewhere of the event when the sun went out Figure 2:6. A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (9 of 12) [2/20/2001 3:14:16 AM] Figure 2:6 We would know about it only after eight minutes, the time it takes light\ to reach us from the sun. Only then would events on earth lie in the future light cone of the event at which\ the sun went out. Similarly, we do not know what is happening at the moment farther away in the universe: the l\ ight that we see from distant galaxies left them millions of years ago, and in the case of the most distant obj\ ect that we have seen, the light left some eight thousand million years ago. Thus, when we look at the universe, we\ are seeing it as it was in the past.

If one neglects gravitational effects, as Einstein and Poincare did in 1\ 905, one has what is called the special theory of relativity. For every event in space-time we may construct a l\ ight cone (the set of all possible paths of light in space-time emitted at that event), and since the speed of ligh\ t is the same at every event and in every direction, all the light cones will be identical and will all point in t\ he same direction. The theory also tells us that nothing can travel faster than light. This means that the path of any ob\ ject through space and time must be represented by a line that lies within the light cone at each event on i\ t (Fig. 2.7). The special theory of relativity was very successful in explaining that the speed of light appears the sa\ me to all observers (as shown by the Michelson-Morley experiment) and in describing what happens when things\ move at speeds close to the speed of light. However, it was inconsistent with the Newtonian theory of grav\ ity, which said that objects attracted each other with a force that depended on the distance between them. This\ meant that if one moved one of the objects, the force on the other one would change instantaneously. Or in \ other gravitational effects should travel with infinite velocity, instead of at or below the speed of light, as th\ e special theory of relativity required.

Einstein made a number of unsuccessful attempts between 1908 and 1914 to\ find a theory of gravity that was consistent with special relativity. Finally, in 1915, he proposed what w\ e now call the general theory of relativity.

Einstein made the revolutionary suggestion that gravity is not a force l\ ike other forces, but is a consequence of the fact that space-time is not flat, as had been previously assumed: it\ is curved, or “warped,” by the distribution A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (10 of 12) [2/20/2001 3:14:16 AM] of mass and energy in it. Bodies like the earth are not made to move on \ curved orbits by a force called gravity; instead, they follow the nearest thing to a straight path in a curved sp\ ace, which is called a geodesic. A geodesic is the shortest (or longest) path between two nearby points. \ For example, the surface of the earth is a two-dimensional curved space. A geodesic on the earth is called a great \ circle, and is the shortest route between two points (Fig. 2.8). As the geodesic is the shortest path be\ tween any two airports, this is the route an airline navigator will tell the pilot to fly along. In general relati\ vity, bodies always follow straight lines in four-dimensional space-time, but they nevertheless appear to us to move \ along curved paths in our three-dimensional space. (This is rather like watching an airplane flyi\ ng over hilly ground. Although it follows a straight line in three-dimensional space, its shadow follows a curved pa\ th on the two-dimensional ground.) The mass of the sun curves space-time in such a way that although the ea\ rth follows a straight path in four-dimensional space-time, it appears to us to move along a circular o\ rbit in three-dimensional space.

Fact, the orbits of the planets predicted by general relativity are almo\ st exactly the same as those predicted by the Newtonian theory of gravity. However, in the case of Mercury, which,\ being the nearest planet to the sun, feels the strongest gravitational effects, and has a rather elongated or\ bit, general relativity predicts that the long axis of the ellipse should rotate about the sun at a rate of about one d\ egree in ten thousand years. Small though this effect is, it had been noticed before 1915 and served as one\ of the first confirmations of Einstein’s theory. In recent years the even smaller deviations of the orbits of the\ other planets from the Newtonian predictions have been measured by radar and found to agree with the pred\ ictions of general relativity.

Light rays too must follow geodesics in space-time. Again, the fact that\ space is curved means that light no longer appears to travel in straight lines in space. So general relativi\ ty predicts that light should be bent by gravitational fields. For example, the theory predicts that the light co\ nes of points near the sun would be slightly bent inward, on account of the mass of the sun. This means that light fr\ om a distant star that happened to pass near the sun would be deflected through a small angle, causing the star \ to appear in a different position to an observer on the earth (Fig. 2.9). Of course, if the light from the sta\ r always passed close to the sun, we would not be able to tell whether the light was being deflected or if instead \ the star was really where we see it.

However, as the earth orbits around the sun, different stars appear to p\ ass behind the sun and have their light deflected. They therefore change their apparent position relative to oth\ er stars. It is normally very difficult to see this effect, because the light from the sun makes it impossible to obser\ ve stars that appear near to the sun the sky. However, it is possible to do so during an eclipse of the sun, when\ the sun’s light is blocked out by the moon. Einstein’s prediction of light deflection could not be tested i\ mmediately in 1915, because the First World War was in progress, and it was not until 1919 that a British expedition\ , observing an eclipse from West Africa, showed that light was indeed deflected by the sun, just as predicted by \ the theory. This proof of a German theory by British scientists was hailed as a great act of reconciliation\ between the two countries after the war. It is ionic, therefore, that later examination of the photographs taken on \ that expedition showed the errors were as great as the effect they were trying to measure. Their measurement had b\ een sheer luck, or a case of knowing the result they wanted to get, not an uncommon occurrence in science. Th\ e light deflection has, however, been accurately confirmed by a number of later observations.

Another prediction of general relativity is that time should appear to s\ lower near a massive body like the earth.

This is because there is a relation between the energy of light and its \ frequency (that is, the number of waves of light per second): the greater the energy, the higher frequency. As lig\ ht travels upward in the earth’s gravitational field, it loses energy, and so its frequency goes down. (\ This means that the length of time between one wave crest and the next goes up.) To someone high up, it would appe\ ar that everything down below was making longer to happen. This prediction was tested in 1962, using a pai\ r of very accurate clocks mounted at the top and bottom of a water tower. The clock at the bottom, which was \ nearer the earth, was found to run slower, in exact agreement with general relativity. The difference in th\ e speed of clocks at different heights above the earth is now of considerable practical importance, with the ad\ vent of very accurate navigation systems based on signals from satellites. If one ignored the predictions\ of general relativity, the position that one calculated would be wrong by several miles!

Newton’s laws of motion put an end to the idea of absolute position i\ n space. The theory of relativity gets rid of absolute time. Consider a pair of twins. Suppose that one twin goes to l\ ive on the top of a mountain while the other stays at sea level. The first twin would age faster than the secon\ d. Thus, if they met again, one would be older than the other. In this case, the difference in ages would be very\ small, but it would be much larger if one A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (11 of 12) [2/20/2001 3:14:16 AM] of the twins went for a long trip in a spaceship at nearly the speed of \ light. When he returned, he would be much younger than the one who stayed on earth. This is known as the twin\ s paradox, but it is a paradox only if one has the idea of absolute time at the back of one’s mind. In the t\ heory of relativity there is no unique absolute time, but instead each individual has his own personal measure \ of time that depends on where he is and how he is moving.

Before 1915, space and time were thought of as a fixed arena in which ev\ ents took place, but which was not affected by what happened in it. This was true even of the special theor\ y of relativity. Bodies moved, forces attracted and repelled, but time and space simply continued, unaffected.\ It was natural to think that space and time went on forever.

The situation, however, is quite different in the general theory of rela\ tivity. Space and time are now dynamic quantities: when a body moves, or a force acts, it affects the curvature\ of space and time – and in turn the structure of space-time affects the way in which bodies move and forces \ act. Space and time not only affect but also are affected by everything that happens in the universe. Just as on\ e cannot talk about events in the universe without the notions of space and time, so in general relativity\ it became meaningless to talk about space and time outside the limits of the universe.

In the following decades this new understanding of space and time was to\ revolutionize our view of the universe. The old idea of an essentially unchanging universe that could \ have existed, and could continue to exist, forever was replaced by the notion of a dynamic, expanding univer\ se that seemed to have begun a finite time ago, and that might end at a finite time in the future. That revolu\ tion forms the subject of the next chapter.

And years later, it was also to be the starting point for my work in the\ oretical physics. Roger Penrose and I showed that Einstein’s general theory of relativity implied that the \ universe must have a beginning and, possibly, an end.

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Chapter 2 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/a.html (12 of 12) [2/20/2001 3:14:16 AM] CHAPTER 3 THE EXPANDING UNIVERSE   If one looks at the sky on a clear, moonless night, the brightest object\ s one sees are likely to be the planets Venus, Mars, Jupiter, and Saturn. There will also be a very large number\ of stars, which are just like our own sun but much farther from us. Some of these fixed stars do, in fact, app\ ear to change very slightly their positions relative to each other as earth orbits around the sun: they ar\ e not really fixed at all! This is because they are comparatively near to us. As the earth goes round the sun, we s\ ee them from different positions against the background of more distant stars. This is fortunate, because\ it enables us to measure directly the distance of these stars from us: the nearer they are, the more they appe\ ar to move. The nearest star, called Proxima Centauri, is found to be about four light-years away (the light\ from it takes about four years to reach earth), or about twenty-three million million miles. Most of the other \ stars that are visible to the naked eye lie within a few hundred light-years of us. Our sun, for comparison, is a me\ re light-minutes away! The visible stars appear spread all over the night sky, but are particularly concentrated \ in one band, which we call the Milky Way. As long ago as 1750, some astronomers were suggesting that the appe\ arance of the Milky Way could be explained if most of the visible stars lie in a single disklike configur\ ation, one example of what we now call a spiral galaxy. Only a few decades later, the astronomer Sir William Hers\ chel confirmed this idea by painstakingly cataloging the positions and distances of vast numbers of \ stars. Even so, the idea gained complete acceptance only early this century.

Our modern picture of the universe dates back to only 1924, when the Ame\ rican astronomer Edwin Hubble demonstrated that ours was not the only galaxy. There were in fact many \ others, with vast tracts of empty space between them. In order to prove this, he needed to determine the d\ istances to these other galaxies, which are so far away that, unlike nearby stars, they really do appear f\ ixed. Hubble was forced, therefore, to use indirect methods to measure the distances. Now, the apparent brightn\ ess of a star depends on two factors:

how much light it radiates (its luminosity), and how far it is from us\ . For nearby stars, we can measure their apparent brightness and their distance, and so we can work out their lum\ inosity. Conversely, if we knew the luminosity of stars in other galaxies, we could work out their distance \ by measuring their apparent brightness.

Hubble noted that certain types of stars always have the same luminosity\ when they are near enough for us to measure; therefore, he argued, if we found such stars in another galaxy,\ we could assume that they had the same luminosity – and so calculate the distance to that galaxy. If we\ could do this for a number of stars in the same galaxy, and our calculations always gave the same distance, we coul\ d be fairly confident of our estimate.

In this way, Edwin Hubble worked out the distances to nine different gal\ axies. We now know that our galaxy is only one of some hundred thousand million that can be seen using modern \ telescopes, each galaxy itself containing some hundred thousand million stars. Figure 3:1 shows a picture of one spiral galaxy that is similar to what we think ours must look like to someone living in another galaxy\ .

A Brief History of Time - Stephen Hawking... Chapter 3 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/b.html (1 of 9) [2/20/2001 3:14:24 AM] Figure 3:1 We live in a galaxy that is about one hundred thousand light-years acros\ s and is slowly rotating; the stars in its spiral arms orbit around its center about once every several hundred mil\ lion years. Our sun is just an ordinary, average-sized, yellow star, near the inner edge of one of the spiral arm\ s. We have certainly come a long way since Aristotle and Ptolemy, when thought that the earth was the center \ of the universe!

Stars are so far away that they appear to us to be just pinpoints of lig\ ht. We cannot see their size or shape. So how can we tell different types of stars apart? For the vast majority of\ stars, there is only one characteristic feature that we can observe – the color of their light. Newton discov\ ered that if light from the sun passes through a triangular-shaped piece of glass, called a prism, it breaks up\ into its component colors (its spectrum) as in a rainbow. By focusing a telescope on an individual star or galaxy\ , one can similarly observe the spectrum of the light from that star or galaxy. Different stars have different sp\ ectra, but the relative brightness of the different colors is always exactly what one would expect to find in the \ light emitted by an object that is glowing red hot. (In fact, the light emitted by any opaque object that is glowi\ ng red hot has a characteristic spectrum that depends only on its temperature – a thermal spectrum. This means\ that we can tell a star’s temperature from the spectrum of its light.) Moreover, we find that certain very sp\ ecific colors are missing from stars’ spectra, and these missing colors may vary from star to star. Since we k\ now that each chemical element absorbs a characteristic set of very specific colors, by matching these \ to those that are missing from a star’s spectrum, we can determine exactly which elements are present in the sta\ r’s atmosphere.

In the 1920s, when astronomers began to look at the spectra of stars in \ other galaxies, they found something most peculiar: there were the same characteristic sets of missing colors\ as for stars in our own galaxy, but they were all shifted by the same relative amount toward the red end of the s\ pectrum. To understand the implications of this, we must first understand the Doppler effect. As we\ have seen, visible light consists of fluctuations, or waves, in the electromagnetic field. The wavelength (o\ r distance from one wave crest to the next) of light is extremely small, ranging from four to seven ten-milli\ onths of a meter. The different wavelengths of light are what the human eye sees as different colors, with the longe\ st wavelengths appearing at the red end of the spectrum and the shortest wavelengths at the blue end. Now imagin\ e a source of light at a constant distance from us, such as a star, emitting waves of light at a constant \ wavelength. Obviously the wavelength of A Brief History of Time - Stephen Hawking... Chapter 3 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/b.html (2 of 9) [2/20/2001 3:14:24 AM] the waves we receive will be the same as the wavelength at which they ar\ e emitted (the gravitational field of the galaxy will not be large enough to have a significant effect). Suppose \ now that the source starts moving toward us. When the source emits the next wave crest it will be nearer to us, s\ o the distance between wave crests will be smaller than when the star was stationary. This means that the wavele\ ngth of the waves we receive is shorter than when the star was stationary. Correspondingly, if the sourc\ e is moving away from us, the wavelength of the waves we receive will be longer. In the case of light,\ therefore, means that stars moving away from us will have their spectra shifted toward the red end of the s\ pectrum (red-shifted) and those moving toward us will have their spectra blue-shifted. This relationship betwee\ n wavelength and speed, which is called the Doppler effect, is an everyday experience. Listen to a car passing o\ n the road: as the car is approaching, its engine sounds at a higher pitch (corresponding to a shorter wavelength \ and higher frequency of sound waves), and when it passes and goes away, it sounds at a lower pitch. The behavi\ or of light or radio waves is similar.

Indeed, the police make use of the Doppler effect to measure the speed o\ f cars by measuring the wavelength of pulses of radio waves reflected off them.

ln the years following his proof of the existence of other galaxies, Rub\ ble spent his time cataloging their distances and observing their spectra. At that time most people expected\ the galaxies to be moving around quite randomly, and so expected to find as many blue-shifted spectra as \ red-shifted ones. It was quite a surprise, therefore, to find that most galaxies appeared red-shifted: ne\ arly all were moving away from us! More surprising still was the finding that Hubble published in 1929: even the size of a galaxy’s red shift is not random, but is directly proportional to the galaxy’s distance from us. Or, in\ other words, the farther a galaxy is, the faster it is moving away! And that meant that the universe could not be static,\ as everyone previously had thought, is in fact expanding; the distance between the different galaxies is changi\ ng all the time.

The discovery that the universe is expanding was one of the great intell\ ectual revolutions of the twentieth century. With hindsight, it is easy wonder why no one had thought of it \ before. Newton, and others should have realized that a static universe would soon start to contract under the i\ nfluence of gravity. But suppose instead that the universe is expanding. If it was expanding fairly slowly, the f\ orce of gravity would cause it eventually to stop expanding and then to start contracting. However, if it was expandi\ ng at more than a certain critical rate, gravity would never be strong enough to stop it, and the universe would \ continue to expand forever. This is a bit like what happens when one fires a rocket upward from the surface of the\ earth. If it has a fairly low speed, gravity will eventually stop the rocket and it will start falling back. \ On the other hand, if the rocket has more than a certain critical speed (about seven miles per second), gravity will \ not be strong enough to pull it back, so it will keep going away from the earth forever. This behavior of the universe co\ uld have been predicted from Newton’s theory of gravity at any time in the nineteenth, the eightee\ nth, or even the late seventeenth century.

Yet so strong was the belief in a static universe that it persisted into\ the early twentieth century. Even Einstein, when he formulated the general theory of relativity in 1915, was so sure\ that the universe had to be static that he modified his theory to make this possible, introducing a so-called co\ smological constant into his equations.

Einstein introduced a new “antigravity” force, which, unlike other\ forces, did not come from any particular source but was built into the very fabric of space-time. He claimed that\ space-time had an inbuilt tendency to expand, and this could be made to balance exactly the attraction of all \ the matter in the universe, so that a static universe would result. Only one man, it seems, was willing to tak\ e general relativity at face value, and while Einstein and other physicists were looking for ways of avoiding ge\ neral relativity’s prediction of a nonstatic universe, the Russian physicist and mathematician Alexander Fr\ iedmann instead set about explaining it.

Friedmann made two very simple assumptions about the universe: that the \ universe looks identical in whichever direction we look, and that this would also be true if we were\ observing the universe from anywhere else. From these two ideas alone, Friedmann showed that we should not ex\ pect the universe to be static. In fact, in 1922, several years before Edwin Hubble’s discovery, Friedma\ nn predicted exactly what Hubble found!

The assumption that the universe looks the same in every direction is cl\ early not true in reality. For example, as we have seen, the other stars in our galaxy form a distinct band of ligh\ t across the night sky, called the Milky Way. But if we look at distant galaxies, there seems to be more or less \ the same number of them. So the universe does seem to be roughly the same in every direction, provided o\ ne views it on a large scale compared to the distance between galaxies, and ignores the differences on small s\ cales. For a long time, this was sufficient justification for Friedmann’s assumption – as a rough a\ pproximation to the real universe. But more recently a lucky accident uncovered the fact that Friedmann’s assumpt\ ion is in fact a remarkably accurate A Brief History of Time - Stephen Hawking... Chapter 3 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/b.html (3 of 9) [2/20/2001 3:14:24 AM] description of our universe.

In 1965 two American physicists at the Bell Telephone Laboratories in Ne\ w Jersey, Arno Penzias and Robert Wilson, were testing a very sensitive microwave detector. (Microwaves a\ re just like light waves, but with a wavelength of around a centimeter.) Penzias and Wilson were worried whe\ n they found that their detector was picking up more noise than it ought to. The noise did not appear to be c\ oming from any particular direction.

First they discovered bird droppings in their detector and checked for o\ ther possible malfunctions, but soon ruled these out. They knew that any noise from within the atmosphere wou\ ld be stronger when the detector was not pointing straight up than when it was, because light rays travel\ through much more atmosphere when received from near the horizon than when received from directly overhead\ . The extra noise was the same whichever direction the detector was pointed, so it must come from outside the atmosphere. It was also the same day and night and throughout the year, even though the earth was ro\ tating on its axis and orbiting around the sun. This showed that the radiation must come from beyond the Solar \ System, and even from beyond the galaxy, as otherwise it would vary as the movement of earth pointed the \ detector in different directions.

In fact, we know that the radiation must have traveled to us across most\ of the observable universe, and since it appears to be the same in different directions, the universe must als\ o be the same in every direction, if only on a large scale. We now know that whichever direction we look, this noi\ se never varies by more than a tiny fraction: so Penzias and Wilson had unwittingly stumbled across a remark\ ably accurate confirmation of Friedmann’s first assumption. However, because the universe is not ex\ actly the same in every direction, but only on average on a large scale, the microwaves cannot be exactly the s\ ame in every direction either. There have to be slight variations between different directions. These were fi\ rst detected in 1992 by the Cosmic Background Explorer satellite, or COBE, at a level of about one part in \ a hundred thousand. Small though these variations are, they are very important, as will be explained in Chapter\ 8.

At roughly the same time as Penzias and Wilson were investigating noise \ in their detector, two American physicists at nearby Princeton University, Bob Dicke and Jim Peebles, we\ re also taking an interest in microwaves. They were working on a suggestion, made by George Gamow (on\ ce a student of Alexander Friedmann), that the early universe should have been very hot and dense\ , glowing white hot. Dicke and Peebles argued that we should still be able to see the glow of the early\ universe, because light from very distant parts of it would only just be reaching us now. However, the exp\ ansion of the universe meant that this light should be so greatly red-shifted that it would appear to us now as\ microwave radiation. Dicke and Peebles were preparing to look for this radiation when Penzias and Wilson heard \ about their work and realized that they had already found it. For this, Penzias and Wilson were awarded the Nobe\ l Prize in 1978 (which seems a bit hard on Dicke and Peebles, not to mention Gamow!).

Now at first sight, all this evidence that the universe looks the same w\ hichever direction we look in might seem to suggest there is something special about our place in the universe. I\ n particular, it might seem that if we observe all other galaxies to be moving away from us, then we must be at\ the center of the universe. There is, however, an alternate explanation: the universe might look the same in e\ very direction as seen from any other galaxy too. This, as we have seen, was Friedmann’s second assumption.\ We have no scientific evidence for, or against, this assumption. We believe it only on grounds of modesty: it w\ ould be most remarkable if the universe looked the same in every direction around us, but not around other point\ s in the universe! In Friedmann’s model, all the galaxies are moving directly away from each other. The si\ tuation is rather like a balloon with a number of spots painted on it being steadily blown up. As the balloon ex\ pands, the distance between any two spots increases, but there is no spot that can be said to be the center \ of the expansion. Moreover, the farther apart the spots are, the faster they will be moving apart. Similarly, in\ Friedmann’s model the speed at which any two galaxies are moving apart is proportional to the distance between th\ em. So it predicted that the red shift of a galaxy should be directly proportional to its distance from us, exactl\ y as Hubble found. Despite the success of his model and his prediction of Hubble’s observations, Friedmann’s\ work remained largely unknown in the West until similar models were discovered in 1935 by the American physicist H\ oward Robertson and the British mathematician Arthur Walker, in response to Hubble’s discovery of the\ uniform expansion of the universe.

Although Friedmann found only one, there are in fact three different kin\ ds of models that obey Friedmann’s two fundamental assumptions. In the first kind (which Friedmann found) the\ universe is expanding sufficiently slowly that the gravitational attraction between the different galaxies \ causes the expansion to slow down and eventually to stop. The galaxies then start to move toward each other an\ d the universe contracts.

A Brief History of Time - Stephen Hawking... Chapter 3 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/b.html (4 of 9) [2/20/2001 3:14:24 AM] Figure 3:2 Figure 3:2 shows how the distance between two neighboring galaxies changes as time\ increases. It starts at zero, increases to a maximum, and then decreases to zero again. In the s\ econd kind of solution, the universe is expanding so rapidly that the gravitational attraction can never stop it\ , though it does slow it down a bit.

Figure 3:3 Figure 3:3 Shows the Separation between neighboring galaxies in this model. It sta\ rts at zero and eventually the galaxies are moving apart at a steady speed. Finally, there is a thi\ rd kind of solution, in which the universe is expanding only just fast enough to avoid recollapse. A Brief History of Time - Stephen Hawking... Chapter 3 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/b.html (5 of 9) [2/20/2001 3:14:24 AM] Figure 3:4 In this case the separation, shown in Figure 3:4, also starts at zero and increases forever. However, the speed at which the galaxies are moving apart gets smaller and smaller, althoug\ h it never quite reaches zero.

A remarkable feature of the first kind of Friedmann model is that in it \ the universe is not infinite in space, but neither does space have any boundary. Gravity is so strong that space is\ bent round onto itself, making it rather like the surface of the earth. If one keeps traveling in a certain direc\ tion on the surface of the earth, one never comes up against an impassable barrier or falls over the edge, but event\ ually comes back to where one started.

In the first kind of Friedmann model, space is just like this, but with \ three dimensions instead of two for the earth’s surface. The fourth dimension, time, is also finite in extent\ , but it is like a line with two ends or boundaries, a beginning and an end. We shall see later that when one com\ bines general relativity with the uncertainty principle of quantum mechanics, it is possible for both spac\ e and time to be finite without any edges or boundaries.

The idea that one could go right round the universe and end up where one\ started makes good science fiction, but it doesn’t have much practical significance, because it can be sh\ own that the universe would recollapse to zero size before one could get round. You would need to travel faster th\ an light in order to end up where you started before the universe came to an end – and that is not allowed!\ In the first kind of Friedmann model, which expands and recollapses, spa\ ce is bent in on itself, like the surface of the earth. It is therefore finite in extent. In the second kind of mo\ del, which expands forever, space is bent the other way, like the surface of a saddle. So in this case space is in\ finite. Finally, in the third kind of Friedmann model, with just the critical rate of expansion, space is flat\ (and therefore is also infinite).

But which Friedmann model describes our universe? Will the universe even\ tually stop expanding and start contracting, or will it expand forever? To answer this question we need \ to know the present rate of expansion of the universe and its present average density. If the density is less tha\ n a certain critical value, determined by the rate of expansion, the gravitational attraction will be too weak to \ halt the expansion. If the density is greater than the critical value, gravity will stop the expansion at some time in\ the future and cause the universe to recollapse.

We can determine the present rate of expansion by measuring the velociti\ es at which other galaxies are moving away from us, using the Doppler effect. This can be done very acc\ urately. However, the distances to the galaxies are not very well known because we can only measure them in\ directly. So all we know is that the universe is expanding by between 5 percent and 10 percent every thousand\ million years. However, our uncertainty about the present average density of the universe is even gr\ eater. If we add up the masses of all A Brief History of Time - Stephen Hawking... Chapter 3 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/b.html (6 of 9) [2/20/2001 3:14:24 AM] the stars that we can see in our galaxy and other galaxies, the total is\ less than one hundredth of the amount required to halt the expansion of the universe, even for the lowest esti\ mate of the rate of expansion. Our galaxy and other galaxies, however, must contain a large amount of “dark mat\ ter” that we cannot see directly, but which we know must be there because of the influence of its gravitationa\ l attraction on the orbits of stars in the galaxies. Moreover, most galaxies are found in clusters, and we can simi\ larly infer the presence of yet more dark matter in between the galaxies in these clusters by its effect on t\ he motion of the galaxies. When we add up all this dark matter, we still get only about one tenth of the amount\ required to halt the expansion. However, we cannot exclude the possibility that there might be some other form of\ matter, distributed almost uniformly throughout the universe, that we have not yet detected and that might st\ ill raise the average density of the universe up to the critical value needed to halt the expansion. The pres\ ent evidence therefore suggests that the universe will probably expand forever, but all we can really be sure of \ is that even if the universe is going to recollapse, it won’t do so for at least another ten thousand million \ years, since it has already been expanding for at least that long. This should not unduly worry us: by that time, u\ nless we have colonized beyond the Solar System, mankind will long since have died out, extinguished along with o\ ur sun!

All of the Friedmann solutions have the feature that at some time in the\ past (between ten and twenty thousand million years ago) the distance between neighboring galaxies must have \ been zero. At that time, which we call the big bang, the density of the universe and the curvature of space-tim\ e would have been infinite. Because mathematics cannot really handle infinite numbers, this means that the g\ eneral theory of relativity (on which Friedmann’s solutions are based) predicts that there is a point in t\ he universe where the theory itself breaks down. Such a point is an example of what mathematicians call a singulari\ ty. In fact, all our theories of science are formulated on the assumption that space-time is smooth and nearly fi\ at, so they break down at the big bang singularity, where the curvature of space-time is infinite. This means t\ hat even if there were events before the big bang, one could not use them to determine what would happen afterwar\ d, because predictability would break down at the big bang.

Correspondingly, if, as is the case, we know only what has happened sinc\ e the big bang, we could not determine what happened beforehand. As far as we are concerned, events b\ efore the big bang can have no consequences, so they should not form part of a scientific model of the \ universe. We should therefore cut them out of the model and say that time had a beginning at the big bang.

Many people do not like the idea that time has a beginning, probably bec\ ause it smacks of divine intervention.

(The Catholic Church, on the other hand, seized on the big bang model a\ nd in 1951officially pronounced it to be in accordance with the Bible.) There were therefore a number of atte\ mpts to avoid the conclusion that there had been a big bang. The proposal that gained widest support was called \ the steady state theory. It was suggested in 1948 by two refugees from Nazi-occupied Austria, Hermann Bo\ ndi and Thomas Gold, together with a Briton, Fred Hoyle, who had worked with them on the development o\ f radar during the war. The idea was that as the galaxies moved away from each other, new galaxies were conti\ nually forming in the gaps in between, from new matter that was being continually created. The univers\ e would therefore look roughly the same at all times as well as at all points of space. The steady state th\ eory required a modification of general relativity to allow for the continual creation of matter, but the rate t\ hat was involved was so low (about one particle per cubic kilometer per year) that it was not in conflict with\ experiment. The theory was a good scientific theory, in the sense described in Chapter 1: it was simple and it made d\ efinite predictions that could be tested by observation. One of these predictions was that the number of galaxies\ or similar objects in any given volume of space should be the same wherever and whenever we look in the univers\ e. In the late 1950s and early 1960s a survey of sources of radio waves from outer space was carried ou\ t at Cambridge by a group of astronomers led by Martin Ryle (who had also worked with Bondi, Gold, a\ nd Hoyle on radar during the war).

The Cambridge group showed that most of these radio sources must lie out\ side our galaxy (indeed many of them could be identified with other galaxies) and also that there were \ many more weak sources than strong ones. They interpreted the weak sources as being the more distant ones, \ and the stronger ones as being nearer. Then there appeared to be less common sources per unit volume of\ space for the nearby sources than for the distant ones. This could mean that we are at the center of a gre\ at region in the universe in which the sources are fewer than elsewhere. Alternatively, it could mean that the \ sources were more numerous in the past, at the time that the radio waves left on their journey to us, than\ they are now. Either explanation contradicted the predictions of the steady state theory. Moreover, the d\ iscovery of the microwave radiation by Penzias and Wilson in 1965 also indicated that the universe must have be\ en much denser in the past. The steady state theory therefore had to be abandoned.

A Brief History of Time - Stephen Hawking... Chapter 3 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/b.html (7 of 9) [2/20/2001 3:14:24 AM] Another attempt to avoid the conclusion that there must have been a big \ bang, and therefore a beginning of time, was made by two Russian scientists, Evgenii Lifshitz and Isaac Kha\ latnikov, in 1963. They suggested that the big bang might be a peculiarity of Friedmann’s models alone, whic\ h after all were only approximations to the real universe. Perhaps, of all the models that were roughly like the\ real universe, only Friedmann’s would contain a big bang singularity. In Friedmann’s models, the galaxies a\ re all moving directly away from each other – so it is not surprising that at some time in the past they we\ re all at the same  place. In the real universe, however, the galaxies are not just moving directly away from each other \ – they also have small sideways velocities. So in reality they need never have been all at exactly the s\ ame place, only very close together.

Perhaps then the current expanding universe resulted not from a big bang\ singularity, but from an earlier contracting phase; as the universe had collapsed the particles in it mig\ ht not have all collided, but had flown past and then away from each other, producing the present expansion of t\ he the universe that were roughly like Friedmann’s models but took account of the irregularities and random \ velocities of galaxies in the real universe.

They showed that such models could start with a big bang, even though th\ e galaxies were no longer always moving directly away from each other, but they claimed that this was sti\ ll only possible in certain exceptional models in which the galaxies were all moving in just the right way. They\ argued that since there seemed to be infinitely more Friedmann-like models without a big bang singularity tha\ n there were with one, we should conclude that there had not in reality been a big bang. They later reali\ zed, however, that there was a much more general class of Friedmann-like models that did have singularities,\ and in which the galaxies did not have to be moving any special way. They therefore withdrew their claim in 197\ 0.

The work of Lifshitz and Khalatnikov was valuable because it showed that\ the universe could have had a singularity, a big bang, if the general theory of relativity was correct\ . However, it did not resolve the crucial question: Does general relativity predict that our universe should have had a big bang, a beginning of time?

The answer to this carne out of a completely different approach introduc\ ed by a British mathematician and physicist, Roger Penrose, in 1965. Using the way light cones behave in g\ eneral relativity, together with the fact that gravity is always attractive, he showed that a star collapsing unde\ r its own gravity is trapped in a region whose surface eventually shrinks to zero size. And, since the surface of\ the region shrinks to zero, so too must its volume. All the matter in the star will be compressed into a region \ of zero volume, so the density of matter and the curvature of space-time become infinite. In other words, one has\ a singularity contained within a region of space-time known as a black hole.

At first sight, Penrose’s result applied only to stars; it didn’t \ have anything to say about the question of whether the entire universe had a big bang singularity in its past. However, at \ the time that Penrose produced his theorem, I was a research student desperately looking for a problem with\ which to complete my Ph.D. thesis.

Two years before, I had been diagnosed as suffering from ALS, commonly k\ nown as Lou Gehrig’s disease, or motor neuron disease, and given to understand that I had only one or two\ more years to live. In these circumstances there had not seemed much point in working on my Ph.D.–\ I did not expect to survive that long.

Yet two years had gone by and I was not that much worse. In fact, things\ were going rather well for me and I had gotten engaged to a very nice girl, Jane Wilde. But in order to get \ married, I needed a job, and in order to get a job, I needed a Ph.D.

In 1965 I read about Penrose’s theorem that any body undergoing gravi\ tational collapse must eventually form a singularity. I soon realized that if one reversed the direction of time \ in Penrose’s theorem, so that the collapse became an expansion, the conditions of his theorem would still hold, pro\ vided the universe were roughly like a Friedmann model on large scales at the present time. Penrose’s theore\ m had shown that any collapsing star must end in a singularity; the time-reversed argument showed that any Friedma\ nn-like expanding universe must have begun with a singularity. For technical reasons, Penrose’s theor\ em required that the universe be infinite in space. So I could in fact, use it to prove that there should\ be a singularity only if the universe was expanding fast enough to avoid collapsing again (since only those Fried\ mann models were infinite in space).

During the next few years I developed new mathematical techniques to rem\ ove this and other technical conditions from the theorems that proved that singularities must occur. \ The final result was a joint paper by Penrose and myself in 1970, which at last proved that there must have be\ en a big bang singularity provided only that general relativity is correct and the universe contains as muc\ h matter as we observe. There was a lot of opposition to our work, partly from the Russians because of their Mar\ xist belief in scientific determinism, and partly from people who felt that the whole idea of singularities was rep\ ugnant and spoiled the beauty of Einstein’s theory. However, one cannot really argue with a mathematic\ al theorem. So in the end our work A Brief History of Time - Stephen Hawking... Chapter 3 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/b.html (8 of 9) [2/20/2001 3:14:24 AM] became generally accepted and nowadays nearly everyone assumes that the \ universe started with a big bang singularity. It is perhaps ironic that, having changed my mind, I am now\ trying to convince other physicists that there was in fact no singularity at the beginning of the universe – a\ s we shall see later, it can disappear once quantum effects are taken into account.

We have seen in this chapter how, in less than half a century, man’s \ view of the universe formed over millennia has been transformed. Hubble’s discovery that the universe was expand\ ing, and the realization of the insignificance of our own planet in the vastness of the universe, were j\ ust the starting point. As experimental and theoretical evidence mounted, it became more and more clear that the\ universe must have had a beginning in time, until in 1970 this was finally proved by Penrose and \ myself, on the basis of Einstein’s general theory of relativity. That proof showed that general relativity is only \ an incomplete theory: it cannot tell us how the universe started off, because it predicts that all physical theories\ , including itself, break down at the beginning of the universe. However, general relativity claims to be only\ a partial theory, so what the singularity theorems really show is that there must have been a time in the very ear\ ly universe when the universe was so small that one could no longer ignore the small-scale effects of the oth\ er great partial theory of the twentieth century, quantum mechanics. At the start of the 1970s, then, we were for\ ced to turn our search for an understanding of the universe from our theory of the extraordinarily vas\ t to our theory of the extraordinarily tiny.

That theory, quantum mechanics, will be described next, before we turn t\ o the efforts to combine the two partial theories into a single quantum theory of gravity.

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Chapter 3 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/b.html (9 of 9) [2/20/2001 3:14:24 AM] CHAPTER 4 THE UNCERTAINTY PRINCIPLE   The success of scientific theories, particularly Newton’s theory of g\ ravity, led the French scientist the Marquis de Laplace at the beginning of the nineteenth century to argue that the universe wa\ s completely deterministic. Laplace suggested that there should be a set of scientific laws that would allow us to pre\ dict everything that would happen in the universe, if only we knew the complete state of the universe at one time. For example\ , if we knew the positions and speeds of the sun and the planets at one time, then we could use Newton’s laws to c\ alculate the state of the Solar System at any other time. Determinism seems fairly obvious in this case, but Laplace went fu\ rther to assume that there were similar laws governing everything else, including human behavior.

The doctrine of scientific determinism was strongly resisted by many peo\ ple, who felt that it infringed God’s freedom to intervene in the world, but it remained the standard assumption of scien\ ce until the early years of this century. One of the first indications that this belief would have to be abandoned came when \ calculations by the British scientists Lord Rayleigh and Sir James Jeans suggested that a hot object, or body, such \ as a star, must radiate energy at an infinite rate. According to the laws we believed at the time, a hot body ought to\ give off electromagnetic waves (such as radio waves, visible light, or X rays) equally at all frequencies. For exampl\ e, a hot body should radiate the same amount of energy in waves with frequencies between one and two million million wav\ es a second as in waves with frequencies between two and three million million waves a second. Now since the numb\ er of waves a second is unlimited, this would mean that the total energy radiated would be infinite.

In order to avoid this obviously ridiculous result, the German scientist\ Max Planck suggested in 1900 that light, X rays, and other waves could not be emitted at an arbitrary rate, but only in c\ ertain packets that he called quanta. Moreover, each quantum had a certain amount of energy that was greater the higher \ the frequency of the waves, so at a high enough frequency the emission of a single quantum would require more ene\ rgy than was available. Thus the radiation at high frequencies would be reduced, and so the rate at which the body los\ t energy would be finite.

The quantum hypothesis explained the observed rate of emission of radiat\ ion from hot bodies very well, but its implications for determinism were not realized until 1926, when another \ German scientist, Werner Heisenberg, formulated his famous uncertainty principle. In order to predict the fut\ ure position and velocity of a particle, one has to be able to measure its present position and velocity accurately. The obviou\ s way to do this is to shine light on the particle.

Some of the waves of light will be scattered by the particle and this wi\ ll indicate its position. However, one will not be able to determine the position of the particle more accurately than the dista\ nce between the wave crests of light, so one needs to use light of a short wavelength in order to measure the position of t\ he particle precisely. Now, by Planck’s quantum hypothesis, one cannot use an arbitrarily small amount of light; one has\ to use at least one quantum. This quantum will disturb the particle and change its velocity in a way that cannot be pre\ dicted. moreover, the more accurately one measures the position, the shorter the wavelength of the light that one \ needs and hence the higher the energy of a single quantum. So the velocity of the particle will be disturbed by a larger a\ mount. In other words, the more accurately you try to measure the position of the particle, the less accurately you can mea\ sure its speed, and vice versa. Heisenberg showed that the uncertainty in the position of the particle times the un\ certainty in its velocity times the mass of the particle can never be smaller than a certain quantity, which is known as\ Planck’s constant. Moreover, this limit does not depend on the way in which one tries to measure the position or velocity\ of the particle, or on the type of particle:

Heisenberg’s uncertainty principle is a fundamental, inescapable prop\ erty of the world.

The uncertainty principle had profound implications for the way in which\ we view the world. Even after more than seventy years they have not been fully appreciated by many philosophers, and are\ still the subject of much controversy. The uncertainty principle signaled an end to Laplace’s dream of a theory \ of science, a model of the universe that would be completely deterministic: one certainly cannot predict future events exa\ ctly if one cannot even measure the present state of the universe precisely! We could still imagine that there is a set of\ laws that determine events completely for some supernatural being, who could observe the present state of the universe \ without disturbing it. However, such models of the universe are not of much interest to us ordinary mortals. It seems b\ etter to employ the principle of economy known as Occam’s razor and cut out all the features of the theory that cannot \ be observed. This approach led Heisenberg, Erwin Schrodinger, and Paul Dirac in the 1920s to reformulate mechanics into a\ new theory called quantum mechanics, based on the uncertainty principle. In this theory particles no longer had sep\ arate, well-defined positions and velocities that could not be observed, Instead, they had a quantum state, which was a co\ mbination of position and velocity.

In general, quantum mechanics does not predict a single definite result \ for an observation. Instead, it predicts a number of different possible outcomes and tells us how likely each of these is.\ That is to say, if one made the same measurement on a large number of similar systems, each of which started off in the s\ ame way, one would find that the result of the A Brief History of Time - Stephen Hawking... Chapter 4 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/c.html (1 of 5) [2/20/2001 3:14:40 AM] measurement would be A in a certain number of cases, B in a different nu\ mber, and so on. One could predict the approximate number of times that the result would be A or B, but one cou\ ld not predict the specific result of an individual measurement. Quantum mechanics therefore introduces an unavoidable eleme\ nt of unpredictability or randomness into science. Einstein objected to this very strongly, despite the important \ role he had played in the development of these ideas. Einstein was awarded the Nobel Prize for his contribution to quan\ tum theory. Nevertheless, Einstein never accepted that the universe was governed by chance; his feelings were sum\ med up in his famous statement “God does not play dice.” Most other scientists, however, were willing to accep\ t quantum mechanics because it agreed perfectly with experiment. Indeed, it has been an outstandingly successful theory and u\ nderlies nearly all of modern science and technology. It governs the behavior of transistors and integrated circui\ ts, which are the essential components of electronic devices such as televisions and computers, and is also the ba\ sis of modern chemistry and biology. The only areas of physical science into which quantum mechanics has not yet been \ properly incorporated are gravity and the large-scale structure of the universe.

Although light is made up of waves, Planck’s quantum hypothesis tells\ us that in some ways it behaves as if it were composed of particles: it can be emitted or absorbed only in packets, or\ quanta. Equally, Heisenberg’s uncertainty principle implies that particles behave in some respects like waves: the\ y do not have a definite position but are “smeared out” with a certain probability distribution. The theory of quantum m\ echanics is based on an entirely new type of mathematics that no longer describes the real world in terms of particle\ s and waves; it is only the observations of the world that may be described in those terms. There is thus a duality betw\ een waves and particles in quantum mechanics:

for some purposes it is helpful to think of particles as waves and for o\ ther purposes it is better to think of waves as particles. An important consequence of this is that one can observe what\ is called interference between two sets of waves or particles. That is to say, the crests of one set of waves may c\ oincide with the troughs of the other set. The two sets of waves then cancel each other out rather than adding up to a stro\ nger wave as one might expect Figure 4:1. Figure 4:1 A familiar example of interference in the case of light is the colors th\ at are often seen in soap bubbles. These are caused by reflection of light from the two sides of the thin film of water form\ ing the bubble. White light consists of light waves of all different wavelengths, or colors, For certain wavelengths the crests\ of the waves reflected from one side of the soap film coincide with the troughs reflected from the other side. The colors\ corresponding to these wavelengths are absent from the reflected light, which therefore appears to be colored. Interfe\ rence can also occur for particles, because of the duality introduced by quantum mechanics. A famous example is the so-call\ ed two-slit experiment Figure 4:2. A Brief History of Time - Stephen Hawking... Chapter 4 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/c.html (2 of 5) [2/20/2001 3:14:40 AM] Figure 4:2 Consider a partition with two narrow parallel slits in it. On one side o\ f the partition one places a source of fight of a particular color (that is, of a particular wavelength). Most of the li\ ght will hit the partition, but a small amount will go through the slits. Now suppose one places a screen on the far side of th\ e partition from the light. Any point on the screen will receive waves from the two slits. However, in general, the distance\ the light has to travel from the source to the screen via the two slits will be different. This will mean that the wave\ s from the slits will not be in phase with each other when they arrive at the screen: in some places the waves will cancel eac\ h other out, and in others they will reinforce each other. The result is a characteristic pattern of light and dark fri\ nges.

The remarkable thing is that one gets exactly the same kind of fringes i\ f one replaces the source of light by a source of particles such as electrons with a definite speed (this means that the \ corresponding waves have a definite length). It seems the more peculiar because if one only has one slit, one does not g\ et any fringes, just a uniform distribution of electrons across the screen. One might therefore think that opening anot\ her slit would just increase the number of electrons hitting each point of the screen, but, because of interference\ , it actually decreases it in some places. If electrons are sent through the slits one at a time, one would expect eac\ h to pass through one slit or the other, and so behave just as if the slit it passed through were the only one there –\ giving a uniform distribution on the screen. In reality, however, even when the electrons are sent one at a time, the fringes sti\ ll appear. Each electron, therefore, must be A Brief History of Time - Stephen Hawking... Chapter 4 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/c.html (3 of 5) [2/20/2001 3:14:40 AM] passing through both slits at the same time!

The phenomenon of interference between particles has been crucial to our\ understanding of the structure of atoms, the basic units of chemistry and biology and the building blocks out of whic\ h we, and everything around us, are made. At the beginning of this century it was thought that atoms were rather like the\ planets orbiting the sun, with electrons (particles of negative electricity) orbiting around a central nucleus, which carri\ ed positive electricity. The attraction between the positive and negative electricity was supposed to keep the electrons in \ their orbits in the same way that the gravitational attraction between the sun and the planets keeps the planets in their or\ bits. The trouble with this was that the laws of mechanics and electricity, before quantum mechanics, predicted that the \ electrons would lose energy and so spiral inward until they collided with the nucleus. This would mean that the at\ om, and indeed all matter, should rapidly collapse to a state of very high density. A partial solution to this problem was \ found by the Danish scientist Niels Bohr in 1913. He suggested that maybe the electrons were not able to orbit at just any di\ stance from the central nucleus but only at certain specified distances. If one also supposed that only one or two electrons\ could orbit at any one of these distances, this would solve the problem of the collapse of the atom, because the electro\ ns could not spiral in any farther than to fill up the orbits with e least distances and energies.

This model explained quite well the structure of the simplest atom, hydr\ ogen, which has only one electron orbiting around the nucleus. But it was not clear how one ought to extend it to more com\ plicated atoms. Moreover, the idea of a limited set of allowed orbits seemed very arbitrary. The new theory of quantum m\ echanics resolved this difficulty. It revealed that an electron orbiting around the nucleus could be thought of as a wave, w\ ith a wavelength that depended on its velocity.

For certain orbits, the length of the orbit would correspond to a whole \ number (as opposed to a fractional number) of wavelengths of the electron. For these orbits the wave crest would be in\ the same position each time round, so the waves would add up: these orbits would correspond to Bohr’s allowed o\ rbits. However, for orbits whose lengths were not a whole number of wavelengths, each wave crest would eventually be cance\ led out by a trough as the electrons went round; these orbits would not be allowed.

A nice way of visualizing the wave/particle duality is the so-called sum\ over histories introduced by the American scientist Richard Feynman. In this approach the particle is not supposed to have a\ single history or path in space-time, as it would in a classical, nonquantum theory. Instead it is supposed to go from A t\ o B by every possible path. With each path there are associated a couple of numbers: one represents the size of a wave an\ d the other represents the position in the cycle (i.e., whether it is at a crest or a trough). The probability of going\ from A to B is found by adding up the waves for all the paths. In general, if one compares a set of neighboring paths, the phase\ s or positions in the cycle will differ greatly. This means that the waves associated with these paths will almost exactly can\ cel each other out. However, for some sets of neighboring paths the phase will not vary much between paths. The waves \ for these paths will not cancel out Such paths correspond to Bohr’s allowed orbits.

With these ideas, in concrete mathematical form, it was relatively strai\ ghtforward to calculate the allowed orbits in more complicated atoms and even in molecules, which are made up of a number o\ f atoms held together by electrons in orbits that go round more than one nucleus. Since the structure of molecules an\ d their reactions with each other underlie all of chemistry and biology, quantum mechanics allows us in principle to predi\ ct nearly everything we see around us, within the limits set by the uncertainty principle. (In practice, however, the\ calculations required for systems containing more than a few electrons are so complicated that we cannot do them.) Einstein’s general theory of relativity seems to govern the large-sca\ le structure of the universe. It is what is called a classical theory; that is, it does not take account of the uncertainty p\ rinciple of quantum mechanics, as it should for consistency with other theories. The reason that this does not lead to a\ ny discrepancy with observation is that all the gravitational fields that we normally experience are very weak. How-ever\ , the singularity theorems discussed earlier indicate that the gravitational field should get very strong in at least\ two situations, black holes and the big bang. In such strong fields the effects of quantum mechanics should be important. Thus\ , in a sense, classical general relativity, by predicting points of infinite density, predicts its own downfall, just a\ s classical (that is, nonquantum) mechanics predicted its downfall by suggesting that atoms should collapse to infinite densit\ y. We do not yet have a complete consistent theory that unifies general relativity and quantum mechanics, but we do know a \ number of the features it should have. The consequences that these would have for black holes and the big bang will\ be described in later chapters. For the moment, however, we shall turn to the recent attempts to bring together \ our understanding of the other forces of nature into a single, unified quantum theory.

  PREVIOUS NEXT   A Brief History of Time - Stephen Hawking... Chapter 4 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/c.html (4 of 5) [2/20/2001 3:14:40 AM]   A Brief History of Time - Stephen Hawking... Chapter 4 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/c.html (5 of 5) [2/20/2001 3:14:40 AM] CHAPTER 5 ELEMENTARY PARTICLES AND THE FORCES OF NATURE   Aristotle believed that all the matter in the universe was made up of fo\ ur basic elements – earth, air, fire, and water.

These elements were acted on by two forces: gravity, the tendency for ea\ rth and water to sink, and levity, the tendency for air and fire to rise. This division of the contents of the universe \ into matter and forces is still used today. Aristotle believed that matter was continuous, that is, one could divide a piece o\ f matter into smaller and smaller bits without any limit: one never came up against a grain of matter that could not be div\ ided further. A few Greeks, however, such as Democritus, held that matter was inherently grainy and that everything w\ as made up of large numbers of various different kinds of atoms. (The word atom means “indivisible” in Greek.) For centuries the argument continu\ ed without any real evidence on either side, but in 1803 the British chemist and physicist J\ ohn Dalton pointed out that the fact that chemical compounds always combined in certain proportions could be explained by t\ he grouping together of atoms to form units called molecules. However, the argument between the two schools of thoug\ ht was not finally settled in favor of the atomists until the early years of this century. One of the important pie\ ces of physical evidence was provided by Einstein.

In a paper written in 1905, a few weeks before the famous paper on speci\ al relativity, Einstein pointed out that what was called Brownian motion – the irregular, random motion of small partic\ les of dust suspended in a liquid – could be explained as the effect of atoms of the liquid colliding with the dust p\ articles.

By this time there were already suspicions that these atoms were not, af\ ter all, indivisible. Several years previously a fellow of Trinity College, Cambridge, J. J. Thomson, had demonstrated th\ e existence of a particle of matter, called the electron, that had a mass less than one thousandth of that of the lighte\ st atom. He used a setup rather like a modern TV picture tube: a red-hot metal filament gave off the electrons, and becau\ se these have a negative electric charge, an electric field could be used to accelerate them toward a phosphor-coated\ screen. When they hit the screen, flashes of light were generated. Soon it was realized that these electrons must be \ coming from within the atoms themselves, and in 1911 the New Zealand physicist Ernest Rutherford finally showed that the\ atoms of matter do have internal structure:

they are made up of an extremely tiny, positively charged nucleus, aroun\ d which a number of electrons orbit. He deduced this by analyzing the way in which alpha-particles, which are positively\ charged particles given off by radioactive atoms, are deflected when they collide with atoms.

At first it was thought that the nucleus of the atom was made up of elec\ trons and different numbers of a positively charged particle called the proton, from the Greek word meaning “firs\ t,” because it was believed to be the fundamental unit from which matter was made. However, in 1932 a colleague of Rutherf\ ord’s at Cambridge, James Chadwick, discovered that the nucleus contained another particle, called the neutr\ on, which had almost the same mass as a proton but no electrical charge. Chadwick received the Nobel Prize for his disc\ overy, and was elected Master of Gonville and Caius College, Cambridge (the college of which I am now a fellow). He \ later resigned as Master because of disagreements with the Fellows. There had been a bitter dispute in the c\ ollege ever since a group of young Fellows returning after the war had voted many of the old Fellows out of the col\ lege offices they had held for a long time. This was before my time; I joined the college in 1965 at the tail end of the \ bitterness, when similar disagreements forced another Nobel Prize – winning Master, Sir Nevill Mott, to resign.

Up to about thirty years ago, it was thought that protons and neutrons w\ ere “elementary” particles, but experiments in which protons were collided with other protons or electrons at high spee\ ds indicated that they were in fact made up of smaller particles. These particles were named quarks by the Caltech phys\ icist Murray Gell-Mann, who won the Nobel Prize in 1969 for his work on them. The origin of the name is an enigmat\ ic quotation from James Joyce: “Three quarks for Muster Mark!” The word quark is supposed to be pronounced like quart, but with a k at the end instead of a t, but is usually pronounced to rhyme with lark.

There are a number of different varieties of quarks: there are six “f\ lavors,” which we call up, down, strange, charmed, bottom, and top. The first three flavors had been known since the 1960s \ but the charmed quark was discovered only in 1974, the bottom in 1977, and the top in 1995. Each flavor comes in thre\ e “colors,” red, green, and blue. (It should be emphasized that these terms are just labels: quarks are much smaller tha\ n the wavelength of visible light and so do not have any color in the normal sense. It is just that modern physicists se\ em to have more imaginative ways of naming new particles and phenomena – they no longer restrict themselves to Greek\ !) A proton or neutron is made up of three quarks, one of each color. A proton contains two up quarks and one down quark; a\ neutron contains two down and one up. We can create particles made up of the other quarks (strange, charmed, bot\ tom, and top), but these all have a much greater mass and decay very rapidly into protons and neutrons.

We now know that neither the atoms nor the protons and neutrons within t\ hem are indivisible. So the question is: what are the truly elementary particles, the basic building blocks from which\ everything is made? Since the wavelength of light A Brief History of Time - Stephen Hawking... Chapter 5 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/d.html (1 of 8) [2/20/2001 3:14:54 AM] is much larger than the size of an atom, we cannot hope to “look” \ at the parts of an atom in the ordinary way. We need to use something with a much smaller wave-length. As we saw in the last cha\ pter, quantum mechanics tells us that all particles are in fact waves, and that the higher the energy of a particl\ e, the smaller the wavelength of the corresponding wave. So the best answer we can give to our question depends on how high\ a particle energy we have at our disposal, because this determines on how small a length scale we can look. These p\ article energies are usually measured in units called electron volts. (In Thomson’s experiments with electrons, we \ saw that he used an electric field to accelerate the electrons. The energy that an electron gains from an electric field of o\ ne volt is what is known as an electron volt.) In the nineteenth century, when the only particle energies that people knew how\ to use were the low energies of a few electron volts generated by chemical reactions such as burning, it was thought th\ at atoms were the smallest unit. In Rutherford’s experiment, the alpha-particles had energies of millions of electron vol\ ts. More recently, we have learned how to use electromagnetic fields to give particles energies of at first millions a\ nd then thousands of millions of electron volts. And so we know that particles that were thought to be “elementary” thirty\ years ago are, in fact, made up of smaller particles.

May these, as we go to still higher energies, in turn be found to be mad\ e from still smaller particles? This is certainly possible, but we do have some theoretical reasons for believing that we \ have, or are very near to, a knowledge of the ultimate building blocks of nature.

Using the wave/particle duality discussed in the last chapter, every-thi\ ng in the universe, including light and gravity, can be described in terms of particles. These particles have a property call\ ed spin. One way of thinking of spin is to imagine the particles as little tops spinning about an axis. However, this can b\ e misleading, because quantum mechanics tells us that the particles do not have any well-defined axis. What the spin of a\ particle really tells us is what the particle looks like from different directions. A particle of spin 0 is like a dot: it looks \ the same from every direction Figure 5:1-i. On the other hand, a particle of spin 1 is like an arrow: it looks different from dif\ ferent directions Figure 5:1-ii. Only if one turns it round a complete revolution (360 degrees) does the particle look the same. A\ particle of spin 2 is like a double-headed arrow Figure 5:1-iii : it looks the same if one turns it round half a revolution (180 degree\ s). Similarly, higher spin particles look the same if one turns them through smaller fractions of a complete revol\ ution. All this seems fairly straightforward, but the remark-able fact is that there are particles that do not look the same i\ f one turns them through just one revolution: you have to turn them through two complete revolutions! Such particles are s\ aid to have spin ½.

A Brief History of Time - Stephen Hawking... Chapter 5 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/d.html (2 of 8) [2/20/2001 3:14:54 AM] Figure 5:1 All the known particles in the universe can be divided into two groups: \ particles of spin ½, which make up the matter in the universe, and particles of spin 0, 1, and 2, which, as we shall see,\ give rise to forces between the matter particles.

The matter particles obey what is called Pauli’s exclusion principle.\ This was discovered in 1925 by an Austrian physicist, Wolfgang Pauli – for which he received the Nobel Prize in 1945. He wa\ s the archetypal theoretical physicist: it was said of him that even his presence in the same town would make experiments go\ wrong! Pauli’s exclusion principle says that two similar particles can-not exist in the same state; that is, they can\ not have both the same position and the same velocity, within the limits given by the uncertainty principle. The excl\ usion principle is crucial because it explains why matter particles do not collapse to a state of very high density under t\ he influence of the forces produced by the particles of spin 0, 1, and 2: if the matter particles have very nearly the same p\ ositions, they must have different velocities, which means that they will not stay in the same position for long. If the worl\ d had been created without the exclusion principle, quarks would not form separate, well-defined protons and neutrons. Nor w\ ould these, together with electrons, form separate, well-defined atoms. They would all collapse to form a roughly \ uniform, dense “soup.” A proper understanding of the electron and other spin-½ particles did\ not come until 1928, when a theory was proposed by Paul Dirac, who later was elected to the Lucasian Professorship of Ma\ thematics at Cambridge (the same professorship that Newton had once held and that I now hold). Dirac’\ s theory was the first of its kind that was consistent with both quantum mechanics and the special theory of relativity. It exp\ lained mathematically why the electron had spin-½; that is, why it didn’t look the same if you turned it thro\ ugh only one complete revolution, but did if you turned it through two revolutions. It also predicted that the electron should have\ a partner: an anti-electron, or positron. The discovery of the positron in 1932 confirmed Dirac’s theory and led to\ his being awarded the Nobel Prize for physics in 1933. We now know that every particle has an antiparticle, with which it\ can annihilate. (In the case of the force-carrying particles, the antiparticles are the same as the particles themselves.)\ There could be whole antiworlds and antipeople made out of antiparticles. However, if you meet your antiself, don’t \ shake hands! You would both vanish in a great flash of light. The question of why there seem to be so many more particles th\ an antiparticles around us is extremely A Brief History of Time - Stephen Hawking... Chapter 5 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/d.html (3 of 8) [2/20/2001 3:14:54 AM] important, and I shall return to it later in the chapter.

In quantum mechanics, the forces or interactions between matter particle\ s are all supposed to be carried by particles of integer spin – 0, 1, or 2. What happens is that a matter particle, su\ ch as an electron or a quark, emits a force-carrying particle. The recoil from this emission changes the velocity of the matt\ er particle. The force-carrying particle then collides with another matter particle and is absorbed. This collision changes the\ velocity of the second particle, just as if there had been a force between the two matter particles. It is an important proper\ ty of ' the force-carrying particles that they do not obey the exclusion principle. This means that there is no limit to the n\ umber that can be exchanged, and so they can give rise to a strong force. However, if the force-carrying particles have a \ high mass, it will be difficult to produce and exchange them over a large distance. So the forces that they carry will \ have only a short range. On the other hand, if the force-carrying particles have no mass of their own, the forces will be l\ ong range. The force-carrying particles exchanged between matter particles are said to be virtual particles because, unlik\ e “real” particles, they cannot be directly detected by a particle detector. We know they exist, however, because they do hav\ e a measurable effect: they give rise to forces between matter particles. Particles of spin 0, 1, or 2 do also exist in \ some circumstances as real particles, when they can be directly detected. They then appear to us as what a classical physici\ st would call waves, such as waves of light or gravitational waves. They may sometimes be emitted when matter particles\ interact with each other by exchanging virtual force-carrying particles. (For example, the electric repulsive force be\ tween two electrons is due to the exchange of virtual photons, which can never be directly detected; but if one electron moves\ past another, real photons may be given off, which we detect as light waves.) Force-carrying particles can be grouped into four categories according t\ o the strength of the force that they carry and the particles with which they interact. It should be emphasized that this di\ vision into four classes is man-made; it is convenient for the construction of partial theories, but it may not corr\ espond to anything deeper. Ultimately, most physicists hope to find a unified theory that will explain all four forc\ es as different aspects of a single force. Indeed, many would say this is the prime goal of physics today. Recently, successful \ attempts have been made to unify three of the four categories of force – and I shall describe these in this chapter\ . The question of the unification of the remaining category, gravity, we shall leave till later.

The first category is the gravitational force. This force is universal, \ that is, every particle feels the force of gravity, according to its mass or energy. Gravity is the weakest of the four forc\ es by a long way; it is so weak that we would not notice it at all were it not for two special properties that it has: it \ can act over large distances, and it is always attractive.

This means that the very weak gravitational forces between the individua\ l particles in two large bodies, such as the earth and the sun, can all add up to produce a significant force. The other th\ ree forces are either short range, or are sometimes attractive and some-times repulsive, so they tend to cancel out. In the \ quantum mechanical way of looking at the gravitational field, the force between two matter particles is pictured \ as being carried by a particle of spin 2 called the graviton. This has no mass of its own, so the force that it carries is l\ ong range. The gravitational force between the sun and the earth is ascribed to the exchange of gravitons between the parti\ cles that make up these two bodies. Although the exchanged particles are virtual, they certainly do produce a measurable \ effect – they make the earth orbit the sun! Real gravitons make up what classical physicists would call gravitational wav\ es, which are very weak – and so difficult to detect that they have not yet been observed.

The next category is the electromagnetic force, which interacts with ele\ ctrically charged particles like electrons and quarks, but not with uncharged particles such as gravitons. It is much s\ tronger than the gravitational force: the electromagnetic force between two electrons is about a million million m\ illion million million million million (1 with forty-two zeros after it) times bigger than the gravitational force. However, the\ re are two kinds of electric charge, positive and negative. The force between two positive charges is repulsive, as is the\ force between two negative charges, but the force is attractive between a positive and a negative charge. A large bo\ dy, such as the earth or the sun, contains nearly equal numbers of positive and negative charges. Thus the attractive and \ repulsive forces between the individual particles nearly cancel each other out, and there is very little net electromagnet\ ic force. However, on the small scales of atoms and molecules, electromagnetic forces dominate. The electromagnetic attr\ action between negatively charged electrons and positively charged protons in the nucleus causes the electrons to or\ bit the nucleus of the atom, just as gravitational attraction causes the earth to orbit the sun. The electromagnetic attrac\ tion is pictured as being caused by the exchange of large numbers of virtual massless particles of spin 1, called photons\ . Again, the photons that are exchanged are virtual particles. However, when an electron changes from one allowed orbit to a\ nother one nearer to the nucleus, energy is released and a real photon is emitted – which can be observed as visi\ ble light by the human eye, if it has the right wave-length, or by a photon detector such as photographic film. Equally,\ if a real photon collides with an atom, it may move an electron from an orbit nearer the nucleus to one farther away. T\ his uses up the energy of the photon, so it is absorbed.

The third category is called the weak nuclear force, which is responsibl\ e for radioactivity and which acts on all matter particles of spin-½, but not on particles of spin 0, 1, or 2, such as\ photons and gravitons. The weak nuclear force was not well understood until 1967, when Abdus Salam at Imperial College, London\ , and Steven Weinberg at Harvard both A Brief History of Time - Stephen Hawking... Chapter 5 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/d.html (4 of 8) [2/20/2001 3:14:54 AM] proposed theories that unified this interaction with the electromagnetic\ force, just as Maxwell had unified electricity and magnetism about a hundred years earlier. They suggested that in addition\ to the photon, there were three other spin-1 particles, known collectively as massive vector bosons, that carried the\ weak force. These were called W + (pronounced W plus), W - (pronounced W minus), and Zº (pronounced Z naught), and each had\ a mass of around 100 GeV (GeV stands for gigaelectron-volt, or one thousand million electron volts). \ The Weinberg-Salam theory exhibits a property known as spontaneous symmetry breaking. This means that what appear to b\ e a number of completely different particles at low energies are in fact found to be all the same type of particle, o\ nly in different states. At high energies all these particles behave similarly. The effect is rather like the behavior of a \ roulette ball on a roulette wheel. At high energies (when the wheel is spun quickly) the ball behaves in essentially only \ one way – it rolls round and round. But as the wheel slows, the energy of the ball decreases, and eventually the ball drops i\ nto one of the thirty-seven slots in the wheel. In other words, at low energies there are thirty-seven different states in \ which the ball can exist. If, for some reason, we could only observe the ball at low energies, we would then think that th\ ere were thirty-seven different types of ball!

In the Weinberg-Salam theory, at energies much greater than 100 GeV, the\ three new particles and the photon would all behave in a similar manner. But at the lower particle energies that occu\ r in most normal situations, this symmetry between the particles would be broken. WE, W, and Zº would acquire la\ rge masses, making the forces they carry have a very short range. At the time that Salam and Weinberg proposed their the\ ory, few people believed them, and particle accelerators were not powerful enough to reach the energies of 100 GeV r\ equired to produce real W +, W -, or Zº particles.

However, over the next ten years or so, the other predictions of the the\ ory at lower energies agreed so well with experiment that, in 1979, Salam and Weinberg were awarded the Nobel Priz\ e for physics, together with Sheldon Glashow, also at Harvard, who had suggested similar unified theories of \ the electromagnetic and weak nuclear forces.

The Nobel committee was spared the embarrassment of having made a mistak\ e by the discovery in 1983 at CERN (European Centre for Nuclear Research) of the three massive partners o\ f the photon, with the correct predicted masses and other properties. Carlo Rubbia, who led the team of several hundred \ physicists that made the discovery, received the Nobel Prize in 1984, along with Simon van der Meer, the CERNengineer who\ developed the antimatter storage system employed. (It is very difficult to make a mark in experimental physics \ these days unless you are already at the top! ) The fourth category is the strong nuclear force, which holds the quarks \ together in the proton and neutron, and holds the protons and neutrons together in the nucleus of an atom. It is believed \ that this force is carried by another spin-1 particle, called the gluon, which interacts only with itself and with the quarks. \ The strong nuclear force has a curious property called confinement: it always binds particles together into combinations\ that have no color. One cannot have a single quark on its own because it would have a color (red, green, or blue). \ Instead, a red quark has to be joined to a green and a blue quark by a “string” of gluons (red + green + blue = white)\ . Such a triplet constitutes a proton or a neutron. Another possibility is a pair consisting of a quark and an antiquark (red + ant\ ired, or green + antigreen, or blue + antiblue = white).

Such combinations make up the particles known as mesons, which are unsta\ ble because the quark and antiquark can annihilate each other, producing electrons and other particles. Similarl\ y, confinement prevents one having a single gluon on its own, because gluons also have color. Instead, one has to have a c\ ollection of gluons whose colors add up to white.

Such a collection forms an unstable particle called a glueball.

The fact that confinement prevents one from observing an isolated quark \ or gluon might seem to make the whole notion of quarks and gluons as particles somewhat metaphysical. However, there \ is another property of the strong nuclear force, called asymptotic freedom, that makes the concept of quarks and g\ luons well defined. At normal energies, the strong nuclear force is indeed strong, and it binds the quarks tightly t\ ogether. However, experiments with large particle accelerators indicate that at high energies the strong force becomes muc\ h weaker, and the quarks and gluons behave almost like free particles. A Brief History of Time - Stephen Hawking... Chapter 5 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/d.html (5 of 8) [2/20/2001 3:14:54 AM] Figure 5:2 Figure 5:2 shows a photograph of a collision between a high-energy proton and anti\ proton. The success of the unification of the electromagnetic and weak nuclear forces led to a number of attemp\ ts to combine these two forces with the strong nuclear force into what is called a grand unified theory (or GUT). Thi\ s title is rather an exaggeration: the resultant theories are not all that grand, nor are they fully unified, as they do not inclu\ de gravity. Nor are they really complete theories, because they contain a number of parameters whose values cannot be predi\ cted from the theory but have to be chosen to fit in with experiment. Nevertheless, they may be a step toward a com\ plete, fully unified theory. The basic idea of GUTs is as follows: as was mentioned above, the strong nuclear force get\ s weaker at high energies. On the other hand, the electromagnetic and weak forces, which are not asymptotically free, \ get stronger at high energies. At some very high energy, called the grand unification energy, these three forces would al\ l have the same strength and so could just be different aspects of a single force. The GUTs also predict that at this \ energy the different spin-½ matter particles, like quarks and electrons, would also all be essentially the same, thus achie\ ving another unification.

The value of the grand unification energy is not very well known, but it\ would probably have to be at least a thousand million million GeV. The present generation of particle accelerators can\ collide particles at energies of about one hundred GeV, and machines are planned that would raise this to a few thousand Ge\ V. But a machine that was powerful enough to accelerate particles to the grand unification energy would have to be as\ big as the Solar System – and would be unlikely to be funded in the present economic climate. Thus it is impossible to t\ est grand unified theories directly in the laboratory.

However, just as in the case of the electromagnetic and weak unified the\ ory, there are low-energy consequences of the theory that can be tested.

The most interesting of these is the prediction that protons, which make\ up much of the mass of ordinary matter, can spontaneously decay into lighter particles such as antielectrons. The re\ ason this is possible is that at the grand unification energy there is no essential difference between a quark and \ an antielectron. The three quarks inside a proton normally do not have enough energy to change into antielectrons, but ver\ y occasionally one of them may acquire A Brief History of Time - Stephen Hawking... Chapter 5 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/d.html (6 of 8) [2/20/2001 3:14:54 AM] sufficient energy to make the transition because the uncertainty princip\ le means that the energy of the quarks inside the proton cannot be fixed exactly. The proton would then decay. The probabi\ lity of a quark gaining sufficient energy is so low that one is likely to have to wait at least a million million millio\ n million million years (1 followed by thirty zeros). This is much longer than the time since the big bang, which is a mere ten thousa\ nd million years or so (1 followed by ten zeros).

Thus one might think that the possibility of spontaneous proton decay co\ uld not be tested experimentally. However, one can increase one’s chances of detecting a decay by observing a large \ amount of matter containing a very large number of protons. (If, for example, one observed a number of protons equal to\ 1 followed by thirty-one zeros for a period of one year, one would expect, according to the simplest GUT, to observe more t\ han one proton decay.) A number of such experiments have been carried out, but none have yielde\ d definite evidence of proton or neutron decay. One experiment used eight thousand tons of water and was performe\ d in the Morton Salt Mine in Ohio (to avoid other events taking place, caused by cosmic rays, that might be confused\ with proton decay). Since no spontaneous proton decay had been observed during the experiment, one can calculate \ that the probable life of the proton must be greater than ten million million million million million years (1 with \ thirty-one zeros). This is longer than the lifetime predicted by the simplest grand unified theory, but there are more elabo\ rate theories in which the predicted lifetimes are longer. Still more sensitive experiments involving even larger quantitie\ s of matter will be needed to test them.

Even though it is very difficult to observe spontaneous proton decay, it\ may be that our very existence is a consequence of the reverse process, the production of protons, or more simply, of qu\ arks, from an initial situation in which there were no more quarks than antiquarks, which is the most natural way to imagine\ the universe starting out. Matter on the earth is made up mainly of protons and neutrons, which in turn are made up of qua\ rks. There are no antiprotons or antineutrons, made up from antiquarks, except for a few that physicists produce in lar\ ge particle accelerators. We have evidence from cosmic rays that the same is true for all the matter in our galaxy: ther\ e are no antiprotons or antineutrons apart from a small number that are produced as particle/ antiparticle pairs in high-e\ nergy collisions. If there were large regions of antimatter in our galaxy, we would expect to observe large quantities of\ radiation from the borders between the regions of matter and antimatter, where many particles would be colliding with thei\ r anti-particles, annihilating each other and giving off high-energy radiation.

We have no direct evidence as to whether the matter in other galaxies is\ made up of protons and neutrons or antiprotons and anti-neutrons, but it must be one or the other: there cannot be a mi\ xture in a single galaxy because in that case we would again observe a lot of radiation from annihilations. We therefore \ believe that all galaxies are composed of quarks rather than antiquarks; it seems implausible that some galaxies should b\ e matter and some antimatter.

Why should there be so many more quarks than antiquarks? Why are there n\ ot equal numbers of each? It is certainly fortunate for us that the numbers are unequal because, if they had been \ the same, nearly all the quarks and antiquarks would have annihilated each other in the early universe and left a unive\ rse filled with radiation but hardly any matter.

There would then have been no galaxies, stars, or planets on which human\ life could have developed. Luckily, grand unified theories may provide an explanation of why the universe should n\ ow contain more quarks than antiquarks, even if it started out with equal numbers of each. As we have seen, GUTs allow q\ uarks to change into antielectrons at high energy. They also allow the reverse processes, antiquarks turning into e\ lectrons, and electrons and antielectrons turning into antiquarks and quarks. There was a time in the very early universe \ when it was so hot that the particle energies would have been high enough for these transformations to take place. But\ why should that lead to more quarks than antiquarks? The reason is that the laws of physics are not quite the sam\ e for particles and antiparticles.

Up to 1956 it was believed that the laws of physics obeyed each of three\ separate symmetries called C, P, and T. The symmetry C means that the laws are the same for particles and antipartic\ les. The symmetry P means that the laws are the same for any situation and its mirror image (the mirror image of a \ particle spinning in a right-handed direction is one spinning in a left-handed direction). The symmetry T means that if you \ reverse the direction of motion of all particles and antiparticles, the system should go back to what it was at earlier times\ ; in other words, the laws are the same in the forward and backward directions of time. In 1956 two American physicists\ , Tsung-Dao Lee and Chen Ning Yang, suggested that the weak force does not in fact obey the symmetry P. In o\ ther words, the weak force would make the universe develop in a different way from the way in which the mirror ima\ ge of the universe would develop. The same year, a colleague, Chien-Shiung Wu, proved their prediction correct. She\ did this by lining up the nuclei of radioactive atoms in a magnetic field, so that they were all spinning in the same di\ rection, and showed that the electrons were given off more in one direction than another. The following year, Lee and Yang\ received the Nobel Prize for their idea. It was also found that the weak force did not obey the symmetry C. That is, it \ would cause a universe composed of antiparticles to behave differently from our universe. Nevertheless, it seemed that th\ e weak force did obey the combined symmetry CP. That is, the universe would develop in the same way as its mirror im\ age if, in addition, every particle was swapped with its antiparticle! However, in 1964 two more Americans, J. W. Cronin\ and Val Fitch, discovered that even the CP symmetry was not obeyed in the decay of certain particles called K-meson\ s. Cronin and Fitch eventually received the Nobel Prize for their work in 1980. (A lot of prizes have been awarded \ for showing that the universe is not as simple as we might have thought!) A Brief History of Time - Stephen Hawking... Chapter 5 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/d.html (7 of 8) [2/20/2001 3:14:54 AM] There is a mathematical theorem that says that any theory that obeys qua\ ntum mechanics and relativity must always obey the combined symmetry CPT. In other words, the universe would have \ to behave the same if one replaced particles by antiparticles, took the mirror image, and also reversed the direction\ of time. But Cronin and Fitch showed that if one replaces particles by antiparticles and takes the mirror image, but does\ not reverse the direction of time, then the universe does not behave the same. The laws of physics, therefore, must change if one reve\ rses the direction of time – they do not obey the symmetry T.

Certainly the early universe does not obey the symmetry T: as time runs \ forward the universe expands – if it ran backward, the universe would be contracting. And since there are forces \ that do not obey the symmetry T, it follows that as the universe expands, these forces could cause more antielectrons to \ turn into quarks than electrons into antiquarks.

Then, as the universe expanded and cooled, the antiquarks would annihila\ te with the quarks, but since there would be more quarks than antiquarks, a small excess of quarks would remain. It i\ s these that make up the matter we see today and out of which we ourselves are made. Thus our very existence could be\ regarded as a confirmation of grand unified theories, though a qualitative one only; the uncertainties are such that\ one cannot predict the numbers of quarks that will be left after the annihilation, or even whether it would be quarks or an\ tiquarks that would remain. (Had it been an excess of antiquarks, however, we would simply have named antiquarks quarks, an\ d quarks antiquarks.) Grand unified theories do not include the force of gravity. This does no\ t matter too much, because gravity is such a weak force that its effects can usually be neglected when we are dealing with\ elementary particles or atoms. However, the fact that it is both long range and always attractive means that its effects \ all add up. So for a sufficiently large number of matter particles, gravitational forces can dominate over all other force\ s. This is why it is gravity that determines the evolution of the universe. Even for objects the size of stars, the attra\ ctive force of gravity can win over all the other forces and cause the star to collapse. My work in the 1970s focused on the blac\ k holes that can result from such stellar collapse and the intense gravitational fields around them. It was this that led t\ o the first hints of how the theories of quantum mechanics and general relativity might affect each other – a glimpse \ of the shape of a quantum theory of gravity yet to come.

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Chapter 5 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/d.html (8 of 8) [2/20/2001 3:14:54 AM] CHAPTER 6 BLACK HOLES   The term black hole is of very recent origin. It was coined in 1969 by the American scientis\ t John Wheeler as a graphic description of an idea that goes back at least two hundred years, to a t\ ime when there were two theories about light:

one, which Newton favored, was that it was composed of particles; the ot\ her was that it was made of waves. We now know that really both theories are correct. By the wave/particle duality\ of quantum mechanics, light can be regarded as both a wave and a particle. Under the theory that light is made up of wa\ ves, it was not clear how it would respond to gravity. But if light is composed of particles, one might expect them to\ be affected by gravity in the same way that cannonballs, rockets, and planets are. At first people thought that part\ icles of light traveled infinitely fast, so gravity would not have been able to slow them down, but the discovery by Roemer \ that light travels at a finite speed meant that gravity might have an important effect.

On this assumption, a Cambridge don, John Michell, wrote a paper in 1783\ in the Philosophical Transactions of the Royal Society of London in which he pointed out that a star that was sufficiently massive and co\ mpact would have such a strong gravitational field that light could not escape: any light emit\ ted from the surface of the star would be dragged back by the star’s gravitational attraction before it could get very \ far. Michell suggested that there might be a large number of stars like this. Although we would not be able to see them bec\ ause the light from them would not reach us, we would still feel their gravitational attraction. Such objects are wha\ t we now call black holes, because that is what they are: black voids in space. A similar suggestion was made a few year\ s later by the French scientist the Marquis de Laplace, apparently independently of Michell. Interestingly enough, Lapl\ ace included it in only the first and second editions of his book The System of the World, and left it out of later editions; perhaps he decided that it was a craz\ y idea. (Also, the particle theory of light went out of favor during the \ nineteenth century; it seemed that everything could be explained by the wave theory, and according to the wave theory, it wa\ s not clear that light would be affected by gravity at all.) In fact, it is not really consistent to treat light like cannonballs in \ Newton’s theory of gravity because the speed of light is fixed. (A cannonball fired upward from the earth will be slowed down by\ gravity and will eventually stop and fall back; a photon, however, must continue upward at a constant speed. How then can \ Newtonian grav-ity affect light?) A consistent theory of how gravity affects light did not come along until \ Einstein proposed general relativity in 1915. And even then it was a long time before the implications of the theory for m\ assive stars were understood.

To understand how a black hole might be formed, we first need an underst\ anding of the life cycle of a star. A star is formed when a large amount of gas (mostly hydrogen) starts to collapse\ in on itself due to its gravitational attraction. As it contracts, the atoms of the gas collide with each other more and more\ frequently and at greater and greater speeds – the gas heats up. Eventually, the gas will be so hot that when the hydro\ gen atoms collide they no longer bounce off each other, but instead coalesce to form helium. The heat released in th\ is reaction, which is like a controlled hydrogen bomb explosion, is what makes the star shine. This additional heat also \ increases the pressure of the gas until it is sufficient to balance the gravitational attraction, and the gas stops co\ ntracting. It is a bit like a balloon – there is a balance between the pressure of the air inside, which is trying to make \ the balloon expand, and the tension in the rubber, which is trying to make the balloon smaller. Stars will remain s\ table like this for a long time, with heat from the nuclear reactions balancing the gravitational attraction. Eventually, ho\ wever, the star will run out of its hydrogen and other nuclear fuels. Paradoxically, the more fuel a star starts off with\ , the sooner it runs out. This is because the more massive the star is, the hotter it needs to be to balance its gravitatio\ nal attraction. And the hotter it is, the faster it will use up its fuel. Our sun has probably got enough fuel for another five t\ housand million years or so, but more massive stars can use up their fuel in as little as one hundred million years, m\ uch less than the age of the universe. When a star runs out of fuel, it starts to cool off and so to contract. What might h\ appen to it then was first understood only at the end of the 1920s.

In 1928 an Indian graduate student, Subrahmanyan Chandrasekhar, set sail\ for England to study at Cambridge with the British astronomer Sir Arthur Eddington, an expert on general relativity\ . (According to some accounts, a journalist told Eddington in the early 1920s that he had heard there were only three peo\ ple in the world who understood general relativity. Eddington paused, then replied, “I am trying to think who\ the third person is.”) During his voyage from India, Chandrasekhar worked out how big a star could be and still support itsel\ f against its own gravity after it had used up all its fuel. The idea was this: when the star becomes small, the matter par\ ticles get very near each other, and so according to the Pauli exclusion principle, they must have very differen\ t velocities. This makes them move away from each other and so tends to make the star expand. A star can therefore ma\ intain itself at a constant radius by a balance between the attraction of gravity and the repulsion that arises from the\ exclusion principle, just as earlier in its life A Brief History of Time - Stephen Hawking... Chapter 6 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/e.html (1 of 9) [2/20/2001 3:15:08 AM] gravity was balanced by the heat.

Chandrasekhar realized, however, that there is a limit to the repulsion \ that the exclusion principle can provide. The theory of relativity limits the maximum difference in the velocities of \ the matter particles in the star to the speed of light.

This means that when the star got sufficiently dense, the repulsion caus\ ed by the exclusion principle would be less than the attraction of gravity. Chandrasekhar calculated that a cold star of \ more than about one and a half times the mass of the sun would not be able to support itself against its own gravity. (T\ his mass is now known as the Chandrasekhar limit.) A similar discovery was made about the same time by the Russian\ scientist Lev Davidovich Landau.

This had serious implications for the ultimate fate of massive stars. If\ a star’s mass is less than the Chandrasekhar limit, it can eventually stop contracting and settle down to a possible final s\ tate as a “white dwarf” with a radius of a few thousand miles and a density of hundreds of tons per cubic inch. A white\ dwarf is supported by the exclusion principle repulsion between the electrons in its matter. We observe a large number\ of these white dwarf stars. One of the first to be discovered is a star that is orbiting around Sirius, the brightest st\ ar in the night sky.

Landau pointed out that there was another possible final state for a sta\ r, also with a limiting mass of about one or two times the mass of the sun but much smaller even than a white dwarf. Thes\ e stars would be supported by the exclusion principle repulsion between neutrons and protons, rather than between el\ ectrons. They were therefore called neutron stars. They would have a radius of only ten miles or so and a density of\ hundreds of millions of tons per cubic inch. At the time they were first predicted, there was no way that neutron stars \ could be observed. They were not actually detected until much later.

Stars with masses above the Chandrasekhar limit, on the other hand, have\ a big problem when they come to the end of their fuel. In some cases they may explode or manage to throw off enough\ matter to reduce their mass below the limit and so avoid catastrophic gravitational collapse, but it was difficult t\ o believe that this always happened, no matter how big the star. How would it know that it had to lose weight? And even if \ every star managed to lose enough mass to avoid collapse, what would happen if you added more mass to a white dwar\ f 'or neutron star to take it over the limit?

Would it collapse to infinite density? Eddington was shocked by that imp\ lication, and he refused to believe Chandrasekhar’s result. Eddington thought it was simply not possible \ that a star could collapse to a point. This was the view of most scientists: Einstein himself wrote a paper in which he clai\ med that stars would not shrink to zero size. The hostility of other scientists, particularly Eddington, his former teache\ r and the leading authority on the structure of stars, persuaded Chandrasekhar to abandon this line of work and turn instead to\ other problems in astronomy, such as the motion of star clusters. However, when he was awarded the Nobel Prize in\ 1983, it was, at least in part, for his early work on the limiting mass of cold stars.

Chandrasekhar had shown that the exclusion principle could not halt the \ collapse of a star more massive than the Chandrasekhar limit, but the problem of understanding what would happen \ to such a star, according to general relativity, was first solved by a young American, Robert Oppenheimer, in\ 1939. His result, however, suggested that there would be no observational consequences that could be detected by t\ he telescopes of the day. Then World War II intervened and Oppenheimer himself became closely involved in the atom b\ omb project. After the war the problem of gravitational collapse was largely forgotten as most scientists became c\ aught up in what happens on the scale of the atom and its nucleus. In the 1960s, however, interest in the large-scale\ problems of astronomy and cosmology was revived by a great increase in the number and range of astronomical obse\ rvations brought about by the application of modern technology. Oppenheimer’s work was then rediscovered and exten\ ded by a number of people.

The picture that we now have from Oppenheimer’s work is as follows. T\ he gravitational field of the star changes the paths of light rays in space-time from what they would have been had the\ star not been present. The light cones, which indicate the paths followed in space and time by flashes of light emitte\ d from their tips, are bent slightly inward near the surface of the star. This can be seen in the bending of light from dista\ nt stars observed during an eclipse of the sun. As the star contracts, the gravitational field at its surface gets stronger\ and the light cones get bent inward more. This makes it more difficult for light from the star to escape, and the light\ appears dimmer and redder to an observer at a distance. Eventually, when the star has shrunk to a certain critical rad\ ius, the gravitational field at the surface becomes so strong that the light cones are bent inward so much that light can no\ longer escape Figure 6:1.

A Brief History of Time - Stephen Hawking... Chapter 6 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/e.html (2 of 9) [2/20/2001 3:15:08 AM] Figure 6:1 According to the theory of relativity, nothing can travel faster than li\ ght. Thus if light cannot escape, neither can anything else; everything is dragged back by the gravitational field. So one has \ a set of events, a region of space-time, from which it is not possible to escape to reach a distant observer. This reg\ ion is what we now call a black hole. Its boundary is called the event horizon and it coincides with the paths of light ray\ s that just fail to escape from the black hole.

In order to understand what you would see if you were watching a star co\ llapse to form a black hole, one has to remember that in the theory of relativity there is no absolute time. Eac\ h observer has his own measure of time. The time for someone on a star will be different from that for someone at a dista\ nce, because of the gravitational field of the star. A Brief History of Time - Stephen Hawking... Chapter 6 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/e.html (3 of 9) [2/20/2001 3:15:08 AM] Suppose an intrepid astronaut on the surface of the collapsing star, col\ lapsing inward with it, sent a signal every second, according to his watch, to his spaceship orbiting about the star\ . At some time on his watch, say 11:00, the star would shrink below the critical radius at which the gravitational field \ becomes so strong nothing can escape, and his signals would no longer reach the spaceship. As 11:00 approached his com\ panions watching from the spaceship would find the intervals between successive signals from the astronaut getting\ longer and longer, but this effect would be very small before 10:59:59. They would have to wait only very slightly more t\ han a second between the astronaut’s 10:59:58 signal and the one that he sent when his watch read 10:59:59, but they w\ ould have to wait forever for the 11:00 signal.

The light waves emitted from the surface of the star between 10:59:59 an\ d 11:00, by the astronaut’s watch, would be spread out over an infinite period of time, as seen from the spaceship. \ The time interval between the arrival of successive waves at the spaceship would get longer and longer, so the li\ ght from the star would appear redder and redder and fainter and fainter. Eventually, the star would be so dim tha\ t it could no longer be seen from the spaceship:

all that would be left would be a black hole in space. The star would, h\ owever, continue to exert the same gravitational force on the spaceship, which would continue to orbit the black hole. Th\ is scenario is not entirely realistic, however, because of the following problem. Gravity gets weaker the farther you ar\ e from the star, so the gravitational force on our intrepid astronaut’s feet would always be greater than the force on h\ is head. This difference in the forces would stretch our astronaut out like spaghetti or tear him apart before the star had c\ ontracted to the critical radius at which the event horizon formed! However, we believe that there are much larger objects i\ n the universe, like the central regions of galaxies, that can also undergo gravitational collapse to produce black \ holes; an astronaut on one of these would not be torn apart before the black hole formed. He would not, in fact, feel any\ thing special as he reached the critical radius, and could pass the point of no return without noticing it However, withi\ n just a few hours, as the region continued to collapse, the difference in the gravitational forces on his head and his\ feet would become so strong that again it would tear him apart.

The work that Roger Penrose and I did between 1965 and 1970 showed that,\ according to general relativity, there must be a singularity of infinite density and space-time curvature within a b\ lack hole. This is rather like the big bang at the beginning of time, only it would be an end of time for the collapsing bo\ dy and the astronaut. At this singularity the laws of science and our ability to predict the future would break down. Howev\ er, any observer who remained outside the black hole would not be affected by this failure of predictability, beca\ use neither light nor any other signal could reach him from the singularity. This remarkable fact led Roger Penrose to prop\ ose the cosmic censorship hypothesis, which might be paraphrased as “God abhors a naked singularity.” In other\ words, the singularities produced by gravitational collapse occur only in places, like black holes, where they are decently\ hidden from outside view by an event horizon.

Strictly, this is what is known as the weak cosmic censorship hypothesis\ : it protects observers who remain outside the black hole from the consequences of the breakdown of predictability that\ occurs at the singularity, but it does nothing at all for the poor unfortunate astronaut who falls into the hole.

There are some solutions of the equations of general relativity in which\ it is possible for our astronaut to see a naked singularity: he may be able to avoid hitting the singularity and instead\ fall through a "wormhole” and come out in another region of the universe. This would offer great possibilities for travel \ in space and time, but unfortunately it seems that these solutions may all be highly unstable; the least disturbance, such \ as the presence of an astronaut, may change them so that the astronaut could not see the singularity until he hit it\ and his time came to an end. In other words, the singularity would always lie in his future and never in his past. The st\ rong version of the cosmic censorship hypothesis states that in a realistic solution, the singularities would always lie \ either entirely in the future (like the singularities of gravitational collapse) or entirely in the past (like the , big bang)\ . I strongly believe in cosmic censorship so I bet Kip Thorne and John Preskill of Cal Tech that it would always hold. I lost t\ he bet on a technicality because examples were produced of solutions with a singularity that was visible from a long wa\ y away. So I had to pay up, which according to the terms of the bet meant I had to clothe their nakedness. But I can cl\ aim a moral victory. The naked singularities were unstable: the least disturbance would cause them either to disappear or \ to be hidden behind an event horizon. So they would not occur in realistic situations.

The event horizon, the boundary of the region of space-time from which i\ t is not possible to escape, acts rather like a one-way membrane around the black hole: objects, such as unwary astronau\ ts, can fall through the event horizon into the black hole, but nothing can ever get out of the black hole through t\ he event horizon. (Remember that the event horizon is the path in space-time of light that is trying to escape from\ the black hole, and nothing can travel faster than light.) One could well say of the event horizon what the poet Dante sai\ d of the entrance to Hell: “All hope abandon, ye who enter here.” Anything or anyone who falls through the event horiz\ on will soon reach the region of infinite density and the end of time.

General relativity predicts that heavy objects that are moving will caus\ e the emission of gravitational waves, ripples in the curvature of space that travel at the speed of light. These are simi\ lar to light waves, which are ripples of the electromagnetic field, but they are much harder to detect. They can be o\ bserved by the very slight change in separation they produce between neighboring freely moving objects. A number of dete\ ctors are being built in the United States, A Brief History of Time - Stephen Hawking... Chapter 6 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/e.html (4 of 9) [2/20/2001 3:15:08 AM] Europe, and Japan that will measure displacements of one part in a thous\ and million million million (1 with twenty-one zeros after it), or less than the nucleus of an atom over a distance of\ ten miles.

Like light, gravitational waves carry energy away from the objects that \ emit them. One would therefore expect a system of massive objects to settle down eventually to a stationary state, beca\ use the energy in any movement would be carried away by the emission of gravitational waves. (It is rather like\ dropping a cork into water: at first it bobs up and down a great deal, but as the ripples carry away its energy, it eventual\ ly settles down to a stationary state.) For example, the movement of the earth in its orbit round the sun produces g\ ravitational waves. The effect of the energy loss will be to change the orbit of the earth so that gradually it gets \ nearer and nearer to the sun, eventually collides with it, and settles down to a stationary state. The rate of energy loss in t\ he case of the earth and the sun is very low – about enough to run a small electric heater. This means it will take about a t\ housand million million million million years for the earth to run into the sun, so there’s no immediate cause for worry! T\ he change in the orbit of the earth is too slow to be observed, but this same effect has been observed over the past few years\ occurring in the system called PSR 1913 + 16 (PSR stands for “pulsar,” a special type of neutron star that emits reg\ ular pulses of radio waves). This system contains two neutron stars orbiting each other, and the energy they are \ losing by the emission of gravitational waves is causing them to spiral in toward each other. This confirmation of genera\ l relativity won J. H. Taylor and R. A. Hulse the Nobel Prize in 1993. It will take about three hundred million . years fo\ r them to collide. Just before they do, they will be orbiting so fast that they will emit enough gravitational waves for dete\ ctors like LIGO to pick up.

During the gravitational collapse of a star to form a black hole, the mo\ vements would be much more rapid, so the rate at which energy is carried away would be much higher. It would therefore no\ t be too long ' before it settled down to a stationary state. What would this final stage look like? One might suppo\ se that it would depend on all the complex features of the star from which it had formed – not only its mass and\ rate of rotation, but also the different densities of various parts of the star, and the complicated movements of the gases wi\ thin the star. And if black holes were as varied as the objects that collapsed to form them, it might be very difficult t\ o make any predictions about black holes in general.

In 1967, however, the study of black holes was revolutionized by Werner \ Israel, a Canadian scientist (who was born in Berlin, brought up in South Africa, and took his doctoral degree in Irel\ and). Israel showed that, according to general relativity, non-rotating black holes must be very simple; they were perf\ ectly spherical, their size depended only on their mass, and any two such black holes with the same mass were identical. Th\ ey could, in fact, be described by a particular solution of Einstein’s equations that had been known since\ 1917, found by Karl Schwarzschild shortly after the discovery of general relativity. At first many people, including Isr\ ael himself, argued that since black holes had to be perfectly spherical, a black hole could only form from the collapse of a\ perfectly spherical object. Any real star – which would never be perfectly spherical – could therefore only collapse to\ form a naked singularity.

There was, however, a different interpretation of Israel’s result, wh\ ich was advocated by Roger Penrose and John Wheeler in particular. They argued that the rapid movements involved in \ a star’s collapse would mean that the gravitational waves it gave off would make it ever more spherical, and b\ y the time it had settled down to a stationary state, it would be precisely spherical. According to this view, any non-\ rotating star, however complicated its shape and internal structure, would end up after gravitational collapse as a perfe\ ctly spherical black hole, whose size would depend only on its mass. Further calculations supported this view, and i\ t soon came to be adopted generally.

Israel’s result dealt with the case of black holes formed from non-ro\ tating bodies only. In 1963, Roy Kerr, a New Zealander, found a set of solutions of the equations of general relativi\ ty that described rotating black holes. These “Kerr” black holes rotate at a constant rate, their size and shape\ depending only on their mass and rate of rotation. If the rotation is zero, the black hole is perfectly round and the solution is \ identical to the Schwarzschild solution. If the rotation is non-zero, the black hole bulges outward near its equator (j\ ust as the earth or the sun bulge due to their rotation), and the faster it rotates, the more it bulges. So, to extend\ Israel’s result to include rotating bodies, it was conjectured that any rotating body that collapsed to form a black hole w\ ould eventually settle down to a stationary state described by the Kerr solution. In 1970 a colleague and fellow research \ student of mine at Cambridge, Brandon Carter, took the first step toward proving this conjecture. He showed that, prov\ ided a stationary rotating black hole had an axis of symmetry, like a spinning top, its size and shape would depend only o\ n its mass and rate of rotation. Then, in 1971, I proved that any stationary rotating black hole would indeed have such an\ axis of symmetry. Finally, in 1973, David Robinson at Kings College, London, used Carter’s and my results to sh\ ow that the conjecture had been correct: such a black hole had indeed to be the Kerr solution. So after gravitational co\ llapse a black hole must settle down into a state in which it could be rotating, but not pulsating. Moreover, its size and\ shape would depend only on its mass and rate of rotation, and not on the nature of the body that had collapsed to form i\ t. This result became known by the maxim: “A black hole has no hair.” The “no hair” theorem is of great prac\ tical importance, because it so greatly restricts the possible types of black holes. One can therefore make detailed models of\ objects that might contain black holes and compare the predictions of the models with observations. It also means t\ hat a very large amount of information about A Brief History of Time - Stephen Hawking... Chapter 6 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/e.html (5 of 9) [2/20/2001 3:15:08 AM] the body that has collapsed must be lost when a black hole is formed, be\ cause afterward all we can possibly measure about the body is its mass and rate of rotation. The significance of thi\ s will be seen in the next chapter.

Black holes are one of only a fairly small number of cases in the histor\ y of science in which a theory was developed in great detail as a mathematical model before there was any evidence from \ observations that it was correct. Indeed, this used to be the main argument of opponents of black holes: how could one \ believe in objects for which the only evidence was calculations based on the dubious theory of general relativ\ ity? In 1963, however, Maarten Schmidt, an astronomer at the Palomar Observatory in California, measured the red sh\ ift of a faint starlike object in the direction of the source of radio waves called 3C273 (that is, source number 273 in t\ he third Cambridge catalogue of radio sources).

He found it was too large to be caused by a gravitational field: if it h\ ad been a gravitational red shift, the object would have to be so massive and so near to us that it would disturb the orbits\ of planets in the Solar System. This suggested that the red shift was instead caused by the expansion of the universe, \ which, in turn, meant that the object was a very long distance away. And to be visible at such a great distance, the obje\ ct must be very bright, must, in other words, be emitting a huge amount of energy. The only mechanism that people could t\ hink of that would produce such large quantities of energy seemed to be the gravitational collapse not just of\ a star but of a whole central region of a galaxy. A number of other similar “quasi-stellar objects,” or quasars, have \ been discovered, all with large red shifts. But they are all too far away and therefore too difficult to observe to provide concl\ usive evidence of black holes.

Further encouragement for the existence of black holes came in 1967 with\ the discovery by a research student at Cambridge, Jocelyn Bell-Burnell, of objects in the sky that were emittin\ g regular pulses of radio waves. At first Bell and her supervisor, Antony Hewish, thought they might have made contact with\ an alien civilization in the galaxy! Indeed, at the seminar at which they announced their discovery, I remember that the\ y called the first four sources to be found LGM 1 – 4, LGM standing for “Little Green Men.” In the end, however, they and eve\ ryone else came to the less romantic conclusion that these objects, which were given the name pulsar\ s, were in fact rotating neutron stars that were emitting pulses of radio waves because of a complicated interaction betw\ een their magnetic fields and surrounding matter. This was bad news for writers of space westerns, but very hopefu\ l for the small number of us who believed in black holes at that time: it was the first positive evidence that neutro\ n stars existed. A neutron star has a radius of about ten miles, only a few times the critical radius at which a star becomes \ a black hole. If a star could collapse to such a small size, it is not unreasonable to expect that other stars could coll\ apse to even smaller size and become black holes.

How could we hope to detect a black hole, as by its very definition it d\ oes not emit any light? It might seem a bit like looking for a black cat in a coal cellar. Fortunately, there is a way. A\ s John Michell pointed out in his pioneering paper in 1783, a black hole still exerts a gravitational fierce on nearby objects\ . Astronomers have observed many systems in which two stars orbit around each other, attracted toward each other by \ gravity. They also observe systems in which there is only one visible star that is orbiting around some unseen compa\ nion. One cannot, of course, immediately conclude that the companion is a black hole: it might merely be a star t\ hat is too faint to be seen. However, some of these systems, like the one called Cygnus X-1 Figure 6:2, are also strong sources of X-rays.

A Brief History of Time - Stephen Hawking... Chapter 6 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/e.html (6 of 9) [2/20/2001 3:15:08 AM] Figure 6:2 The best explanation for this phenomenon is that matter has been blown o\ ff the surface of the visible star. As it falls toward the unseen companion, it develops a spiral motion (rather like w\ ater running out of a bath), and it gets very hot, emitting X-rays Figure 6:3. A Brief History of Time - Stephen Hawking... Chapter 6 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/e.html (7 of 9) [2/20/2001 3:15:08 AM] Figure 6:3 For this mechanism to work, the unseen object has to be very small, like\ a white dwarf, neutron star, or black hole.

From the observed orbit of the visible star, one can determine the lowes\ t possible mass of the unseen object. In the case of Cygnus X-l, this is about six times the mass of the sun, which, \ according to Chandrasekhar’r result, is too great for the unseen object to be a white dwarf. It is also too large a mass t\ o be a neutron star. It seems, therefore, that it must be a black hole.

There are other models to explain Cygnus X-1 that do not include a black\ hole, but they are all rather far-fetched. A black hole seems to be the only really natural explanation of the observ\ ations. Despite this, I had a bet with Kip Thorne of the California Institute of Technology that in fact Cygnus X-1 does n\ ot contain a black hole! This was a form f insurance policy for me. I have done a lot of work on black holes, and i\ t would all be wasted if it turned out that black holes do not exist. But in that case, I would have the consolation of wi\ nning my bet, which would bring me four years of the magazine Private Eye. In fact, although the situation with Cygnus X-1 has not changed much sin\ ce we made the bet in 1975, there is now so much other observational evidence in favor of b\ lack holes that I have conceded the bet. I paid the specified penalty, which was a one-year subscription to Penthouse, to the outrage of Kip’s liberated wife.

We also now have evidence for several other black holes in systems like \ Cygnus X-1 in our galaxy and in two neighboring galaxies called the Magellanic Clouds. The number of black h\ oles, however, is almost certainly very much higher; in the long history of the universe, many stars must have burned\ all their nuclear fuel and have had to collapse.

The number of black holes may well be greater even than the number of vi\ sible stars, which totals about a hundred thousand million in our galaxy alone. The extra gravitational attraction\ of such a large number of black holes could explain why our galaxy rotates at the rate it does: the mass of the visi\ ble stars is insufficient to account for this. We also A Brief History of Time - Stephen Hawking... Chapter 6 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/e.html (8 of 9) [2/20/2001 3:15:08 AM] have some evidence that there is a much larger black hole, with a mass o\ f about a hundred thousand times that of the sun, at the center of our galaxy. Stars in the galaxy that come too near\ this black hole will be torn apart by the difference in the gravitational forces on their near and far sides. Thei\ r remains and gas that is thrown off other stars, will fall toward the black hole. As in the case of Cygnus X-l, the gas will s\ piral inward and will heat up, though not as much as in that case. It will not get hot enough to emit X rays, but it could\ account for the very compact source of radio waves and infrared rays that is observed at the galactic center.

It is thought that similar but even larger black holes, with masses of a\ bout a hundred million times the mass of the sun, occur at the centers of quasars. For example, observations with the Hubb\ le telescope of the galaxy known as M87 reveal that it contains a disk of gas 130 light-years across rotating ab\ out a central object two thousand million times the mass of the sun. This can only be a black hole. Matter falling into such\ a supermassive black hole would provide the only source of power great enough to explain the enormous amounts of ene\ rgy that these objects are emitting. As the matter spirals into the black hole, it would make the black hole rotate \ in the same direction, causing it to develop a magnetic field rather like that of the earth. Very high-energy particles\ would be generated near the black hole by the in-falling matter. The magnetic field would be so strong that it could f\ ocus these particles into jets ejected outward along the axis of rotation of the black hole, that is, in the directions of it\ s north and south poles. Such jets are indeed observed in a number of galaxies and quasars. One can also consider the possibili\ ty that there might be black holes with masses much less than that of the sun. Such black holes could not be formed by \ gravitational collapse, because their masses are below the Chandrasekhar mass limit: stars of this low mass can suppo\ rt themselves against the force of gravity even when they have exhausted their nuclear fuel. Low-mass black holes c\ ould form only if matter was compressed to enormous densities by very large external pressures. Such conditions cou\ ld occur in a very big hydrogen bomb: the physicist John Wheeler once calculated that if one took all the heavy wa\ ter in all the oceans of the world, one could build a hydrogen bomb that would compress matter at the center so much t\ hat a black hole would be created. (Of course, there would be no one left to observe it!) A more practical pos\ sibility is that such low-mass black holes might have been formed in the high temperatures and pressures of the very earl\ y universe. Black holes would have been formed only if the early universe had not been perfectly smooth and unif\ orm, because only a small region that was denser than average could be compressed in this way to form a black hole\ . But we know that there must have been some irregularities, because otherwise the matter in the universe would \ still be perfectly uniformly distributed at the present epoch, instead of being clumped together in stars and galaxies.

Whether the irregularities required to account for stars and galaxies wo\ uld have led to the formation of a significant number of “primordial” black holes clearly depends on the details \ of the conditions in the early universe. So if we could determine how many primordial black holes there are now, we would learn \ a lot about the very early stages of the universe. Primordial black holes with masses more than a thousand millio\ n tons (the mass of a large mountain) could be detected only by their gravitational influence on other, visible matt\ er or on the expansion of the universe. However, as we shall learn in the next chapter, black holes are not really black \ after all: they glow like a hot body, and the smaller they are, the more they glow. So, paradoxically, smaller black holes mig\ ht actually turn out to be easier to detect than large ones!

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Chapter 6 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/e.html (9 of 9) [2/20/2001 3:15:08 AM] CHAPTER 7 BLACK HOLES AIN’T SO BLACK   Before 1970, my research on general relativity had concentrated mainly o\ n the question of whether or not there had been a big bang singularity. However, one evening in November that year,\ shortly after the birth of my daughter, Lucy, I started to think about black holes as I was getting into bed. My disab\ ility makes this rather a slow process, so I had plenty of time. At that date there was no precise definition of which po\ ints in space-time lay inside a black hole and which lay outside. I had already discussed with Roger Penrose the idea o\ f defining a black hole as the set of events from which it was not possible to escape to a large distance, which is n\ ow the generally accepted definition. It means that the boundary of the black hole, the event horizon, is formed by the\ light rays that just fail to escape from the black hole, hovering forever just on the edge Figure 7:1. It is a bit like running away from the police and just managing to keep one step ahead but not being able to get clear away! Figure 7:1 Suddenly I realized that the paths of these light rays could never appro\ ach one another. If they did they must eventually run into one another. It would be like meeting someone else r\ unning away from the police in the opposite direction – you would both be caught! (Or, in this case, fall into a\ black hole.) But if these light rays were swallowed up by the black hole, then they could not have been on the boundary of the \ black hole. So the paths of light rays in the A Brief History of Time - Stephen Hawking... Chapter 7 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/f.html (1 of 8) [2/20/2001 3:15:18 AM] event horizon had always to be moving parallel to, or away from, each ot\ her. Another way of seeing this is that the event horizon, the boundary of the black hole, is like the edge of a sha\ dow – the shadow of impending doom. If you look at the shadow cast by a source at a great distance, such as the sun\ , you will see that the rays of light in the edge are not approaching each other.

If the rays of light that form the event horizon, the boundary of the bl\ ack hole, can never approach each other, the area of the event horizon might stay the same or increase with time, but it c\ ould never decrease because that would mean that at least some of the rays of light in the boundary would have to be\ approaching each other. In fact, the area would increase whenever matter or radiation fell into the black hole Figure 7:2 . Figures 7:2 & 7:3 Or if two black holes collided and merged together to form a single blac\ k hole, the area of the event horizon of the final black hole would be greater than or equal to the sum of the areas of the\ event horizons of the original black holes Figure 7:3 . This nondecreasing property of the event horizon’s area placed an i\ mportant restriction on the possible behavior of black holes. I was so excited with my discovery that I did n\ ot get much sleep that night. The next day I rang up Roger Penrose. He agreed with me. I think, in fact, that he had been \ aware of this property of the area. However, he had been using a slightly different definition of a black hole. He ha\ d not realized that the boundaries of the black A Brief History of Time - Stephen Hawking... Chapter 7 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/f.html (2 of 8) [2/20/2001 3:15:18 AM] hole according to the two definitions would be the same, and hence so wo\ uld their areas, provided the black hole had settled down to a state in which it was not changing with time.

The nondecreasing behavior of a black hole’s area was very reminiscen\ t of the behavior of a physical quantity called entropy, which measures the degree of disorder of a system. It is a matt\ er of common experience that disorder will tend to increase if things are left to themselves. (One has only to sto\ p making repairs around the house to see that!) One can create order out of disorder (for example, one can paint the ho\ use), but that requires expenditure of effort or energy and so decreases the amount of ordered energy available.

A precise statement of this idea is known as the second law of thermodyn\ amics. It states that the entropy of an isolated system always increases, and that when two systems are joined together, \ the entropy of the combined system is greater than the sum of the entropies of the individual systems. For exa\ mple, consider a system of gas molecules in a box. The molecules can be thought of as little billiard balls continuall\ y colliding with each other and bouncing off the walls of the box. The higher the temperature of the gas, the faster the \ molecules move, and so the more frequently and harder they collide with the walls of the box and the greater the outwar\ d pressure they exert on the walls. Suppose that initially the molecules are all confined to the left-hand side of the bo\ x by a partition. If the partition is then removed, the molecules will tend to spread out and occupy both halves of the box. At \ some later time they could, by chance, all be in the right half or back in the left half, but it is overwhelmingly more p\ robable that there will be roughly equal numbers in the two halves. Such a state is less ordered, or more disordered, than t\ he original state in which all the molecules were in one half. One therefore says that the entropy of the gas has gone up.\ Similarly, suppose one starts with two boxes, one containing oxygen molecules and the other containing nitrogen molecu\ les. If one joins the boxes together and removes the intervening wall, the oxygen and the nitrogen molecules will\ start to mix. At a later time the most probable state would be a fairly uniform mixture of oxygen and nitrogen molecules\ throughout the two boxes. This state would be less ordered, and hence have more entropy, than the initial state of \ two separate boxes.

The second law of thermodynamics has a rather different status than that\ of other laws of science, such as Newton's law of gravity, for example, because it does not hold always, just in th\ e vast majority of cases. The probability of all the gas molecules in our first box found in one half of the box at a later time is many millions of million\ s to one, but it can happen. However, if one has a black hole around there seems to be a rather easier way of violating the\ second law: just throw some matter with a lot of entropy such as a box of gas, down the black hole. The total entropy \ of matter outside the black hole would go down. One could, of course, still say that the total entropy, including \ the entropy inside the black hole, has not gone down - but since there is no way to look inside the black hole, we canno\ t see how much entropy the matter inside it has. It would be nice, then, if there was some feature of the black hole\ by which observers outside the black hole could tell its entropy, and which would increase whenever matter carrying entr\ opy fell into the black hole. Following the discovery, described above, that the area of the event horizon increased\ whenever matter fell into a black hole, a research student at Princeton named Jacob Bekenstein suggested that the \ area of the event horizon was a measure of the entropy of the black hole. As matter carrying entropy fell into a bl\ ack hole, the area of its event horizon would go up, so that the sum of the entropy of matter outside black holes and the\ area of the horizons would never go down.

This suggestion seemed to prevent the second law of thermodynamics from \ being violated in most situations.

However, there was one fatal flaw. If a black hole has entropy, then it \ ought to also have a temperature. But a body with a particular temperature must emit radiation at a certain rate. It \ is a matter of common experience that if one heats up a poker in a fire it glows red hot and emits radiation, but bodies at\ lower temperatures emit radiation too; one just does not normally notice it because the amount is fairly small. This rad\ iation is required in order to prevent violation of the second law. So black holes ought to emit radiation. But by their ver\ y definition, black holes are objects that are not supposed to emit anything. It therefore seemed that the area of the even\ t horizon of a black hole could not be regarded as its entropy. In 1972 I wrote a paper with Brandon Carter and an Ameri\ can colleague, Jim Bardeen, in which we pointed out that although there were many similarities between entropy a\ nd the area of the event horizon, there was this apparently fatal difficulty. I must admit that in writing this pape\ r I was motivated partly by irritation with Bekenstein, who, I felt, had misused my discovery of the increase of the area of the\ event horizon. However, it turned out in the end that he was basically correct, though in a manner he had certainly not e\ xpected.

In September 1973, while I was visiting Moscow, I discussed black holes \ with two leading Soviet experts, Yakov Zeldovich and Alexander Starobinsky. They convinced me that, according t\ o the quantum mechanical uncertainty principle, rotating black holes should create and emit particles. I beli\ eved their arguments on physical grounds, but I did not like the mathematical way in which they calculated the emission. I t\ herefore set about devising a better mathematical treatment, which I described at an informal seminar in Oxfo\ rd at the end of November 1973. At that time I had not done the calculations to find out how much would actually be emi\ tted. I was expecting to discover just the radiation that Zeldovich and Starobinsky had predicted from rotating bla\ ck holes. However, when I did the calculation, I A Brief History of Time - Stephen Hawking... Chapter 7 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/f.html (3 of 8) [2/20/2001 3:15:18 AM] found, to my surprise and annoyance, that even non-rotating black holes \ should apparently create and emit particles at a steady rate. At first I thought that this emission indicated that one \ of the approximations I had used was not valid. I was afraid that if Bekenstein found out about it, he would use it as a f\ urther argument to support his ideas about the entropy of black holes, which I still did not like. However, the more I \ thought about it, the more it seemed that the approximations really ought to hold. But what finally convinced me that \ the emission was real was that the spectrum of the emitted particles was exactly that which would be emitted by a hot b\ ody, and that the black hole was emitting particles at exactly the correct rate to prevent violations of the secon\ d law. Since then the calculations have been repeated in a number of different forms by other people. They all confir\ m that a black hole ought to emit particles and radiation as if it were a hot body with a temperature that depends only \ on the black hole’s mass: the higher the mass, the lower the temperature.

How is it possible that a black hole appears to emit particles when we k\ now that nothing can escape from within its event horizon? The answer, quantum theory tells us, is that the particle\ s do not come from within the black hole, but from the “empty” space just outside the black hole’s event hori\ zon! We can understand this in the following way: what we think of as “empty” space cannot be completely empty because th\ at would mean that all the fields, such as the gravitational and electromagnetic fields, would have to be exactly zero.\ However, the value of a field and its rate of change with time are like the position and velocity of a particle: the u\ ncertainty principle implies that the more accurately one knows one of these quantities, the less accurately one ca\ n know the other. So in empty space the field cannot be fixed at exactly zero, because then it would have both a preci\ se value (zero) and a precise rate of change (also zero). There must be a certain minimum amount of uncertainty, or\ quantum fluctuations, in the value of the field.

One can think of these fluctuations as pairs of particles of light or gr\ avity that appear together at some time, move apart, and then come together again and annihilate each other. These par\ ticles are virtual particles like the particles that carry the gravitational force of the sun: unlike real particles, th\ ey cannot be observed directly with a particle detector. However, their indirect effects, such as small changes in the \ energy of electron orbits in atoms, can be measured and agree with the theoretical predictions to a remarkable degr\ ee of accuracy. The uncertainty principle also predicts that there will be similar virtual pairs of matter particles, s\ uch as electrons or quarks. In this case, however, one member of the pair will be a particle and the other an antiparticle \ (the antiparticles of light and gravity are the same as the particles).

Because energy cannot be created out of nothing, one of the partners in \ a particle/antiparticle pair will have positive energy, and the other partner negative energy. The one with negative ene\ rgy is condemned to be a short-lived virtual particle because real particles always have positive energy in normal si\ tuations. It must therefore seek out its partner and annihilate with it. However, a real particle close to a massive body\ has less energy than if it were far away, because it would take energy to lift it far away against the gravitation\ al attraction of the body. Normally, the energy of the particle is still positive, but the gravitational field inside a bla\ ck hole is so strong that even a real particle can have negative energy there. It is therefore possible, if a black hole is pres\ ent, for the virtual particle with negative energy to fall into the black hole and become a real particle or antiparticle. In \ this case it no longer has to annihilate with its partner. Its forsaken partner may fall into the black hole as well. Or, \ having positive energy, it might also escape from the vicinity of the black hole as a real particle or antiparticle Figure 7:4.

A Brief History of Time - Stephen Hawking... Chapter 7 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/f.html (4 of 8) [2/20/2001 3:15:18 AM] Figure 7:4 To an observer at a distance, it will appear to have been emitted from t\ he black hole. The smaller the black hole, the shorter the distance the particle with negative energy will have to go b\ efore it becomes a real particle, and thus the greater the rate of emission, and the apparent temperature, of the black\ hole.

The positive energy of the outgoing radiation would be balanced by a flo\ w of negative energy particles into the black hole. By Einstein’s equation E = mc 2 (where E is energy, m is mass, and c is the speed of light), energy is proportional to mass. A flow of negative energy into the black hole therefore reduces\ its mass. As the black hole loses mass, the area of its event horizon gets smaller, but this decrease in the entropy\ of the black hole is more than compensated for by the entropy of the emitted radiation, so the second law is never viol\ ated.

Moreover, the lower the mass of the black hole, the higher its temperatu\ re. So as the black hole loses mass, its temperature and rate of emission increase, so it loses mass more quickly\ . What happens when the mass of the black hole eventually becomes extremely small is not quite clear, but the most\ reasonable guess is that it would disappear completely in a tremendous final burst of emission, equivalent to the ex\ plosion of millions of H-bombs.

A black hole with a mass a few times that of the sun would have a temper\ ature of only one ten millionth of a degree above absolute zero. This is much less than the temperature of the micro\ wave radiation that fills the universe (about A Brief History of Time - Stephen Hawking... Chapter 7 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/f.html (5 of 8) [2/20/2001 3:15:18 AM] 2.7º above absolute zero), so such black holes would emit even less \ than they absorb. If the universe is destined to go on expanding forever, the temperature of the microwave radiation will ev\ entually decrease to less than that of such a black hole, which will then begin to lose mass. But, even then, its temp\ erature would be so low that it would take about a million million million million million million million million millio\ n million million years (1 with sixty-six zeros after it) to evaporate completely. This is much longer than the age of the universe, \ which is only about ten or twenty thousand million years (1 or 2 with ten zeros after it). On the other hand, as \ mentioned in Chapter 6, there might be primordial black holes with a very much smaller mass that were made by the collapse\ of irregularities in the very early stages of the universe. Such black holes would have a much higher temperature and \ would be emitting radiation at a much greater rate. A primordial black hole with an initial mass of a thousand\ million tons would have a lifetime roughly equal to the age of the universe. Primordial black holes with initial masses l\ ess than this figure would already have completely evaporated, but those with slightly greater masses would stil\ l be emitting radiation in the form of X rays and gamma rays. These X rays and gamma rays are like waves of light, but wit\ h a much shorter wavelength. Such holes hardly deserve the epithet black: they really are white hot and are emit\ ting energy at a rate of about ten thousand megawatts.

One such black hole could run ten large power stations, if only we could\ harness its power. This would be rather difficult, however: the black hole would have the mass of a mountain com\ pressed into less than a million millionth of an inch, the size of the nucleus of an atom! If you had one of these black \ holes on the surface of the earth, there would be no way to stop it from falling through the floor to the center of the ea\ rth. It would oscillate through the earth and back, until eventually it settled down at the center. So the only place to put\ such a black hole, in which one might use the energy that it emitted, would be in orbit around the earth – and the \ only way that one could get it to orbit the earth would be to attract it there by towing a large mass in front of it, rath\ er like a carrot in front of a donkey. This does not sound like a very practical proposition, at least not in the immediate f\ uture.

But even if we cannot harness the emission from these primordial black h\ oles, what are our chances of observing them? We could look for the gamma rays that the primordial black holes e\ mit during most of their lifetime. Although the radiation from most would be very weak because they are far away, the to\ tal from all of them might be detectable. We do observe such a background of gamma rays: Figure 7:5 shows how the observed intensity differs at different frequencies (the number of waves per second). However, this background\ could have been, and probably was, generated by processes other than primordial black holes. The dotted lin\ e in Figure 7:5 shows how the intensity should vary with frequency for gamma rays given off by primordial black holes, \ if there were on average 300 per cubic light-year. One can therefore say that the observations of the gamma ray\ background do not provide any positive evidence for primordial black holes, but they do tell us that on average\ there cannot be more than 300 in every cubic light-year in the universe. This limit means that primordial black holes\ could make up at most one millionth of the matter in the universe.

A Brief History of Time - Stephen Hawking... Chapter 7 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/f.html (6 of 8) [2/20/2001 3:15:18 AM] Figure 7:5 With primordial black holes being so scarce, it might seem unlikely that\ there would be one near enough for us to observe as an individual source of gamma rays. But since gravity would d\ raw primordial black holes toward any matter, they should be much more common in and around galaxies. So although the \ gamma ray background tells us that there can be no more than 300 primordial black holes per cubic light-year on a\ verage, it tells us nothing about how common they might be in our own galaxy. If they were, say, a million times more\ common than this, then the nearest black hole to us would probably be at a distance of about a thousand million kilome\ ters, or about as far away as Pluto, the farthest known planet. At this distance it would still be very difficult to detec\ t the steady emission of a black hole, even if it was ten thousand megawatts. In order to observe a primordial black hole one \ would have to detect several gamma ray quanta coming from the same direction within a reasonable space of time,\ such as a week. Otherwise, they might simply be part of the background. But Planck’s quantum principle tell\ s us that each gamma ray quantum has a very high energy, because gamma rays have a very high frequency, so it would \ not take many quanta to radiate even ten thousand megawatts. And to observe these few coming from the distance of\ Pluto would require a larger gamma ray detector than any that have been constructed so far. Moreover, the detec\ tor would have to be in space, because gamma rays cannot penetrate the atmosphere.

Of course, if a black hole as close as Pluto were to reach the end of it\ s life and blow up, it would be easy to detect the final burst of emission. But if the black hole has been emitting for the\ last ten or twenty thousand million years, the chance of it reaching the end of its life within the next few years, rat\ her than several million years in the past or future, is really rather small! So in order to have a reasonable chance of seein\ g an explosion before your research grant ran out, you would have to find a way to detect any explosions within a dist\ ance of about one light-year. In fact bursts of A Brief History of Time - Stephen Hawking... Chapter 7 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/f.html (7 of 8) [2/20/2001 3:15:18 AM] gamma rays from space have been detected by satellites originally constr\ ucted to look for violations of the Test Ban Treaty. These seem to occur about sixteen times a month and to be roughl\ y uniformly distributed in direction across the sky. This indicates that they come from outside the Solar System sin\ ce otherwise we would expect them to be concentrated toward the plane of the orbits of the planets. The uniform \ distribution also indicates that the sources are either fairly near to us in our galaxy or right outside it at cosmologic\ al distances because otherwise, again, they would be concentrated toward the plane of the galaxy. In the latter case, the \ energy required to account for the bursts would be far too high to have been produced by tiny black holes, but if the so\ urces were close in galactic terms, it might be possible that they were exploding black holes. I would very much like th\ is to be the case but I have to recognize that there are other possible explanations for the gamma ray bursts, such as \ colliding neutron stars. New observations in the next few years, particularly by gravitational wave detectors like LI\ GO, should enable us to discover the origin of the gamma ray bursts.

Even if the search for primordial black holes proves negative, as it see\ ms it may, it will still give us important information about the very early stages of the universe. If the early un\ iverse had been chaotic or irregular, or if the pressure of matter had been low, one would have expected it to produce m\ any more primordial black holes than the limit already set by our observations of the gamma ray background. Only \ if the early universe was very smooth and uniform, with a high pressure, can one explain the absence of observable\ numbers of primordial black holes.

The idea of radiation from black holes was the first example of a predic\ tion that depended in an essential way on both the great theories of this century, general relativity and quantum mecha\ nics. It aroused a lot of opposition initially because it upset the existing viewpoint: “How can a black hole emit a\ nything?” When I first announced the results of my calculations at a conference at the Rutherford-Appleton Laboratory ne\ ar Oxford, I was greeted with general incredulity. At the end of my talk the chairman of the session, John G. \ Taylor from Kings College, London, claimed it was all nonsense. He even wrote a paper to that effect. However, in the \ end most people, including John Taylor, have come to the conclusion that black holes must radiate like hot bodies if \ our other ideas about general relativity and quantum mechanics are correct. Thus, even though we have not yet managed\ to find a primordial black hole, there is fairly general agreement that if we did, it would have to be emitting a \ lot of gamma rays and X rays.

The existence of radiation from black holes seems to imply that gravitat\ ional collapse is not as final and irreversible as we once thought. If an astronaut falls into a black hole, its mass will \ increase, but eventually the energy equivalent of that extra mass will be returned to the universe in the form of radiatio\ n. Thus, in a sense, the astronaut will be “recycled.” It would be a poor sort of immortality, however, becau\ se any personal concept of time for the astronaut would almost certainly come to an end as he was torn apart inside the bl\ ack hole! Even the types of particles that were eventually emitted by the black hole would in general be different from \ those that made up the astronaut: the only feature of the astronaut that would survive would be his mass or energy.\ The approximations I used to derive the emission from black holes should\ work well when the black hole has a mass greater than a fraction of a gram. However, they will break down at the \ end of the black hole’s life when its mass gets very small. The most likely outcome seems to be that the black hole will\ just disappear, at least from our region of the universe, taking with it the astronaut and any singularity there might b\ e inside it, if indeed there is one. This was the first indication that quantum mechanics might remove the singularities t\ hat were predicted by general relativity.

However, the methods that I and other people were using in 1974 were not\ able to answer questions such as whether singularities would occur in quantum gravity. From 1975 onward I therefo\ re started to develop a more powerful approach to quantum gravity based on Richard Feynrnan’s idea of a sum\ over histories. The answers that this approach suggests for the origin and fate of the universe and its conten\ ts, such as astronauts, will be de-scribed in the next two chapters. We shall see that although the uncertainty principle \ places limitations on the accuracy of all our predictions, it may at the same time remove the fundamental unpredictabi\ lity that occurs at a space-time singularity.

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Chapter 7 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/f.html (8 of 8) [2/20/2001 3:15:18 AM] CHAPTER 8 THE ORIGIN AND FATE OF THE UNIVERSE   Einstein’s general theory of relativity, on its own, predicted that s\ pace-time began at the big bang singularity and would come to an end either at the big crunch singularity (if the whole\ universe recollapsed), or at a singularity inside a black hole (if a local region, such as a star, were to collapse). An\ y matter that fell into the hole would be destroyed at the singularity, and only the gravitational effect of its mass would \ continue to be felt outside. On the other hand, when quantum effects were taken into account, it seemed that the mass or\ energy of the matter would eventually be returned to the rest of the universe, and that the black hole, along wit\ h any singularity inside it, would evaporate away and finally disappear. Could quantum mechanics have an equally dram\ atic effect on the big bang and big crunch singularities? What really happens during the very early or late \ stages of the universe, when gravitational fields are so strong that quantum effects cannot be ignored? Does the un\ iverse in fact have a beginning or an end?

And if so, what are they like?

Throughout the 1970s I had been mainly studying black holes, but in 1981\ my interest in questions about the origin and fate of the universe was reawakened when I attended a conference on \ cosmology organized by the Jesuits in the Vatican. The Catholic Church had made a bad mistake with Galileo whe\ n it tried to lay down the law on a question of science, declaring that the sun went round the earth. Now, c\ enturies later, it had decided to invite a number of experts to advise it on cosmology. At the end of the conferenc\ e the participants were granted an audience with the Pope. He told us that it was all right to study the evolution o\ f the universe after the big bang, but we should not inquire into the big bang itself because that was the moment of Crea\ tion and therefore the work of God. I was glad then that he did not know the subject of the talk I had just given \ at the conference – the possibility that space-time was finite but had no boundary, which means that it had no be\ ginning, no moment of Creation. I had no desire to share the fate of Galileo, with whom I feel a strong sense of \ identity, partly because of the coincidence of having been born exactly 300 years after his death!

In order to explain the ideas that I and other people have had about how\ quantum mechanics may affect the origin and fate of the universe, it is necessary first to understand the genera\ lly accepted history of the universe, according to what is known as the “hot big bang model.” This assumes that th\ e universe is described by a Friedmann model, right back to the big bang. In such models one finds that as the univers\ e expands, any matter or radiation in it gets cooler. (When the universe doubles in size, its temperature falls by ha\ lf.) Since temperature is simply a measure of the average energy – or speed – of the particles, this cooling of \ the universe would have a major effect on the matter in it. At very high temperatures, particles would be moving around so fa\ st that they could escape any attraction toward each other due to nuclear or electromagnetic forces, but as they \ cooled off one would expect particles that attract each other to start to clump together. Moreover, even the types \ of particles that exist in the universe would depend on the temperature. At high enough temperatures, particles have s\ o much energy that whenever they collide many different particle/antiparticle pairs would be produced – and al\ though some of these particles would annihilate on hitting antiparticles, they would be produced more rap-idly than they\ could annihilate. At lower temperatures, however, when colliding particles have less energy, particle/antiparticl\ e pairs would be produced less quickly – and annihilation would become faster than production.

At the big bang itself the universe is thought to have had zero size, an\ d so to have been infinitely hot. But as the universe expanded, the temperature of the radiation decreased. One secon\ d after the big bang, it would have fallen to about ten thousand million degrees. This is about a thousand times th\ e temperature at the center of the sun, but temperatures as high as this are reached in H-bomb explosions. At this t\ ime the universe would have contained mostly photons, electrons, and neutrinos (extremely light particles tha\ t are affected only by the weak force and gravity) and their antiparticles, together with some protons and neutro\ ns. As the universe continued to expand and the temperature to drop, the rate at which electron/antielectron pairs w\ ere being produced in collisions would have fallen below the rate at which they were being destroyed by annihilation\ . So most of the electrons and antielectrons would have annihilated with each other to produce more photons, leaving \ only a few electrons left over. The neutrinos and antineutrinos, however, would not have annihilated with ea\ ch other, because these particles interact with themselves and with other particles only very weakly. So they shoul\ d still be around today. If we could observe them, it would provide a good test of this picture of a very hot early s\ tage of the universe. Unfortunately, their energies nowadays would be too low for us to observe them directly. Howe\ ver, if neutrinos are not massless, but have a small mass of their own, as suggested by some recent experiments,\ we might be able to detect them indirectly: they could be a form of “dark matter,” like that menti\ oned earlier, with sufficient gravitational attraction to stop the expansion of the universe and cause it to collapse again.

A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (1 of 11) [2/20/2001 3:15:29 AM] About one hundred seconds after the big bang, the temperature would have\ fallen to one thousand million degrees, the temperature inside the hottest stars. At this temperature protons an\ d neutrons would no longer have sufficient energy to escape the attraction of the strong nuclear force, and would h\ ave started to combine together to produce the nuclei of atoms of deuterium (heavy hydrogen), which contain one p\ roton and one neutron. The deuterium nuclei would then have combined with more protons and neutrons to make helium n\ uclei, which contain two protons and two neutrons, and also small amounts of a couple of heavier elements, li\ thium and beryllium. One can calculate that in the hot big bang model about a quarter of the protons and neutrons wo\ uld have been converted into helium nuclei, along with a small amount of heavy hydrogen and other elements. The rema\ ining neutrons would have decayed into protons, which are the nuclei of ordinary hydrogen atoms.

This picture of a hot early stage of the universe was first put forward \ by the scientist George Gamow in a famous paper written in 1948 with a student of his, Ralph Alpher. Gamow had qui\ te a sense of humor – he persuaded the nuclear scientist Hans Bethe to add his name to the paper to make the li\ st of authors “Alpher, Bethe, Gamow,” like the first three letters of the Greek alphabet, alpha, beta, gamma: parti\ cularly appropriate for a paper on the beginning of the universe! In this paper they made the remarkable prediction that \ radiation (in the form of photons) from the very hot early stages of the universe should still be around today, but \ with its temperature reduced to only a few degrees above absolute zero (–273ºC). It was this radiation that\ Penzias and Wilson found in 1965. At the time that Alpher, Bethe, and Gamow wrote their paper, not much was known about the\ nuclear reactions of protons and neutrons. Predictions made for the proportions of various elements in th\ e early universe were therefore rather inaccurate, but these calculations have been repeated in the light of be\ tter knowledge and now agree very well with what we observe. It is, moreover, very difficult to explain in any other\ way why there should be so much helium in the universe. We are therefore fairly confident that we have the right pictu\ re, at least back to about one second after the big bang.

Within only a few hours of the big bang, the production of helium and ot\ her elements would have stopped. And after that, for the next million years or so, the universe would have just con\ tinued expanding, without anything much happening. Eventually, once the temperature had dropped to a few thousan\ d degrees, and electrons and nuclei no longer had enough energy to overcome the electromagnetic attraction betw\ een them, they would have started combining to form atoms. The universe as a whole would have continued ex\ panding and cooling, but in regions that were slightly denser than average, the expansion would have been slowed \ down by the extra gravitational attraction.

This would eventually stop expansion in some regions and cause them to s\ tart to recollapse. As they were collapsing, the gravitational pull of matter outside these regions might\ start them rotating slightly. As the collapsing region got smaller, it would spin faster – just as skaters spinning o\ n ice spin faster as they draw in their arms.

Eventually, when the region got small enough, it would be spinning fast \ enough to balance the attraction of gravity, and in this way disklike rotating galaxies were born. Other regions, whi\ ch did not happen to pick up a rotation, would become oval-shaped objects called elliptical galaxies. In these, the reg\ ion would stop collapsing because individual parts of the galaxy would be orbiting stably round its center, but the g\ alaxy would have no overall rotation.

As time went on, the hydrogen and helium gas in the galaxies would break\ up into smaller clouds that would collapse under their own gravity. As these contracted, and the atoms within them \ collided with one another, the temperature of the gas would increase, until eventually it became hot enough to star\ t nuclear fusion reactions. These would convert the hydrogen into more helium, and the heat given off would rais\ e the pressure, and so stop the clouds from contracting any further. They would remain stable in this state for a lo\ ng time as stars like our sun, burning hydrogen into helium and radiating the resulting energy as heat and light. More m\ assive stars would need to be hotter to balance their stronger gravitational attraction, making the nuclear fusi\ on reactions proceed so much more rapidly that they would use up their hydrogen in as little as a hundred million years\ . They would then contract slightly, and as they heated up further, would start to convert helium into heavier eleme\ nts like carbon or oxygen. This, however, would not release much more energy, so a crisis would occur, as was desc\ ribed in the chapter on black holes. What happens next is not completely clear, but it seems likely that the centr\ al regions of the star would collapse to a very dense state, such as a neutron star or black hole. The outer regions of \ the star may sometimes get blown off in a tremendous explosion called a supernova, which would outshine all the ot\ her stars in its galaxy. Some of the heavier elements produced near the end of the star’s life would be flung back\ into the gas in the galaxy, and would provide some of the raw material for the next generation of stars. Our own sun c\ ontains about 2 percent of these heavier elements, because it is a second- or third-generation star, formed some \ five thousand million years ago out of a cloud of rotating gas containing the debris of earlier supernovas. Most \ of the gas in that cloud went to form the sun or got blown away, but a small amount of the heavier elements collected tog\ ether to form the bodies that now orbit the sun as planets like the earth.

The earth was initially very hot and without an atmosphere. In the cours\ e of time it cooled and acquired an A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (2 of 11) [2/20/2001 3:15:29 AM] atmosphere from the emission of gases from the rocks. This early atmosph\ ere was not one in which we could have survived. It contained no oxygen, but a lot of other gases that are pois\ onous to us, such as hydrogen sulfide (the gas that gives rotten eggs their smell). There are, however, other primitiv\ e forms of life that can flourish under such conditions. It is thought that they developed in the oceans, possibly as\ a result of chance combinations of atoms into large structures, called macromolecules, which were capable of assemblin\ g other atoms in the ocean into similar structures. They would thus have reproduced themselves and multiplied. I\ n some cases there would be errors in the reproduction. Mostly these errors would have been such that the new macr\ omolecule could not reproduce itself and eventually would have been destroyed. However, a few of the errors would\ have produced new macromolecules that were even better at reproducing themselves. They would have therefore ha\ d an advantage and would have tended to replace the original macromolecules. In this way a process of evoluti\ on was started that led to the development of more and more complicated, self-reproducing organisms. The first primiti\ ve forms of life consumed various materials, including hydrogen sulfide, and released oxygen. This gradually changed \ the atmosphere to the composition that it has today, and allowed the development of higher forms of life such as f\ ish, reptiles, mammals, and ultimately the human race.

This picture of a universe that started off very hot and cooled as it ex\ panded is in agreement with all the observational evidence that we have today. Nevertheless, it leaves a num\ ber of important questions unanswered:

1. Why was the early universe so hot?

2. Why is the universe so uniform on a large scale? Why does it look the\ same at all points of space and in all directions? In particular, why is the temperature of the microwave back-\ ground radiation so nearly the same when we look in different directions? It is a bit like asking a number of studen\ ts an exam question. If they all give exactly the same answer, you can be pretty sure they have communicated with each oth\ er. Yet, in the model described above, there would not have been time since the big bang for light to get from \ one distant region to another, even though the regions were close together in the early universe. According to the theo\ ry of relativity, if light cannot get from one region to another, no other information can. So there would be no way in\ which different regions in the early universe could have come to have the same temperature as each other, unless for s\ ome unexplained reason they happened to start out with the same temperature.

3. Why did the universe start out with so nearly the critical rate of expan\ sion that separates models that recollapse from those that go on expanding forever, that even now, ten thousand mil\ lion years later, it is still expanding at nearly the critical rate? If the rate of expansion one second after the big ban\ g had been smaller by even one part in a hundred thousand million million, the universe would have recollapsed be\ fore it ever reached its present size.

4. Despite the fact that the universe is so uniform and homogeneous on a\ large scale, it contains local irregularities, such as stars and galaxies. These are thought to have developed from sma\ ll differences in the density of the early universe from one region to another. What was the origin of these densit\ y fluctuations?

The general theory of relativity, on its own, cannot explain these featu\ res or answer these questions because of its prediction that the universe started off with infinite density at the bi\ g bang singularity. At the singularity, general relativity and all other physical laws would break down: one couldn’t\ predict what would come out of the singularity.

As explained before, this means that one might as well cut the big bang,\ and any events before it, out of the theory, because they can have no effect on what we observe. Space-time w ould have a boundary – a beginning at the big bang.

Science seems to have uncovered a set of laws that, within the limits se\ t by the uncertainty principle, tell us how the universe will develop with time, if we know its state at any one time. T\ hese laws may have originally been decreed by God, but it appears that he has since left the universe to evolve accord\ ing to them and does not now intervene in it.

But how did he choose the initial state or configuration of the universe\ ? What were the “boundary conditions” at the beginning of time?

One possible answer is to say that God chose the initial configuration o\ f the universe for reasons that we cannot hope to understand. This would certainly have been within the power of a\ n omnipotent being, but if he had started it off in such an incomprehensible way, why did he choose to let it evolve \ according to laws that we could understand?

The whole history of science has been the gradual realization that event\ s do not happen in an arbitrary manner, but that they reflect a certain underlying order, which may or may not be di\ vinely inspired. It would be only natural to suppose that this order should apply not only to the laws, but also to t\ he conditions at the boundary of space-time that specify the initial state of the universe. There may be a large num\ ber of models of the universe with different initial conditions that all obey the laws. There ought to be some princi\ ple that picks out one initial state, and hence A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (3 of 11) [2/20/2001 3:15:29 AM] one model, to represent our universe.

One such possibility is what are called chaotic boundary conditions. The\ se implicitly assume either that the universe is spatially infinite or that there are infinitely many universes. Under\ chaotic boundary conditions, the probability of finding any particular region of space in any given configuration just a\ fter the big bang is the same, in some sense, as the probability of finding it in any other configuration: the initial\ state of the universe is chosen purely randomly.

This would mean that the early universe would have probably been very ch\ aotic and irregular because there are many more chaotic and disordered configurations for the universe than th\ ere are smooth and ordered ones. (If each configuration is equally probable, it is likely that the universe starte\ d out in a chaotic and disordered state, simply because there are so many more of them.) It is difficult to see how suc\ h chaotic initial conditions could have given rise to a universe that is so smooth and regular on a large scale as our\ s is today. One would also have expected the density fluctuations in such a model to have led to the formation of man\ y more primordial black holes than the upper limit that has been set by observations of the gamma ray background.

If the universe is indeed spatially infinite, or if there are infinitely\ many universes, there would probably be some large regions somewhere that started out in a smooth and uniform manner. It is\ a bit like the well-known horde of monkeys hammering away on typewriters – most of what they write will be garba\ ge, but very occasionally by pure chance they will type out one of Shakespeare’s sonnets. Similarly, in the case of\ the universe, could it be that we are living in a region that just happens by chance to be smooth and uniform? At first si\ ght this might seem very improbable, because such smooth regions would be heavily outnumbered by chaotic and \ irregular regions. However, suppose that only in the smooth regions were galaxies and stars formed and were \ conditions right for the development of complicated self-replicating organisms like ourselves who were capable o\ f asking the question: why is the universe so smooth.? This is an example of the application of what is known as th\ e anthropic principle, which can be paraphrased as “We see the universe the way it is because we exist.”\ There are two versions of the anthropic principle, the weak and the stro\ ng. The weak anthropic principle states that in a universe that is large or infinite in space and/or time, the condit\ ions necessary for the development of intelligent life will be met only in certain regions that are limited in space and t\ ime. The intelligent beings in these regions should therefore not be surprised if they observe that their locality in\ the universe satisfies the conditions that are necessary for their existence. It is a bit like a rich person living in \ a wealthy neighborhood not seeing any poverty.

One example of the use of the weak anthropic principle is to “explain\ ” why the big bang occurred about ten thousand million years ago – it takes about that long for intelligent beings t\ o evolve. As explained above, an early generation of stars first had to form. These stars converted some of the original hydr\ ogen and helium into elements like carbon and oxygen, out of which we are made. The stars then exploded as supernovas,\ and their debris went to form other stars and planets, among them those of our Solar System, which is about five t\ housand million years old. The first one or two thousand million years of the earth’s existence were too hot for \ the development of anything complicated. The remaining three thousand million years or so have been taken up by the s\ low process of biological evolution, which has led from the simplest organisms to beings who are capable of measuri\ ng time back to the big bang.

Few people would quarrel with the validity or utility of the weak anthro\ pic principle. Some, however, go much further and propose a strong version of the principle. According to this theory,\ there are either many different universes or many different regions of a single universe, each with its own initial c\ onfiguration and, perhaps, with its own set of laws of science. In most of these universes the conditions would not be \ right for the development of complicated organisms; only in the few universes that are like ours would intelligen\ t beings develop and ask the question, “Why is the universe the way we see it?” The answer is then simple: if it had\ been different, we would not be here!

The laws of science, as we know them at present, contain many fundamenta\ l numbers, like the size of the electric charge of the electron and the ratio of the masses of the proton and the\ electron. We cannot, at the moment at least, predict the values of these numbers from theory – we have to find the\ m by observation. It may be that one day we shall discover a complete unified theory that predicts them all, but it \ is also possible that some or all of them vary from universe to universe or within a single universe. The remarkable fa\ ct is that the values of these numbers seem to have been very finely adjusted to make possible the development of li\ fe. For example, if the electric charge of the electron had been only slightly different, stars either would have been \ unable to burn hydrogen and helium, or else they would not have exploded. Of course, there might be other forms of i\ ntelligent life, not dreamed of even by writers of science fiction, that did not require the light of a star lik\ e the sun or the heavier chemical elements that are made in stars and are flung back into space when the stars explode. Neve\ rtheless, it seems clear that there are relatively few ranges of values for the numbers that would allow the dev\ elopment of any form of intelligent life. Most sets of values would give rise to universes that, although they might be\ very beautiful, would contain no one able to wonder at that beauty. One can take this either as evidence of a divine \ purpose in Creation and the choice of the A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (4 of 11) [2/20/2001 3:15:29 AM] laws of science or as support for the strong anthropic principle.

There are a number of objections that one can raise to the strong anthro\ pic principle as an explanation of the observed state of the universe. First, in what sense can all these diffe\ rent universes be said to exist? If they are really separate from each other, what happens in another universe can ha\ ve no observable consequences in our own universe. We should therefore use the principle of economy and cut t\ hem out of the theory. If, on the other hand, they are just different regions of a single universe, the laws of \ science would have to be the same in each region, because otherwise one could not move continuously from one regio\ n to another. In this case the only difference between the regions would be their initial configurations and\ so the strong anthropic principle would reduce to the weak one.

A second objection to the strong anthropic principle is that it runs aga\ inst the tide of the whole history of science. We have developed from the geocentric cosmologies of Ptolemy and his forebe\ ars, through the heliocentric cosmology of Copernicus and Galileo, to the modern picture in which the earth is a\ medium-sized planet orbiting around an average star in the outer suburbs of an ordinary spiral galaxy, which is\ itself only one of about a million million galaxies in the observable universe. Yet the strong anthropic principle \ would claim that this whole vast construction exists simply for our sake. This is very hard to believe. Our Solar Syst\ em is certainly a prerequisite for our existence, hand one might extend this to the whole of our galaxy to allow for an ea\ rlier generation of stars that created the heavier elements. But there does not seem to be any need for all those o\ ther galaxies, nor for the universe to be so uniform and similar in every direction on the large scale.

One would feel happier about the anthropic principle, at least in its we\ ak version, if one could show that quite a number of different initial configurations for the universe would have e\ volved to produce a universe like the one we observe. If this is the case, a universe that developed from some sort o\ f random initial conditions should contain a number of regions that are smooth and uniform and are suitable for the e\ volution of intelligent life. On the other hand, if the initial state of the universe had to be chosen extremely carefull\ y to lead to something like what we see around us, the universe would be unlikely to contain any region in which life w\ ould appear. In the hot big bang model described above, there was not enough time in the early universe for hea\ t to have flowed from one region to another.

This means that the initial state of the universe would have to have had\ exactly the same temperature everywhere in order to account for the fact that the microwave back-ground has the sam\ e temperature in every direction we look.

The initial rate of expansion also would have had to be chosen very prec\ isely for the rate of expansion still to be so close to the critical rate needed to avoid recollapse. This means that t\ he initial state of the universe must have been very carefully chosen indeed if the hot big bang model was correct right\ back to the beginning of time. It would be very difficult to explain why the universe should have begun in just thi\ s way, except as the act of a God who intended to create beings like us.

In an attempt to find a model of the universe in which many different in\ itial configurations could have evolved to something like the present universe, a scientist at the Massachusetts In\ stitute of Technology, Alan Guth, suggested that the early universe might have gone through a period of very rapid e\ xpansion. This expansion is said to be “inflationary,” meaning that the universe at one time expanded at \ an increasing rate rather than the decreasing rate that it does today. According to Guth, the radius of the universe increa\ sed by a million million million million million (1 with thirty zeros after it) times in only a tiny fraction of a second.

Guth suggested that the universe started out from the big bang in a very\ hot, but rather chaotic, state. These high temperatures would have meant that the particles in the universe would b\ e moving very fast and would have high energies. As we discussed earlier, one would expect that at such high te\ mperatures the strong and weak nuclear forces and the electromagnetic force would all be unified into a single \ force. As the universe expanded, it would cool, and particle energies would go down. Eventually there would be what is c\ alled a phase transition and the symmetry between the forces would be broken: the strong force would become differ\ ent from the weak and electromagnetic forces. One common example of a phase transition is the freezing of wate\ r when you cool it down. Liquid water is symmetrical, the same at every point and in every direction. However, wh\ en ice crystals form, they will have definite positions and will be lined up in some direction. This breaks water’s\ symmetry.

In the case of water, if one is careful, one can “supercool” it: t\ hat is, one can reduce the temperature below the freezing point (OºC) without ice forming. Guth suggested that the u\ niverse might behave in a similar way: the temperature might drop below the critical value without the symmetry bet\ ween the forces being broken. If this happened, the universe would be in an unstable state, with more energy t\ han if the symmetry had been broken. This special extra energy can be shown to have an antigravitational effect: i\ t would have acted just like the cosmological constant that Einstein introduced into general relativity when he was tr\ ying to construct a static model of the universe. Since the universe would already be expanding just as in the h\ ot big bang model, the repulsive effect of A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (5 of 11) [2/20/2001 3:15:29 AM] this cosmological constant would therefore have made the universe expand\ at an ever-increasing rate. Even in regions where there were more matter particles than average, the gravita\ tional attraction of the matter would have been outweighed by the repulsion of the effective cosmological constant.\ Thus these regions would also expand in an accelerating inflationary manner. As they expanded and the matter par\ ticles got farther apart, one would be left with an expanding universe that contained hardly any particles and was s\ till in the supercooled state. Any irregularities in the universe would simply have been smoothed out by th\ e expansion, as the wrinkles in a balloon are smoothed away when you blow it up. Thus the present smooth and uniform s\ tate of the universe could have evolved from many different non-uniform initial states.

In such a universe, in which the expansion was accelerated by a cosmolog\ ical constant rather than slowed down by the gravitational attraction of matter, there would be enough time for l\ ight to travel from one region to another in the early universe. This could provide a solution to the problem, raised ear\ lier, of why different regions in the early universe have the same properties. Moreover, the rate of expansion of th\ e universe would automatically become very close to the critical rate determined by the energy density of the \ universe. This could then explain why the rate of expansion is still so close to the critical rate, without having to a\ ssume that the initial rate of expansion of the universe was very carefully chosen.

The idea of inflation could also explain why there is so much matter in \ the universe. There are something like ten million million million million million million million million million \ million million million million million (1 with eighty zeros after it) particles in the region of the universe that we can obs\ erve. Where did they all come from? The answer is that, in quantum theory, particles can be created out of energy in th\ e form of particle/antiparticle pairs. But that just raises the question of where the energy came from. The answer is that th\ e total energy of the universe is exactly zero. The matter in the universe is made out of positive energy. However\ , the matter is all attracting itself by gravity.

Two pieces of matter that are close to each other have less energy than \ the same two pieces a long way apart, because you have to expend energy to separate them against the gravitati\ onal force that is pulling them together.

Thus, in a sense, the gravitational field has negative energy. In the ca\ se of a universe that is approximately uniform in space, one can show that this negative gravitational energy exactly c\ ancels the positive energy represented by the matter. So the total energy of the universe is zero.

Now twice zero is also zero. Thus the universe can double the amount of \ positive matter energy and also double the negative gravitational energy without violation of the conservation of e\ nergy. This does not happen in the normal expansion of the universe in which the matter energy density goes down a\ s the universe gets bigger. It does happen, however, in the inflationary expansion because the energy density of the\ supercooled state remains constant while the universe expands: when the universe doubles in size, the positive ma\ tter energy and the negative gravitational energy both double, so the total energy remains zero. During the inflati\ onary phase, the universe increases its size by a very large amount. Thus the total amount of energy available to mak\ e particles becomes very large. As Guth has remarked, “It is said that there’s no such thing as a free lun\ ch. But the universe is the ultimate free lunch.” The universe is not expanding in an inflationary way today. Thus there h\ as to be some mechanism that would eliminate the very large effective cosmological constant and so change t\ he rate of expansion from an accelerated one to one that is slowed down by gravity, as we have today. In the infl\ ationary expansion one might expect that eventually the symmetry between the forces would be broken, just as supe\ r-cooled water always freezes in the end.

The extra energy of the unbroken symmetry state would then be released a\ nd would reheat the universe to a temperature just below the critical temperature for symmetry between the\ forces. The universe would then go on to expand and cool just like the hot big bang model, but there would now be\ an explanation of why the universe was expanding at exactly the critical rate and why different regions had the\ same temperature.

In Guth’s original proposal the phase transition was supposed to occu\ r suddenly, rather like the appearance of ice crystals in very cold water. The idea was that “bubbles” of the ne\ w phase of broken symmetry would have formed in the old phase, like bubbles of steam surrounded by boiling water. The bu\ bbles were supposed to expand and meet up with each other until the whole universe was in the new phase. The tr\ ouble was, as I and several other people pointed out, that the universe was expanding so fast that even if the bu\ bbles grew at the speed of light, they would be moving away from each other and so could not join up. The universe wo\ uld be left in a very non-uniform state, with some regions still having symmetry between the different forces. Su\ ch a model of the universe would not correspond to what we see.

In October 1981, I went to Moscow for a conference on quantum gravity. A\ fter the conference I gave a seminar on the inflationary model and its problems at the Sternberg Astronomical In\ stitute. Before this, I had got someone else to give my lectures for me, because most people could not understand my \ voice. But there was not time to prepare this seminar, so I gave it myself, with one of my graduate students repe\ ating my words. It worked well, and gave me A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (6 of 11) [2/20/2001 3:15:29 AM] much more contact with my audience. In the audience was a young Russian,\ Andrei Linde, from the Lebedev Institute in Moscow. He said that the difficulty with the bubbles not jo\ ining up could be avoided if the bubbles were so big that our region of the universe is all contained inside a single bub\ ble. In order for this to work, the change from symmetry to broken symmetry must have taken place very slowly inside the\ bubble, but this is quite possible according to grand unified theories. Linde’s idea of a slow breaking \ of symmetry was very good, but I later realized that his bubbles would have to have been bigger than the size of the uni\ verse at the time! I showed that instead the symmetry would have broken everywhere at the same time, rather than just\ inside bubbles. This would lead to a uniform universe, as we observe. I was very excited by this idea and dis\ cussed it with one of my students, Ian Moss.

As a friend of Linde’s, I was rather embarrassed, however, when I was\ later sent his paper by a scientific journal and asked whether it was suitable for publication. I replied that there was \ this flaw about the bubbles being bigger than the universe, but that the basic idea of a slow breaking of symmetry was\ very good. I recommended that the paper ¿ published as it was because it would take Linde several months to correc\ t it, since anything he sent to the West would have to be passed by Soviet censorship, which was neither very ski\ llful nor very quick with scientific papers.

Instead, I wrote a short paper with Ian Moss in the same journal in whic\ h we pointed out this problem with the bubble and showed how it could be resolved.

The day after I got back from Moscow I set out for Philadelphia, where I\ was due to receive a medal from the Franklin Institute. My secretary, Judy Fella, had used her not inconside\ rable charm to persuade British Airways to give herself and me free seats on a Concorde as a publicity venture. How\ ever, I .was held up on my way to the airport by heavy rain and I missed the plane. Nevertheless, I got to Phi\ ladelphia in the end and received my medal. I was then asked to give a seminar on the inflationary universe at Drexel \ University in Philadelphia. I gave the same seminar about the problems of the inflationary universe, just as in Mosc\ ow.

A very similar idea to Linde’s was put forth independently a few mont\ hs later by Paul Steinhardt and Andreas Albrecht of the University of Pennsylvania. They are now given joint cre\ dit with Linde for what is called “the new inflationary model,” based on the idea of a slow breaking of symmetry\ . (The old inflationary model was Guth’s original suggestion of fast symmetry breaking with the formation of bubb\ les.) The new inflationary model was a good attempt to explain why the univers\ e is the way it is. However, I and several other people showed that, at least in its original form, it predicted mu\ ch greater variations in the temperature of the microwave background radiation than are observed. Later work has also ca\ st doubt on whether there could be a phase transition in the very early universe of the kind required. In my \ personal opinion, the new inflationary model is now dead as a scientific theory, although a lot of people do not seem to\ have heard of its demise and are still writing papers as if it were viable. A better model, called the chaotic inflatio\ nary model, was put forward by Linde in 1983. In this there is no phase transition or supercooling. Instead, there is a s\ pin 0 field, which, because of quantum fluctuations, would have large values in some regions of the early unive\ rse. The energy of the field in those regions would behave like a cosmological constant. It would have a repulsive gra\ vitational effect, and thus make those regions expand in an inflationary manner. As they expanded, the energy o\ f the field in them would slowly decrease until the inflationary expansion changed to an expansion like that in th\ e hot big bang model. One of these regions would become what we now see as the observable universe. This model has \ all the advantages of the earlier inflationary models, but it does not depend on a dubious phase transitio\ n, and it can moreover give a reasonable size for the fluctuations in the temperature of the microwave background that\ agrees with observation.

This work on inflationary models showed that the present state of the un\ iverse could have arisen from quite a large number of different initial configurations. This is important, because i\ t shows that the initial state of the part of the universe that we inhabit did not have to be chosen with great care. So w\ e may, if we wish, use the weak anthropic principle to explain why the universe looks the way it does now. It cann\ ot be the case, however, that every initial configuration would have led to a universe like the one we observe. One \ can show this by considering a very different state for the universe at the present time, say, a very lumpy \ and irregular one. One could use the laws of science to evolve the universe back in time to determine its configurati\ on at earlier times. According to the singularity theorems of classical general relativity, there would still have been a \ big bang singularity. If you evolve such a universe forward in time according to the laws of science, you will end \ up with the lumpy and irregular state you started with. Thus there must have been initial configurations that woul\ d not have given rise to a universe like the one we see today. So even the inflationary model does not tell us why th\ e initial configuration was not such as to produce something very different from what we observe. Must we turn to t\ he anthropic principle for an explanation?

Was it all just a lucky chance? That would seem a counsel of despair, a \ negation of all our hopes of understanding the underlying order of the universe.

In order to predict how the universe should have started off, one needs \ laws that hold at the beginning of time. If the classical theory of general relativity was correct, the singularity theo\ rems that Roger Penrose and I proved show that A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (7 of 11) [2/20/2001 3:15:29 AM] the beginning of time would have been a point of infinite density and in\ finite curvature of space-time. All the known laws of science would break down at such a point. One might suppose that\ there were new laws that held at singularities, but it would be very difficult even to formulate such law\ s at such badly behaved points, and we would have no guide from observations as to what those laws might be. However,\ what the singularity theorems really indicate is that the gravitational field becomes so strong that quantum \ gravitational effects become important:

classical theory is no longer a good description of the universe. So one\ has to use a quantum theory of gravity to discuss the very early stages of the universe. As we shall see, it is po\ ssible in the quantum theory for the ordinary laws of science to hold everywhere, including at the beginning of time: \ it is not necessary to postulate new laws for singularities, because there need not be any singularities in the quantu\ m theory.

We don’t yet have a complete and consistent theory that combines quan\ tum mechanics and gravity. However, we are fairly certain of some features that such a unified theory should ha\ ve. One is that it should incorporate Feynman’s proposal to formulate quantum theory in terms of a sum over\ histories. In this approach, a particle does not have just a single history, as it would in a classical theory. Inste\ ad, it is supposed to follow every possible path in space-time, and with each of these histories there are associated a coup\ le of numbers, one represent-ing the size of a wave and the other representing its position in the cycle (its phase)\ . The probability that the particle, say, passes through some particular point is found by adding up the waves associated\ with every possible history that passes through that point. When one actually tries to perform these sums, howev\ er, one runs into severe technical problems. The only way around these is the following peculiar prescripti\ on: one must add up the waves for particle histories that are not in the “real” time that you and I experienc\ e but take place in what is called imaginary time.

Imaginary time may sound like science fiction but it is in fact a well-d\ efined mathematical concept. If we take any ordinary (or “real”) number and multiply it by itself, the resul\ t is a positive number. (For example, 2 times 2 is 4, but so is – 2 times – 2.) There are, however, special numbers (calle\ d imaginary numbers) that give negative numbers when multiplied by themselves. (The one called i, when multiplied by it\ self, gives – 1, 2i multiplied by itself gives – 4, and so on.) One can picture real and imaginary numbers in the following way: The rea\ l numbers can be represented by a line going from left to right, with zero in the middle, negative numbers like\ – 1, – 2, etc. on the left, and positive numbers, 1, 2, etc. on the right. Then imaginary numbers are represented by a lin\ e going up and down the page, with i, 2i, etc.

above the middle, and – i, – 2i, etc. below. Thus imaginary number\ s are in a sense numbers at right angles to ordinary real numbers.

To avoid the technical difficulties with Feynman’s sum over histories\ , one must use imaginary time. That is to say, for the purposes of the calculation one must measure time using imaginary nu\ mbers, rather than real ones. This has an interesting effect on space-time: the distinction between time and space\ disappears completely. A space-time in which events have imaginary values of the time coordinate is said to be \ Euclidean, after the ancient Greek Euclid, who founded the study of the geometry of two-dimensional surfaces. What \ we now call Euclidean space-time is very similar except that it has four dimensions instead of two. In Euclidean \ space-time there is no difference between the time direction and directions in space. On the other hand, in real space\ -time, in which events are labeled by ordinary, real values of the time coordinate, it is easy to tell the difference –\ the time direction at all points lies within the light cone, and space directions lie outside. In any case, as far as everyday \ quantum mechanics is concerned, we may regard our use of imaginary time and Euclidean space-time as merely a ma\ thematical device (or trick) to calculate answers about real space-time.

A second feature that we believe must be part of any ultimate theory is \ Einstein’s idea that the gravitational field is represented by curved space-time: particles try to follow the nearest th\ ing to a straight path in a curved space, but because space-time is not flat their paths appear to be bent, as if by a\ gravitational field. When we apply Feynman’s sum over histories to Einstein’s view of gravity, the analogue of the\ history of a particle is now a complete curved space-time that represents the history of the whole universe. To avoid t\ he technical difficulties in actually performing the sum over histories, these curved space-times must be taken to be Euc\ lidean. That is, time is imaginary and is indistinguishable from directions in space. To calculate the probability\ of finding a real space-time with some certain property, such as looking the same at every point and in every direction\ , one adds up the waves associated with all the histories that have that property.

In the classical theory of general relativity, there are many different \ possible curved space-times, each corresponding to a different initial state of the universe. If we knew the initial sta\ te of our universe, we would know its entire history.

Similarly, in the quantum theory of gravity, there are many different po\ ssible quantum states for the universe. Again, if we knew how the Euclidean curved space-times in the sum over historie\ s behaved at early times, we would know the quantum state of the universe.

A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (8 of 11) [2/20/2001 3:15:29 AM] In the classical theory of gravity, which is based on real space-time, t\ here are only two possible ways the universe can behave: either it has existed for an infinite time, or else it had a\ beginning at a singularity at some finite time in the past. In the quantum theory of gravity, on the other hand, a third p\ ossibility arises. Because one is using Euclidean space-times, in which the time direction is on the same footin\ g as directions in space, it is possible for space-time to be finite in extent and yet to have no singularities that \ formed a boundary or edge. Space-time would be like the surface of the earth, only with two more dimensions. The sur\ face of the earth is finite in extent but it doesn’t have a boundary or edge: if you sail off into the sunset, you\ don’t fall off the edge or run into a singularity. (I know, because I have been round the world!) If Euclidean space-time stretches back to infinite imaginary time, or el\ se starts at a singularity in imaginary time, we have the same problem as in the classical theory of specifying the initi\ al state of the universe: God may know how the universe began, but we cannot give any particular reason for thinkin\ g it began one way rather than another. On the other hand, the quantum theory of gravity has opened up a new possib\ ility, in which there would be no boundary to space-time and so there would be no need to specify the behavior at t\ he boundary. There would be no singularities at which the laws of science broke down, and no edge of sp\ ace-time at which one would have to appeal to God or some new law to set the boundary conditions for space-time. On\ e could say: “The boundary condition of the universe is that it has no boundary.” The universe would be compl\ etely self-contained and not affected by anything outside itself. It would neither be created nor destroyed, It w\ ould just BE.

It was at the conference in the Vatican mentioned earlier that I first p\ ut forward the suggestion that maybe time and space together formed a surface that was finite in size but did not have\ any boundary or edge. My paper was rather mathematical, however, so its implications for the role of God in the cr\ eation of the universe were not generally recognized at the time (just as well for me). At the time of the Vatic\ an conference, I did not know how to use the “no boundary” idea to make predictions about the universe. However, I spe\ nt the following sum-mer at the University of California, Santa Barbara. There a friend and colleague of mine, Jim Har\ tle, worked out with me what conditions the universe must satisfy if space-time had no boundary. When I returned to \ Cambridge, I continued this work with two of my research students, Julian Luttrel and Jonathan Halliwell.

I’d like to emphasize that this idea that time and space should be fi\ nite “without boundary” is just a proposal: it cannot be deduced from some other principle. Like any other scientific theory, \ it may initially be put forward for aesthetic or metaphysical reasons, but the real test is whether it makes predictions \ that agree with observation. This, how-ever, is difficult to determine in the case of quantum gravity, for two reasons. \ First, as will be explained in Chapter 11, we are not yet sure exactly which theory successfully combines general relativi\ ty and quantum mechanics, though we know quite a lot about the form such a theory must have. Second, any model th\ at described the whole universe in detail would be much too complicated mathematically for us to be able to calcul\ ate exact predictions. One therefore has to make simplifying assumptions and approximations – and even then, the \ problem of extracting predictions remains a formidable one.

Each history in the sum over histories will describe not only the space-\ time but everything in it as well, including any complicated organisms like human beings who can observe the history of t\ he universe. This may provide another justification for the anthropic principle, for if all the histories are \ possible, then so long as we exist in one of the histories, we may use the anthropic principle to explain why the univers\ e is found to be the way it is. Exactly what meaning can be attached to the other histories, in which we do not exist\ , is not clear. This view of a quantum theory of gravity would be much more satisfactory, however, if one could show t\ hat, using the sum over histories, our universe is not just one of the possible histories but one of the most p\ robable ones. To do this, we must perform the sum over histories for all possible Euclidean space-times that have no b\ oundary.

Under the “no boundary” proposal one learns that the chance of the\ universe being found to be following most of the possible histories is negligible, but there is a particular family of hi\ stories that are much more probable than the others. These histories may be pictured as being like the surface of the\ earth, with the distance from the North Pole representing imaginary time and the size of a circle of constant distanc\ e from the North Pole representing the spatial size of the universe. The universe starts at the North Pole as a single \ point. As one moves south, the circles of latitude at constant distance from the North Pole get bigger, correspond\ ing to the universe expanding with imaginary time Figure 8:1 . The universe would reach a maximum size at the equator and would contr\ act with increasing imaginary time to a single point at the South Pole. Ever though the univ\ erse would have zero size at the North and South Poles, these points would not be singularities, any more than the \ North aid South Poles on the earth are singular. The laws of science will hold at them, just as they do at the \ North and South Poles on the earth.

A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (9 of 11) [2/20/2001 3:15:29 AM] Figure 8:1 The history of the universe in real time, however, would look very diffe\ rent. At about ten or twenty thousand million years ago, it would have a minimum size, which was equal to the maximum \ radius of the history in imaginary time. At later real times, the universe would expand like the chaotic inflationar\ y model proposed by Linde (but one would not now have to assume that the universe was created somehow in the right so\ rt of state). The universe would expand to a very large size Figure 8:1 and eventually it would collapse again into what looks like a singulari\ ty in real time. Thus, in a sense, we are still all doomed, even if we keep away from black hol\ es. Only if we could picture the universe in terms of imaginary time would there be no singularities.

If the universe really is in such a quantum state, there would be no sin\ gularities in the history of the universe in imaginary time. It might seem therefore that my more recent work had com\ pletely undone the results of my earlier work on singularities. But, as indicated above, the real importance of t\ he singularity theorems was that they showed that the gravitational field must become so strong that quantum gravitat\ ional effects could not be ignored. This in turn led to the idea that the universe could be finite in imaginary time but \ without boundaries or singularities. When one goes back to the real time in which we live, however, there will still a\ ppear to be singularities. The poor astronaut who falls into a black hole will still come to a sticky end; only if he \ lived in imaginary time would he encounter no singularities.

This might suggest that the so-called imaginary time is really the real \ time, and that what we call real time is just a figment of our imaginations. In real time, the universe has a beginning \ and an end at singularities that form a boundary to space-time and at which the laws of science break down. But \ in imaginary time, there are no singularities or boundaries. So maybe what we call imaginary time is rea\ lly more basic, and what we call real is just an idea that we invent to help us describe what we think the universe is\ like. But according to the approach I described in Chapter 1, a scientific theory is just a mathematical model\ we make to describe our observations: it exists only in our minds. So it is meaningless to ask: which is real, “\ real” or “imaginary” time? It is simply a matter of which is the more useful description.

One can also use the sum over histories, along with the no boundary prop\ osal, to find which properties of the universe are likely to occur together. For example, one can calculate th\ e probability that the universe is expanding at nearly the same rate in all different directions at a time when the dens\ ity of the universe has its present value. In the simplified models that have been examined so far, this probability turns\ out to be high; that is, the proposed no boundary condition leads to the prediction that it is extremely probable\ that the present rate of expansion of the universe is almost the same in each direction. This is consistent with t\ he observations of the microwave background radiation, which show that it has almost exactly the same intensity in a\ ny direction. If the universe were expanding faster in some directions than in others, the intensity of the radiation\ in those directions would be reduced by an A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (10 of 11) [2/20/2001 3:15:29 AM] additional red shift.

Further predictions of the no boundary condition are currently being wor\ ked out. A particularly interesting problem is the size of the small departures from uniform density in the early unive\ rse that caused the formation first of the galaxies, then of stars, and finally of us. The uncertainty principle im\ plies that the early universe cannot have been completely uniform because there must have been some uncertainties or fl\ uctuations in the positions and velocities of the particles. Using the no boundary condition, we find that the univ\ erse must in fact have started off with just the minimum possible non-uniformity allowed by the uncertainty principle. Th\ e universe would have then undergone a period of rapid expansion, as in the inflationary models. During this pe\ riod, the initial non-uniformities would have been amplified until they were big enough to explain the origin of the s\ tructures we observe around us. In 1992 the Cosmic Background Explorer satellite (COBE) first detected very slight\ variations in the intensity of the microwave background with direction. The way these non-uniformities depend on dire\ ction seems to agree with the predictions of the inflationary model and the no boundary proposal. Thus the no boun\ dary proposal is a good scientific theory in the sense of Karl Popper: it could have been falsified by observations b\ ut instead its predictions have been confirmed. In an expanding universe in which the density of matter varie\ d slightly from place to place, gravity would have caused the denser regions to slow down their expansion and start co\ ntracting. This would lead to the formation of galaxies, stars, and eventually even insignificant creatures like our\ selves. Thus all the complicated structures that we see in the universe might be explained by the no boundary condition f\ or the universe together with the uncertainty principle of quantum mechanics.

The idea that space and time may form a closed surface without boundary \ also has profound implications for the role of God in the affairs of the universe. With the success of scientific th\ eories in describing events, most people have come to believe that God allows the universe to evolve according to a se\ t of laws and does not intervene in the universe to break these laws. However, the laws do not tell us what the \ universe should have looked like when it started – it would still be up to God to wind up the clockwork and ch\ oose how to start it off. So long as the universe had a beginning, we could suppose it had a creator. But if the universe \ is really completely self-contained, having no boundary or edge, it would have neither beginning nor end: it would simp\ ly be. What place, then, for a creator?

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Chapter 8 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/g.html (11 of 11) [2/20/2001 3:15:29 AM] CHAPTER 9 THE ARROW OF TIME   In previous chapters we have seen how our views of the nature of time ha\ ve changed over the years. Up to the beginning of this century people believed in an absolute time. That is, \ each event could be labeled by a number called “time” in a unique way, and all good clocks would agree on \ the time interval between two events.

However, the discovery that the speed of light appeared the same to ever\ y observer, no matter how he was moving, led to the theory of relativity – and in that one had to aban\ don the idea that there was a unique absolute time. Instead, each observer would have his own measure of time\ as recorded by a clock that he carried: clocks carried by different observers would not necessarily agr\ ee. Thus time became a more personal concept, relative to the observer who measured it.

When one tried to unify gravity with quantum mechanics, one had to intro\ duce the idea of “imaginary” time.

Imaginary time is indistinguishable from directions in space. If one can\ go north, one can turn around and head south; equally, if one can go forward in imaginary time, one ought to be\ able to turn round and go backward.

This means that there can be no important difference between the forward\ and backward directions of imaginary time. On the other hand, when one looks at “real” time, \ there’s a very big difference between the forward and backward directions, as we all know. Where does this differe\ nce between the past and the future come from? Why do we remember the past but not the future?

The laws of science do not distinguish between the past and the future. \ More precisely, as explained earlier, the laws of science are unchanged under the combination of operations (\ or symmetries) known as C, P, and T.

(C means changing particles for antiparticles. P means taking the mirror image, so left and right are interchanged. And T means reversing the direction of motion of all particles: in effect, run\ ning the motion backward.) The laws of science that govern the behavior of matter under\ all normal situations are unchanged under the combination of the two operations C and P on their own. In oth\ er words, life would be just the same for the inhabitants of another planet who were both mirror images of us \ and who were made of antimatter, rather than matter.

If the laws of science are unchanged by the combination of operations C \ and P, and also by the combination C, P, and T, they must also be unchanged under the operation T alone. Yet t\ here is a big difference between the forward and backward directions of real time in ordinary life. Imagine a\ cup of water falling off a table and breaking into pieces on the floor. If you take a film of this, you can e\ asily tell whether it is being run forward or backward. If you run it backward you will see the pieces suddenly gather\ themselves together off the floor and jump back to form a whole cup on the table. You can tell that the film i\ s being run backward because this kind of behavior is never observed in ordinary life. If it were, crockery man\ ufacturers would go out of business.

The explanation that is usually given as to why we don’t see broken c\ ups gathering themselves together off the floor and jumping back onto the table is that it is forbidden by the sec\ ond law of thermodynamics. This says that in any closed system disorder, or entropy, always increases with time. I\ n other words, it is a form of Murphy’s law: things always tend to go wrong! An intact cup on the table is a sta\ te of high order, but a broken cup on the floor is a disordered state. One can go readily from the cup on the tabl\ e in the past to the broken cup on the floor in the future, but not the other way round.

The increase of disorder or entropy with time is one example of what is \ called an arrow of time, something that distinguishes the past from the future, giving a direction to time. Ther\ e are at least three different arrows of time. First, there is the thermodynamic arrow of time, the direction of \ time in which disorder or entropy increases. Then, there is the psychological arrow of time. This is the d\ irection in which we feel time passes, the direction in which we remember the past but not the future. Finally, the\ re is the cosmological arrow of time. This is the direction of time in which the universe is expanding rather than \ contracting.

In this chapter I shall argue that the no boundary condition for the uni\ verse, together with the weak anthropic principle, can explain why all three arrows point in the same direction \ – and moreover, why a well-defined arrow of time should exist at all. I shall argue that the psychological arrow \ is determined by the thermodynamic arrow, A Brief History of Time - Stephen Hawking... Chapter 9 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/h.html (1 of 5) [2/20/2001 3:15:38 AM] and that these two arrows necessarily always point in the same direction\ . If one assumes the no boundary condition for the universe, we shall see that there must be well-defined\ thermodynamic and cosmological arrows of time, but they will not point in the same direction for the wh\ ole history of the universe. However, I shall argue that it is only when they do point in the same direction tha\ t conditions are suitable for the development of intelligent beings who can ask the question: why does dis\ order increase in the same direction of time as that in which the universe expands?

I shall discuss first the thermodynamic arrow of time. The second law of\ thermodynamics results from the fact that there are always many more disordered states than there are ordered\ ones. For example, consider the pieces of a jigsaw in a box. There is one, and. only one, arrangement in\ which the pieces make a complete picture. On the other hand, there are a very large number of arrangement\ s in which the pieces are disordered and don’t make a picture.

Suppose a system starts out in one of the small number of ordered states\ . As time goes by, the system will evolve according to the laws of science and its state will change. At a \ later time, it is more probable that the system will be in a disordered state than in an ordered one because ther\ e are more disordered states. Thus disorder will tend to increase with time if the system obeys an initial \ condition of high order.

Suppose the pieces of the jigsaw start off in a box in the ordered arran\ gement in which they form a picture. If you shake the box, the pieces will take up another arrangement. This wil\ l probably be a disordered arrangement in which the pieces don’t form a proper picture, simply b\ ecause there are so many more disordered arrangements. Some groups of pieces may still form parts of t\ he picture, but the more you shake the box, the more likely it is that these groups will get broken up and \ the pieces will be in a completely jumbled state in which they don’t form any sort of picture. So the disorder o\ f the pieces will probably increase with time if the pieces obey the initial condition that they start off in a condition\ of high order.

Suppose, however, that God decided that the universe should finish up in\ a state of high order but that it didn’t matter what state it started in. At early times the universe would proba\ bly be in a disordered state. This would mean that disorder would decrease with time. You would see broken cups gathering themselves together and jumping back onto the table. However, any human beings who were observin\ g the cups would be living in a universe in which disorder decreased with time. I shall argue that such \ beings would have a psychological arrow of time that was backward. That is, they would remember events in \ the future, and not remember events in their past. When the cup was broken, they would remember it being on \ the table, but when it was on the table, they would not remember it being on the floor.

It is rather difficult to talk about human memory because we don’t kn\ ow how the brain works in detail. We do, however, know all about how computer memories work. I shall therefore di\ scuss the psychological arrow of time for computers. I think it is reasonable to assume that the arrow fo\ r computers is the same as that for humans. If it were not, one could make a killing on the stock exchange b\ y having a computer that would remember tomorrow’s prices! A computer memory is basically a device c\ ontaining elements that can exist in either of two states. A simple example is an abacus. In its simplest for\ m, this consists of a number of wires; on each wire there are a number of beads that can be put in one of two posi\ tions. Before an item is recorded in a computer’s memory, the memory is in a disordered state, with equal pr\ obabilities for the two possible states.

(The abacus beads are scattered randomly on the wires of the abacus.) \ After the memory interacts with the system to be remembered, it will definitely be in one state or the other\ , according to the state of the system.

(Each abacus bead will be at either the left or the right of the abacus\ wire.) So the memory has passed from a disordered state to an ordered one. However, in order to make sure that \ the memory is in the right state, it is necessary to use a certain amount of energy (to move the bead or to pow\ er the computer, for example). This energy is dissipated as heat, and increases the amount of disorder in th\ e universe. One can show that this increase in disorder is always greater than the increase in the order of\ the memory itself. Thus the heat expelled by the computer’s cooling fan means that when a computer rec\ ords an item in memory, the total amount of disorder in the universe still goes up. The direction of time \ in which a computer remembers the past is the same as that in which disorder increases.

Our subjective sense of the direction of time, the psychological arrow o\ f time, is therefore determined within our brain by the thermodynamic arrow of time. Just like a computer, we must \ remember things in the order in which entropy increases. This makes the second law of thermodynamics almost tr\ ivial. Disorder increases with time A Brief History of Time - Stephen Hawking... Chapter 9 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/h.html (2 of 5) [2/20/2001 3:15:38 AM] because we measure time in the direction in which disorder increases You\ can’t have a safer bet than that!

But why should the thermodynamic arrow of time exist at all? Or, in othe\ r words, why should the universe be in a state of high order at one end of time, the end that we call the past?\ Why is it not in a state of complete disorder at all times? After all, this might seem more probable. And why\ is the direction of time in which disorder increases the same as that in which the universe expands?

In the classical theory of general relativity one cannot predict how the\ universe would have begun because all the known laws of science would have broken down at the big bang singula\ rity. The universe could have started out in a very smooth and ordered state. This would have led to w\ ell-defined thermodynamic and cosmological arrows of time, as we observe. But it could equally well ha\ ve started out in a very lumpy and disordered state. In that case, the universe would already be in a state\ of complete disorder, so disorder could not increase with time. It would either stay constant, in which case the\ re would be no well-defined thermodynamic arrow of time, or it would decrease, in which case the the\ rmodynamic arrow of time would point in the opposite direction to the cosmological arrow. Neither of these po\ ssibilities agrees with what we observe.

However, as we have seen, classical general relativity predicts its own \ downfall. When the curvature of space-time becomes large, quantum gravitational effects will become impo\ rtant and the classical theory will cease to be a good description of the universe. One has to use a quantum\ theory of gravity to understand how the universe began.

In a quantum theory of gravity, as we saw in the last chapter, in order \ to specify the state of the universe one would still have to say how the possible histories of the universe would\ behave at the boundary of space-time in the past. One could avoid this difficulty of having to describe what we \ do not and cannot know only if the histories satisfy the no boundary condition: they are finite in extent b\ ut have no boundaries, edges, or singularities. In that case, the beginning of time would be a regular, s\ mooth point of space-time and the universe would have begun its expansion in a very smooth and ordered sta\ te. It could not have been completely uniform, because that would violate the uncertainty principle\ of quantum theory. There had to be small fluctuations in the density and velocities of particles. The no bo\ undary condition, however, implied that these fluctuations were as small as they could be, consistent with the u\ ncertainty principle.

The universe would have started off with a period of exponential or “\ inflationary” expansion in which it would have increased its size by a very large factor. During this expansion, t\ he density fluctuations would have remained small at first, but later would have started to grow. Regions i\ n which the density was slightly higher than average would have had their expansion slowed down by the gravitati\ onal attraction of the extra mass.

Eventually, such regions would stop expanding and collapse to form galax\ ies, stars, and beings like us. The universe would have started in a smooth and ordered state, and would bec\ ome lumpy and disordered as time went on. This would explain the existence of the thermodynamic arrow of \ time.

But what would happen if and when the universe stopped expanding and beg\ an to contract? Would the thermodynamic arrow reverse and disorder begin to decrease with time? Th\ is would lead to all sorts of science-fiction-like possibilities for people who survived from the expa\ nding to the contracting phase. Would they see broken cups gathering themselves together off the floor and jum\ ping back onto the table? Would they be able to remember tomorrow’s prices and make a fortune on the stock\ market? It might seem a bit academic to worry about what will happen when the universe collapses again, as it\ will not start to contract for at least another ten thousand million years. But there is a quicker way to find o\ ut what will happen: jump into a black hole. The collapse of a star to form a black hole is rather like the lat\ er stages of the collapse of the whole universe. So if disorder were to decrease in the contracting phase of th\ e universe, one might also expect it to decrease inside a black hole. So perhaps an astronaut who fell into a bl\ ack hole would be able to make money at roulette by remembering where the ball went before he placed his bet.\ (Unfortunately, however, he would not have long to play before he was turned to spaghetti. Nor would he be abl\ e to let us know about the reversal of the thermodynamic arrow, or even bank his winnings, because he would be \ trapped behind the event horizon of the black hole.) At first, I believed that disorder would decrease when the universe reco\ llapsed. This was because I thought that the universe had to return to a smooth and ordered state when it became \ small again. This would mean that the contracting phase would be like the time reverse of the expanding ph\ ase. People in the contracting phase would live their lives backward: they would die before they were born an\ d get younger as the universe A Brief History of Time - Stephen Hawking... Chapter 9 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/h.html (3 of 5) [2/20/2001 3:15:38 AM] contracted.

This idea is attractive because it would mean a nice symmetry between th\ e expanding and contracting phases.

However, one cannot adopt it on its own, independent of other ideas abou\ t the universe. The question is: is it implied by the no boundary condition, or is it inconsistent with that co\ ndition? As I said, I thought at first that the no boundary condition did indeed imply that disorder would decrease in t\ he contracting phase. I was misled partly by the analogy with the surface of the earth. If one took the beg\ inning of the universe to correspond to the North Pole, then the end of the universe should be similar to the be\ ginning, just as the South Pole is similar to the North. However, the North and South Poles correspond to the begin\ ning and end of the universe in imaginary time. The beginning and end in real time can be very different\ from each other. I was also misled by work I had done on a simple model of the universe in which the collapsin\ g phase looked like the time reverse of the expanding phase. However, a colleague of mine, Don Page, of Penn Sta\ te University, pointed out that the no boundary condition did not require the contracting phase necessarily \ to be the time reverse of the expanding phase. Further, one of my students, Raymond Laflamme, found that in a sl\ ightly more complicated model, the collapse of the universe was very different from the expansion. I realiz\ ed that I had made a mistake: the no boundary condition implied that disorder would in fact continue to incre\ ase during the contraction. The thermodynamic and psychological arrows of time would not reverse when th\ e universe begins to recontract, or inside black holes.

What should you do when you find you have made a mistake like that? Some\ people never admit that they are wrong and continue to find new, and often mutually inconsistent, argumen\ ts to support their case – as Eddington did in opposing black hole theory. Others claim to have never \ really supported the incorrect view in the first place or, if they did, it was only to show that it was inconsi\ stent. It seems to me much better and less confusing if you admit in print that you were wrong. A good example of t\ his was Einstein, who called the cosmological constant, which he introduced when he was trying to make a \ static model of the universe, the biggest mistake of his life.

To return to the arrow of time, there remains the question: why do we ob\ serve that the thermodynamic and cosmological arrows point in the same direction? Or in other words, why \ does disorder increase in the same direction of time as that in which the universe expands? If one believes\ that the universe will expand and then contract again, as the no boundary proposal seems to imply, this becomes\ a question of why we should be in the expanding phase rather than the contracting phase.

One can answer this on the basis of the weak anthropic principle. Condit\ ions in the contracting phase would not be suitable for the existence of intelligent beings who could ask the qu\ estion: why is disorder increasing in the same direction of time as that in which the universe is expanding? The i\ nflation in the early stages of the universe, which the no boundary proposal predicts, means that the univer\ se must be expanding at very close to the critical rate at which it would just avoid recollapse, and so will n\ ot recollapse for a very long time. By then all the stars will have burned out and the protons and neutrons in them will\ probably have decayed into light particles and radiation. The universe would be in a state of almost comp\ lete disorder. There would be no strong thermodynamic arrow of time. Disorder couldn’t increase much because \ the universe would be in a state of almost complete disorder already. However, a strong thermodynamic arrow \ is necessary for intelligent life to operate. In order to survive, human beings have to consume food, which i\ s an ordered form of energy, and convert it into heat, which is a disordered form of energy. Thus intelli\ gent life could not exist in the contracting phase of the universe. This is the explanation of why we observe that th\ e thermodynamic and cosmological arrows of time point in the same direction. It is not that the expansion\ of the universe causes disorder to increase. Rather, it is that the no boundary condition causes disorder t\ o increase and the conditions to be suitable for intelligent life only in the expanding phase.

To summarize, the laws of science do not distinguish between the forward\ and backward directions of time.

However, there are at least three arrows of time that do distinguish the\ past from the future. They are the thermodynamic arrow, the direction of time in which disorder increases; \ the psychological arrow, the direction of time in which we remember the past and not the future; and the cosmol\ ogical arrow, the direction of time in which the universe expands rather than contracts. I have shown that the \ psychological arrow is essentially the same as the thermodynamic arrow, so that the two would always point in t\ he same direction. The no boundary proposal for the universe predicts the existence of a well-defined therm\ odynamic arrow of time because the universe must start off in a smooth and ordered state. And the reason we\ observe this thermodynamic arrow to A Brief History of Time - Stephen Hawking... Chapter 9 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/h.html (4 of 5) [2/20/2001 3:15:38 AM] agree with the cosmological arrow is that intelligent beings can exist o\ nly in the expanding phase. The contracting phase will be unsuitable because it has no strong thermodyna\ mic arrow of time.

The progress of the human race in understanding the universe has establi\ shed a small corner of order in an increasingly disordered universe. If you remember every word in this boo\ k, your memory will have recorded about two million pieces of information: the order in your brain will ha\ ve increased by about two million units.

However, while you have been reading the book, you will have converted a\ t least a thousand calories of ordered energy, in the form of food, into disordered energy, in the form\ of heat that you lose to the air around you by convection and sweat. This will increase the disorder of the univ\ erse by about twenty million million million million units – or about ten million million million times th\ e increase in order in your brain – and that’s if you remember everything in this book. In the next chapter but one I will try to increase the ord\ er in our neck of the woods a little further by explaining how people are trying to fit to\ gether the partial theories I have described to form a complete unified theory that would cover everything in the uni\ verse.

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Chapter 9 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/h.html (5 of 5) [2/20/2001 3:15:38 AM] CHAPTER 10 WORMHOLES AND TIME TRAVEL   The last chapter discussed why we see time go forward: why disorder incr\ eases and why we remember the past but not the future. Time was treated as if it were a straight railw\ ay line on which one could only go one way or the other.

But what if the railway line had loops and branches so that a train coul\ d keep going forward but come back to a station it had already passed? In other words, might it be possible for \ someone to travel into the future or the past?

H. G. Wells in The Time Machine explored these possibilities as have countless other writers of science \ fiction.

Yet many of the ideas of science fiction, like submarines and travel to \ the moon, have become matters of science fact. So what are the prospects for time travel?

The first indication that the laws of physics might really allow people \ to travel in time came in 1949 when Kurt Godel discovered a new space-time allowed by general relativity. Godel w\ as a mathematician who was famous for proving that it is impossible to prove all true statements, even if \ you limit yourself to trying to prove all the true statements in a subject as apparently cut and dried as arithmetic. \ Like the uncertainty principle, Godel’s incompleteness theorem may be a fundamental limitation on our ability to\ understand and predict the universe, but so far at least it hasn’t seemed to be an obstacle in our search \ for a complete unified theory.

Godel got to know about general relativity when he and Einstein spent th\ eir later years at the Institute for Advanced Study in Princeton. His space-time had the curious property tha\ t the whole universe was rotating.

One might ask: “Rotating with respect to what?” The answer is that\ distant matter would be rotating with respect to directions that little tops or gyroscopes point in.

This had the side effect that it would be possible for someone to go off\ in a rocket ship and return to earth before he set out. This property really upset Einstein, who had thought \ that general relativity wouldn’t allow time travel. However, given Einstein’s record of ill-founded opposition to\ gravitational collapse and the uncertainty principle, maybe this was an encouraging sign. The solution Godel found \ doesn’t correspond to the universe we live in because we can show that the universe is not rotating. It als\ o had a non-zero value of the cosmological constant that Einstein introduced when he thought the unive\ rse was unchanging. After Hubble discovered the expansion of the universe, there was no need for a cosmol\ ogical constant and it is now generally believed to be zero. However, other more reasonable space-time\ s that are allowed by general relativity and which permit travel into the past have since been found. \ One is in the interior of a rotating black hole. Another is a space-time that contains two cosmic strings moving pa\ st each other at high speed. As their name suggests, cosmic strings are objects that are like string in that t\ hey have length but a tiny cross section.

Actually, they are more like rubber bands because they are under enormou\ s tension, something like a million million million million tons. A cosmic string attached to the earth coul\ d accelerate it from 0 to 60 mph in 1/30th of a second. Cosmic strings may sound like pure science fiction but ther\ e are reasons to believe they could have formed in the early universe as a result of symmetry-breaking of th\ e kind discussed in Chapter 5.

Because they would be under enormous tension and could start in any conf\ iguration, they might accelerate to very high speeds when they straighten out.

The Godel solution and the cosmic string space-time start out so distort\ ed that travel into the past was always possible. God might have created such a warped universe but we have no r\ eason to believe he did.

Observations of the microwave background and of the abundances of the li\ ght elements indicate that the early universe did not have the kind of curvature required to allow time trave\ l. The same conclusion follows on theoretical grounds if the no boundary proposal is correct. So the quest\ ion is: if the universe starts out without the kind of curvature required for time travel, can we subsequently warp\ local regions of space-time sufficiently to allow it?

A closely related problem that is also of concern to writers of science \ fiction is rapid interstellar or intergalactic A Brief History of Time - Stephen Hawking... Chapter 10 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/i.html (1 of 5) [2/20/2001 3:15:45 AM] travel. According to relativity, nothing can travel faster than light. I\ f we therefore sent a spaceship to our nearest neighboring star, Alpha Centauri, which is about four light-years away, \ it would take at least eight years before we could expect the travelers to return and tell us what they had found.\ If the expedition were to the center of our galaxy, it would be at least a hundred thousand years before it came\ back. The theory of relativity does allow one consolation. This is the so-called twins paradox mentioned in \ Chapter 2.

Because there is no unique standard of time, but rather observers each h\ ave their own time as measured by clocks that they carry with them, it is possible for the journey to seem\ to be much shorter for the space travelers than for those who remain on earth. But there would not be much joy in r\ eturning from a space voyage a few years older to find that everyone you had left behind was dead and gone \ thousands of years ago. So in order to have any human interest in their stories, science fiction writers had to\ suppose that we would one day discover how to travel faster than light. What most of thee authors don’t seem\ to have realized is that if you can travel faster than light, the theory of relativity implies you can also travel \ back in the, as the following limerick says:

There was a young lady of Wight Who traveled much faster than light.

She departed one day, In a relative way, And arrived on the previous night The point is that the theory of relativity says hat there is no unique m\ easure of time that all observers will agree on Rather, each observer has his or her own measure of time. If it is po\ ssible for a rocket traveling below the speed of light to get from event A (say, the final of the 100-meter rac\ e of the Olympic Games in 202) to event B (say, the opening of the 100,004th meeting of the Congress of Alpha Cen\ tauri), then all observers will agree that event A happened before event B according to their times. Suppose, \ however, that the spaceship would have to travel faster than light to carry the news of the race to the Co\ ngress. Then observers moving at different speeds can disagree about whether event A occurred before B or\ vice versa. According to the time of an observer who is at rest with respect to the earth, it may be that the\ Congress opened after the race. Thus this observer would think that a spaceship could get from A to B in time\ if only it could ignore the speed-of-light speed limit. However, to an observer at Alpha Centauri moving away from \ the earth at nearly the speed of light, it would appear that event B, the opening of the Congress, would occur b\ efore event A, the 100-meter race.

The theory of relativity says that the laws of physics appear the same t\ o observers moving at different speeds.

This has been well tested by experiment and is likely to remain a featur\ e even if we find a more advanced theory to replace relativity Thus the moving observer would say that if \ faster-than-light travel is possible, it should be possible to get from event B, the opening of the Congress, to \ event A, the 100-meter race. If one went slightly faster, one could even get back before the race and place \ a bet on it in the sure knowledge 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 to accelerate a spaceship gets greater and greater the nearer it \ gets to the speed of light. We have experimental evidence for this, not with spaceships but with elementary \ particles in particle accelerators like those at Fermilab or CERN (European Centre for Nuclear Research). We c\ an accelerate particles to 99.99 percent of the speed of light, but however much power we feed in, we can\ ’t get them beyond the speed-of-light barrier. Similarly with spaceships: no matter how much rocket power they\ have, they can’t accelerate beyond the speed of light.

That might seem to rule out both rapid space travel and travel back in t\ ime. However, there is a possible way out. It might be that one could warp space-time so that there was a shor\ tcut between A and B One way of doing this would be to create a wormhole between A and B. As its name suggests\ , a wormhole is a thin tube of space-time which can connect two nearly flat regions far apart.

There need be no relation between the distance through the wormhole and \ the separation of its ends in the nearly Hat background. Thus one could imagine that one could create or f\ ind a wormhole that world lead from the vicinity of the Solar System to Alpha Centauri. The distance through\ the wormhole might be only a few million miles even though earth and Alpha Centauri are twenty million mi\ llion miles apart in ordinary space. This would allow news of the 100-meter race to reach the opening of the Congr\ ess. But then an observer moving toward 6e earth should also be able to find another wormhole that would \ enable him to get from the opening of A Brief History of Time - Stephen Hawking... Chapter 10 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/i.html (2 of 5) [2/20/2001 3:15:45 AM] the Congress on Alpha Centauri back to earth before the start of the rac\ e. So wormholes, like any other possible form of travel faster than light, would allow one to travel int\ o the past.

The idea of wormholes between different regions of space-time was not an\ invention of science fiction writers but came from a very respectable source.

In 1935, Einstein and Nathan Rosen wrote a paper in which they showed th\ at general relativity allowed what they called “bridges,” but which are now known as wormholes. The E\ instein-Rosen bridges didn’t last long enough for a spaceship to get through: the ship would run into a singula\ rity as the wormhole pinched off.

However, it has been suggested that it might be possible for an advanced\ civilization to keep a wormhole open.

To do this, or to warp space-time in any other way so as to permit time \ travel, one can show that one needs a region of space-time with negative curvature, like the surface of a sadd\ le. Ordi-nary matter, which has a positive energy density, gives space-time a positive curvature, like the\ surface of a sphere. So what one needs, in order to warp space-time in a way that will allow travel into the pas\ t, is matter with negative energy density.

Energy is a bit like money: if you have a positive balance, you can dist\ ribute it in various ways, but according to the classical laws that were believed at the beginning of the century, y\ ou weren’t allowed to be overdrawn. So these classical laws would have ruled out any possibility of time travel\ . However, as has been described in earlier chapters, the classical laws were superseded by quantum laws bas\ ed on the uncertainty principle. The quantum laws are more liberal and allow you to be overdrawn on one or tw\ o accounts provided the total balance is positive. In other words, quantum theory allows the energy de\ nsity to be negative in some places, provided that this is made up for by positive energy densities in other \ places, so that the total energy re-mains positive. An example of how quantum theory can allow negative energy den\ sities is provided by what is called the Casimir effect. As we saw in Chapter 7, even what we think of as “\ empty” space is filled with pairs of virtual particles and antiparticles that appear together, move apart, and come b\ ack together and annihilate each other.

Now, suppose one has two parallel metal plates a short distance apart. T\ he plates will act like mirrors for the virtual photons or particles of light. In fact they will form a cavity b\ etween them, a bit like an organ pipe that will resonate only at certain notes. This means that virtual photons can occu\ r in the space between the plates only if their wavelengths (the distance between the crest of one wave and th\ e next) fit a whole number of times into the gap between the plates. If the width of a cavity is a whole number o\ f wavelengths plus a fraction of a wave-length, then after some reflections backward and forward between th\ e plates, the crests of one wave will coincide with the troughs of another and the waves will cancel out.

Because the virtual photons between the plates can have only the resonan\ t wavelengths, there will be slightly fewer of them than in the region outside the plates where virtual photon\ s can have any wavelength. Thus there will be slightly fewer virtual photons hitting the inside surfaces of th\ e plates than the outside surfaces. One would therefore expect a force on the plates, pushing them toward each o\ ther. This force has actually been detected and has the predicted value. Thus we have experimental evidence\ that virtual particles exist and have real effects.

The fact that there are fewer virtual photons between the plates means t\ hat their energy density will be less than elsewhere. But the total energy density in “empty” space far \ away from the plates must be zero, because otherwise the energy density would warp the space and it would not be al\ most flat. So, if the energy density between the plates is less than the energy density far away, it must be \ negative.

We thus have experimental evidence both that space-time can be warped (\ from the bending of light during eclipses) and that it can be curved in the way necessary to allow time \ travel (from the Casimir effect). One might hope therefore that as we advance in science and technology, we wo\ uld eventually manage to build a time machine. But if so, why hasn’t anyone come back from the future \ and told us how to do it? There might be good reasons why it would be unwise to give us the secret of time travel\ at our present primitive state of development, but unless human nature changes radically, it is difficult \ to believe that some visitor from the future wouldn’t spill the beans. Of course, some people would claim t\ hat sightings of UFOs are evidence that we are being visited either by aliens or by people from the future. (If\ the aliens were to get here in reasonable time, they would need faster-than-light travel, so the two possibilities\ may be equivalent.) However, I think that any visit by aliens or people from the future woul\ d be much more obvious and, probably, much more unpleasant. If they are going to reveal themselves at all, why\ do so only to those who are not A Brief History of Time - Stephen Hawking... Chapter 10 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/i.html (3 of 5) [2/20/2001 3:15:45 AM] regarded as reliable witnesses? If they are trying to warn us of some gr\ eat danger, they are not being very effective.

A possible way to explain the absence of visitors from the future would \ be to say that the past is fixed because we have observed it and seen that it does not have the kind of warping n\ eeded to allow travel back from the future. On the other hand, the future is unknown and open, so it might w\ ell have the curvature required. This would mean that any time travel would be confined to the future. There w\ ould be no chance of Captain Kirk and the Starship Enterprise turning up at the present time.

This might explain why we have not yet been overrun by tourists from the\ future, but it would not avoid the problems that would arise if one were able to go back and change history\ . Suppose, for example, you went back and killed your great-great-grandfather while he was still a child.\ There are many versions of this paradox but they are essentially equivalent: one would get contradictions if one\ were free to change the past.

There seem to be two possible resolutions to the paradoxes posed by time\ travel. One I shall call the consistent histories approach. It says that even if space-time is warped so that it\ would be possible to travel into the past, what happens in space-time must be a consistent solution of the laws of \ physics. According to this viewpoint, you could not go back in time unless history showed that you had already\ arrived in the past and, while there, had not killed your great-great-grandfather or committed any other acts \ that would conflict with your current situation in the present. Moreover, when you did go back, you wouldn’\ t be able to change recorded history.

That means you wouldn’t have free will to do what you wanted. Of cour\ se, one could say that free will is an illusion anyway. If there really is a complete unified theory that gover\ ns everything, it presumably also determines your actions. But it does so in a way that is impossible to c\ alculate for an organism that is as complicated as a human being. The reason we say that humans have free wi\ ll is because we can’t predict what they will do. However, if the human then goes off in a rocket ship and c\ omes back before he or she set off, we will be able to predict what he or she will do because it will be part of rec\ orded history. Thus, in that situation, the time traveler would have no free will.

The other possible way to resolve the paradoxes of time travel might be \ called the alternative histories hypothesis. The idea here is that when time travelers go back to the pas\ t, they enter alternative histories which differ from recorded history. Thus they can act freely, without the cons\ traint of consistency with their previous history. Steven Spiel-berg had fun with this notion in the Back to the Future films: Marty McFly was able to go back and change his parents’ courtship to a more satisfactory history\ .

The alternative histories hypothesis sounds rather like Richard Feynman’\ s way of expressing quantum theory as a sum over histories, which was described in Chapters 4 and 8. This s\ aid that the universe didn’t just have a single history: rather it had every possible history, each with its own \ probability. However, there seems to be an important difference between Feynman’s proposal and alternative histo\ ries. In Feynman’s sum, each history comprises a complete space-time and everything in it. The space-time may\ be so warped that it is possible to travel in a rocket into the past. But the rocket would remain in the sam\ e space-time and therefore the same history, which would have to be consistent. Thus Feynman’s sum over h\ istories proposal seems to support the consistent histories hypothesis rather than the alternative histories.

The Feynman sum over histories does allow travel into the past on a microscopic scale. In Chapter 9 we saw that the laws of science are unchanged by combinations of the operations\ C, P, and T. This means that an antiparticle spinning in the anticlockwise direction and moving from A t\ o B can also be viewed as an ordinary particle spinning clockwise and moving backward in time from B to A. Sim\ ilarly, an ordinary particle moving forward in time is equivalent to an antiparticle moving backward in time\ . As has been discussed in this chapter and Chapter 7, “empty” space is filled with pairs of virtual parti\ cles and antiparticles that appear together, move apart, and then come back together and annihilate each other.

So, one can regard the pair of particles as a single particle moving on \ a closed loop in space-time. When the pair is moving forward in time (from the event at which it appears to t\ hat at which it annihilates), it is called a particle. But when the particle is traveling back in time (from the eve\ nt at which the pair annihilates to that at which it appears), it is said to be an antiparticle traveling forward i\ n time.

The explanation of how black holes can emit particles and radiation (gi\ ven in Chapter 7) was that one member A Brief History of Time - Stephen Hawking... Chapter 10 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/i.html (4 of 5) [2/20/2001 3:15:45 AM] of a virtual particle/ antiparticle pair (say, the antiparticle) might\ fall into the black hole, leaving the other member without a partner with which to annihilate. The forsaken particle\ might fall into the hole as well, but it might also escape from the vicinity of the black hole. If so, to an obse\ rver at a distance it would appear to be a particle emitted by the black hole.

One can, however, have a different but equivalent intuitive picture of t\ he mechanism for emission from black holes. One can regard the member of the virtual pair that fell into the \ black hole (say, the antiparticle) as a particle traveling backward in time out of the hole. When it gets to the\ point at which the virtual particle/antiparticle pair appeared together, it is scattered by the gra\ vitational field into a particle traveling forward in time and escaping from the black hole. If, instead, it were t\ he particle member of the virtual pair that fell into the hole, one could regard it as an antiparticle traveling bac\ k in time and coming out of the black hole.

Thus the radiation by black holes shows that quantum theory allows trave\ l back in time on a microscopic scale and that such time travel can produce observable effects.

One can therefore ask: does quantum theory allow time travel on a macros\ copic scale, which people could use? At first sight, it seems it should. The Feynman sum over histories \ proposal is supposed to be over all histories. Thus it should include histories in which space-time is so wa\ rped that it is possible to travel into the past. Why then aren’t we in trouble with history? Suppose, for exampl\ e, someone had gone back and given the Nazis the secret of the atom bomb?

One would avoid these problems if what I call the chronology protection \ conjecture holds. This says that the laws of physics conspire to prevent macroscopic bodies from carrying information into the past. Like the cosmic censorship conjecture, it has not been proved but there are reasons to b\ elieve it is true.

The reason to believe that chronology protection operates is that when s\ pace-time is warped enough to make travel into the past possible, virtual particles moving on closed loops \ in space-time can become real particles traveling forward in time at or below the speed of light. As these parti\ cles can go round the loop any number of times, they pass each point on their route many times. Thus their energy\ is counted over and over again and the energy density will become very large. This could give space-time a \ positive curvature that would not allow travel into the past. It is not yet clear whether these particles would \ cause positive or negative curvature or whether the curvature produced by some kinds of virtual particles might \ cancel that produced by other kinds.

Thus the possibility of time travel remains open. But I’m not going t\ o bet on it. My opponent might have the unfair advantage of knowing the future.

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Chapter 10 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/i.html (5 of 5) [2/20/2001 3:15:45 AM] CHAPTER 11 THE UNIFICATION OF PHYSICS   As was explained in the first chapter, it would be very difficult to con\ struct a complete unified theory of everything in the universe all at one go. So instead we have made progre\ ss by finding partial theories that describe a limited range of happenings and by neglecting other effects o\ r approximating them by certain numbers. (Chemistry, for example, allows us to calculate the interactio\ ns of atoms, without knowing the internal structure of an atom’s nucleus.) Ultimately, however, one would hope\ to find a complete, consistent, unified theory that would include all these partial theories as approximations, \ and that did not need to be adjusted to fit the facts by picking the values of certain arbitrary numbers in the theo\ ry. The quest for such a theory is known as “the unification of physics.” Einstein spent most of his later \ years unsuccessfully searching for a unified theory, but the time was not ripe: there were partial theories for gravi\ ty and the electromagnetic force, but very little was known about the nuclear forces. Moreover, Einstein refused to\ believe in the reality of quantum mechanics, despite the important role he had played in its development. \ Yet it seems that the uncertainty principle is a fundamental feature of the universe we live in. A success\ ful unified theory must, therefore, necessarily incorporate this principle.

As I shall describe, the prospects for finding such a theory seem to be \ much better now because we know so much more about the universe. But we must beware of overconfidence – \ we have had false dawns before! At the beginning of this century, for example, it was thought that everythi\ ng could be explained in terms of the properties of continuous matter, such as elasticity and heat conduction.\ The discovery of atomic structure and the uncertainty principle put an emphatic end to that. Then again, in 19\ 28, physicist and Nobel Prize winner Max Born told a group of visitors to Gottingen University, “Physics, \ as we know it, will be over in six months.” His confidence was based on the recent discovery by Dirac of the equatio\ n that governed the electron. It was thought that a similar equation would govern the proton, which was the o\ nly other particle known at the time, and that would be the end of theoretical physics. However, the discovery\ of the neutron and of nuclear forces knocked that one on the head too. Having said this, I still believe ther\ e are grounds for cautious optimism that we may now be near the end of the search for the ultimate laws of nature\ .

In previous chapters I have described general relativity, the partial th\ eory of gravity, and the partial theories that govern the weak, the strong, and the electromagnetic forces. The last th\ ree may be combined in so-called grand unified theories, or GUTs, which are not very satisfactory because\ they do not include gravity and because they contain a number of quantities, like the relative masses of\ different particles, that cannot be predicted from the theory but have to be chosen to fit observations. The\ main difficulty in finding a theory that unifies gravity with the other forces is that general relativity is a “\ classical” theory; that is, it does not incorporate the uncertainty principle of quantum mechanics. On the other hand, the o\ ther partial theories depend on quantum mechanics in an essential way. A necessary first step, therefore\ , is to combine general relativity with the uncertainty principle. As we have seen, this can produce some remark\ able consequences, such as black holes not being black, and the universe not having any singularities but\ being completely self-contained and without a boundary. The trouble is, as explained in Chapter 7, that the \ uncertainty principle means that even “empty” space is filled with pairs of virtual particles and antipa\ rticles. These pairs would have an infinite amount of energy and, therefore, by Einstein’s famous equation E = mc 2, they would have an infinite amount of mass.

Their gravitational attraction would thus curve up the universe to infin\ itely small size.

Rather similar, seemingly absurd infinities occur in the other partial t\ heories, but in all these cases the infinities can be canceled out by a process called renormalization. This involves c\ anceling the infinities by introducing other infinities. Although this technique is rather dubious mathematical\ ly, it does seem to work in practice, and has been used with these theories to make predictions that agree with ob\ servations to an extraordinary degree of accuracy. Renormalization, however, does have a serious drawback from\ the point of view of trying to find a complete theory, because it means that the actual values of the masses a\ nd the strengths of the forces cannot be predicted from the theory, but have to be chosen to fit the observati\ ons.

In attempting to incorporate the uncertainty principle into general rela\ tivity, one has only two quantities that can be adjusted: the strength of gravity and the value of the cosmological c\ onstant. But adjusting these is not A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (1 of 11) [2/20/2001 3:15:56 AM] sufficient to remove all the infinities. One therefore has a theory that\ seems to predict that certain quantities, such as the curvature of space-time, are really infinite, yet these quan\ tities can be observed and measured to be perfectly finite! This problem in combining general relativity and th\ e uncertainty principle had been suspected for some time, but was finally confirmed by detailed calculati\ ons in 1972. Four years later, a possible solution, called “supergravity,” was suggested. The idea was to co\ mbine the spin-2 particle called the graviton, which carries the gravitational force, with certain other particles of s\ pin 3/2, 1, ½, and 0. In a sense, all these particles could then be regarded as different aspects of the same “su\ perparticle,” thus unifying the matter particles with spin ½ and 3/2 with the force-carrying particles of sp\ in 0, 1, and 2. The virtual particle/antiparticle pairs of spin ½ and 3/2 would have negative energy, and so would tend\ to cancel out the positive energy of the spin 2, 1, and 0 virtual pairs. This would cause many of the possible in\ finities to cancel out, but it was suspected that some infinities might still remain. However, the calculat\ ions required to find out whether or not there were any infinities left uncancelled were so long and difficult th\ at no one was prepared to undertake them.

Even with a computer it was reckoned it would take at least four years, \ and the chances were very high that one would make at least one mistake, probably more. So one would know on\ e had the right answer only if someone else repeated the calculation and got the same answer, and that \ did not seem very likely!

Despite these problems, and the fact that the particles in the super-gra\ vity theories did not seem to match the observed particles, most scientists believed that supergravity was proba\ bly the right answer to the problem of the unification of physics. It seemed the best way of unifying gravity w\ ith the other forces. However, in 1984 there was a remarkable change of opinion in favor of what are called str\ ing theories. In these theories the basic objects are not particles, which occupy a single point of space, but thi\ ngs that have a length but no other dimension, like an infinitely thin piece of string. These strings may ha\ ve ends (the so-called open strings) or they may be joined up with themselves in closed loops (closed strings)\ Figure 11:1 and Figure 11:2 .

A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (2 of 11) [2/20/2001 3:15:56 AM] Figures 11:1 & 11:2 A particle occupies one point of space at each instant of time. Thus its\ history can be represented by a line in space-time (the “world-line”). A string, on the other hand, occu\ pies a line in space at each moment of time. So its history in space-time is a two-dimensional surface called the world-\ sheet. (Any point on such a world-sheet can be described by two numbers, one specifying the time and the other t\ he position of the point on the string.) The world-sheet of an open string is a strip: its edges represent the pa\ ths through space-time of the ends of the string Figure 11:1 . The world-sheet of a closed string is a cylinder or tube Figure 11:2: a slice through the tube is a circle, which represents the position of the string at one particul\ ar time.

Two pieces of string can join together to form a single string; in the c\ ase of open strings they simply join at the ends Figure 11:3 , while in the case of closed strings it is like the two legs joining on\ a pair of trousers Figure 11:4 . A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (3 of 11) [2/20/2001 3:15:56 AM] Figure 11:3   A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (4 of 11) [2/20/2001 3:15:56 AM] Figure 11:4 Similarly, a single piece of string can divide into two strings. In stri\ ng theories, what were previously thought of as particles are now pictured as waves traveling down the string, like w\ aves on a vibrating kite string. The emission or absorption of one particle by another corresponds to the div\ iding or joining together of strings. For example, the gravitational force of the sun on the earth was pictured in\ particle theories as being caused by the emission of a graviton by a particle in the sun and its absorption by a \ particle in the earth Figure 11:5. A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (5 of 11) [2/20/2001 3:15:56 AM] Figures 11:5 & 11:6 In string theory, this process corresponds to an H-shaped tube or pipe Figure 11:6 (string theory is rather like plumbing, in a way). The two vertical sides of the H correspond to the \ particles in the sun and the earth, and the horizontal crossbar corresponds to the graviton that travels between the\ m.

String theory has a curious history. It was originally invented in the l\ ate 1960s in an attempt to find a theory to describe the strong force. The idea was that particles like the proton a\ nd the neutron could be regarded as waves on a string. The strong forces between the particles would corresp\ ond to pieces of string that went between other bits of string, as in a spider’s web. For this theory t\ o give the observed value of the strong force between particles, the strings had to be like rubber bands with a pull o\ f about ten tons.

In 1974 Joel Scherk from Paris and John Schwarz from the California Inst\ itute of Technology published a paper in which they showed that string theory could describe the gravitational\ force, but only if the tension in the string were very much higher, about a thousand million million million million \ million million tons (1 with thirty-nine zeros after it). The predictions of the string theory would be just the\ same as those of general relativity on normal length scales, but they would differ at very small distances, les\ s than a thousand million million million million millionth of a centimeter (a centimeter divided by 1 with thirt\ y-three zeros after it). Their work did not receive much attention, however, because at just about that time most pe\ ople abandoned the original string theory of the strong force in favor of the theory based on quarks and gl\ uons, which seemed to fit much better with observations. Scherk died in tragic circumstances (he suffered fro\ m diabetes and went into a coma when no one was around to give him an injection of insulin). So Schwarz was \ left alone as almost the only supporter of string theory, but now with the much higher proposed value of the str\ ing tension.

In 1984 interest in strings suddenly revived, apparently for two reasons\ . One was that people were not really making much progress toward showing that supergravity was finite or that\ it could explain the kinds of particles that we observe. The other was the publication of a paper by John Schwar\ z and Mike Green of Queen Mary College, London, that showed that string theory might be able to explain\ the existence of particles that have a built-in left-handedness, like some of the particles that we observe. Wh\ atever the reasons, a large number of people soon began to work on string theory and a new version was develop\ ed, the so-called heterotic string, which seemed as if it might be able to explain the types of particles th\ at we observe.

String theories also lead to infinities, but it is thought they will all\ cancel out in versions like the heterotic string A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (6 of 11) [2/20/2001 3:15:56 AM] (though this is not yet known for certain). String theories, however, \ have a bigger problem: they seem to be consistent only if space-time has either ten or twenty-six dimensions, i\ nstead of the usual four! Of course, extra space-time dimensions are a commonplace of science fiction indeed, they \ provide an ideal way of overcoming the normal restriction of general relativity that one cannot travel fast\ er than light or back in time (see Chapter 10). The idea is to take a shortcut through the extra dimensions. One c\ an picture this in the following way.

Imagine that the space we live in has only two dimensions and is curved \ like the surface of an anchor ring or torus Figure 11:7 . Figure 11:7 If you were on one side of the inside edge of the ring and you wanted to\ get to a point on the other side, you would have to go round the inner edge of the ring. However, if you were \ able to travel in the third dimension, you could cut straight across.

Why don’t we notice all these extra dimensions, if they are really th\ ere? Why do we see only three space dimensions and one time dimension? The suggestion is that the other dime\ nsions are curved up into a space of very small size, something like a million million million million millio\ nth of an inch. This is so small that we just don’t notice it: we see only one time dimension and three space dimen\ sions, in which space-time is fairly flat. It is like the surface of a straw. If you look at it closely, you see it is\ two-dimensional (the position of a point on the straw is described by two numbers, the length along the straw and the di\ stance round the circular direction).

But if you look at it from a distance, you don’t see the thickness of\ the straw and it looks one-dimensional (the position of a point is specified only by the length along the straw). S\ o it is with space-time: on a very small scale it is ten-dimensional and highly curved, but on bigger scales you don’\ t see the curvature or the extra dimensions. If this picture is correct, it spells bad news for would-be \ space travelers: the extra dimensions would be far too small to allow a spaceship through. However, it raises \ another major problem. Why should some, but not all, of the dimensions be curled up into a small ball? Pre\ sumably, in the very early universe all the dimensions would have been very curved. Why did one time dimension a\ nd three space dimensions flatten A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (7 of 11) [2/20/2001 3:15:56 AM] out, while the other dimensions remain tightly curled up?

One possible answer is the anthropic principle. Two space dimensions do \ not seem to be enough to allow for the development of complicated beings like us. For example, two-dimensio\ nal animals living on a one-dimensional earth would have to climb over each other in order to ge\ t past each other. If a two-dimensional creature ate something it could not digest completely, it would have to \ bring up the remains the same way it swallowed them, because if there were a passage right through its body, \ it would divide the creature into two separate halves: our two-dimensional being would fall apart Figure 11:8. Similarly, it is difficult to see how there could be any circulation of the blood in a two-dimensional creature. Figure 11:8 There would also be problems with more than three space dimensions. The \ gravitational force between two bodies would decrease more rapidly with distance than it does in three d\ imensions. (In three dimensions, the gravitational force drops to 1/4 if one doubles the distance. In four di\ mensions it would drop to 1/5, in five dimensions to 1/6, and so on.) The significance of this is that the orb\ its of planets, like the earth, around the sun would be unstable: the least disturbance from a circular orbit (such as\ would be caused by the gravitational attraction of other planets) would result in the earth spiraling away f\ rom or into the sun. We would either freeze or be burned up. In fact, the same behavior of gravity with distance in \ more than three space dimensions means that the sun would not be able to exist in a stable state with pre\ ssure balancing gravity. It would either fall apart or it would collapse to form a black hole. In either case, it\ would not be of much use as a source of heat and light for life on earth. On a smaller scale, the electrical for\ ces that cause the electrons to orbit round the nucleus in an atom would behave in the same way as gravitational for\ ces. Thus the electrons would either escape from the atom altogether or would spiral into the nucleus. In eit\ her case, one could not have atoms as we know them.

It seems clear then that life, at least as we know it, can exist only in\ regions of space-time in which one time dimension and three space dimensions are not curled up small. This would\ mean that one could appeal to the weak anthropic principle, provided one could show that string theory doe\ s at least allow there to be such regions of the universe – and it seems that indeed string theory does\ . There may well be other regions of the universe, or other universes (whatever that may mean), in which all the dimensions are curled up small or in which more than four dimensions are nearly flat, but there would be no i\ ntelligent beings in such regions to A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (8 of 11) [2/20/2001 3:15:56 AM] observe the different number of effective dimensions.

Another problem is that there are at least four different string theorie\ s (open strings and three different closed string theories) and millions of ways in which the extra dimensions pre\ dicted by string theory could be curled up. Why should just one string theory and one kind of curling up be pick\ ed out? For a time there seemed no answer, and progress got bogged down. Then, from about 1994, people star\ ted discovering what are called dualities: different string theories and different ways of curling up th\ e extra dimensions could lead to the same results in four dimensions. Moreover, as well as particles, which occupy\ a single point of space, and strings, which are lines, there were found to be other objects called p-branes, w\ hich occupied two-dimensional or higher-dimensional volumes in space. (A particle can be regarded as a 0\ -brane and a string as a 1-brane but there were also p-branes for p=2 to p=9.) What this seems to indicate i\ s that there is a sort of democracy among supergravity, string, and p-brane theories: they seem to fit toget\ her but none can be said to be more fundamental than the others. They appear to be different approximations \ to some fundamental theory that are valid in different situations.

People have searched for this underlying theory, but without any success\ so far. However, I believe there may not be any single formulation of the fundamental theory any more than, a\ s Godel showed, one could formulate arithmetic in terms of a single set of axioms. Instead it may be like ma\ ps – you can’t use a single map to describe the surface of the earth or an anchor ring: you need at least t\ wo maps in the case of the earth and four for the anchor ring to cover every point. Each map is valid only in a li\ mited region, but different maps will have a region of overlap. The collection of maps provides a complete descriptio\ n of the surface. Similarly, in physics it may be necessary to use different formulations in different situations, \ but two different formulations would agree in situations where they can both be applied. The whole collection of di\ fferent formulations could be regarded as a complete unified theory, though one that could not be expressed in \ terms of a single set of postulates.

But can there really be such a unified theory? Or are we perhaps just ch\ asing a mirage? There seem to be three possibilities:

1. There really is a complete unified theory (or a collection of overla\ pping formulations), which we will someday discover if we are smart enough.

2. There is no ultimate theory of the universe, just an infinite sequenc\ e of theories that describe the universe more and more accurately.

3. There is no theory of the universe: events cannot be predicted beyond a \ certain extent but occur in a random and arbitrary manner.

Some would argue for the third possibility on the grounds that if there \ were a complete set of laws, that would infringe God’s freedom to change his mind and intervene in the world.\ It’s a bit like the old paradox: can God make a stone so heavy that he can’t lift it? But the idea that God mi\ ght want to change his mind is an example of the fallacy, pointed out by St. Augustine, of imagining God as a bein\ g existing in time: time is a property only of the universe that God created. Presumably, he knew what he intended w\ hen he set it up!

With the advent of quantum mechanics, we have come to recognize that eve\ nts cannot be predicted with complete accuracy but that there is always a degree of uncertainty. If o\ ne likes, one could ascribe this randomness to the intervention of God, but it would be a very strange ki\ nd of intervention: there is no evidence that it is directed toward any purpose. Indeed, if it were, it would by \ definition not be random. In modern times, we have effectively removed the third possibility above by redefining th\ e goal of science: our aim is to formulate a set of laws that enables us to predict events only up to the limit set\ by the uncertainty principle.

The second possibility, that there is an infinite sequence of more and m\ ore refined theories, is in agreement with all our experience so far. On many occasions we have increased the \ sensitivity of our measurements or made a new class of observations, only to discover new phenomena that we\ re not predicted by the existing theory, and to account for these we have had to develop a more advanced \ theory. It would therefore not be very surprising if the present generation of grand unified theories was \ wrong in claiming that nothing essentially new will happen between the electroweak unification energy of about 100 \ GeV and the grand unification energy of about a thousand million million GeV. We might indeed expect to find \ several new layers of structure more A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (9 of 11) [2/20/2001 3:15:56 AM] basic than the quarks and electrons that we now regard as “elementary\ ” particles.

However, it seems that gravity may provide a limit to this sequence of “\ boxes within boxes.” If one had a particle with an energy above what is called the Planck energy, ten mill\ ion million million GeV (1 followed by nineteen zeros), its mass would be so concentrated that it would cut it\ self off from the rest of the universe and form a little black hole. Thus it does seem that the sequence of more an\ d more refined theories should have some limit as we go to higher and higher energies, so that there should \ be some ultimate theory of the universe. Of course, the Planck energy is a very long way from the energ\ ies of around a hundred GeV, which are the most that we can produce in the laboratory at the present time. \ We shall not bridge that gap with particle accelerators in the foreseeable future! The very early stages o\ f the universe, however, are an arena where such energies must have occurred. I think that there is a good cha\ nce that the study of the early universe and the requirements of mathematical consistency will lead us t\ o a complete unified theory within the lifetime of some of us who are around today, always presuming we don’\ t blow ourselves up first.

What would it mean if we actually did discover the ultimate theory of th\ e universe? As was explained in Chapter 1, we could never be quite sure that we had indeed found the correct the\ ory, since theories can’t be proved.

But if the theory was mathematically consistent and always gave predicti\ ons that agreed with observations, we could be reasonably confident that it was the right one. It would bring \ to an end a long and glorious chapter in the history of humanity’s intellectual struggle to understand the uni\ verse. But it would also revolutionize the ordinary person’s understanding of the laws that govern the universe.\ In Newton’s time it was possible for an educated person to have a grasp of the whole of human knowledge, at leas\ t in outline. But since then, the pace of the development of science has made this impossible. Because theories\ are always being changed to account for new observations, they are never properly digested or simpli\ fied so that ordinary people can understand them. You have to be a specialist, and even then you can only\ hope to have a proper grasp of a small proportion of the scientific theories. Further, the rate of progre\ ss is so rapid that what one learns at school or university is always a bit out of date. Only a few people can keep up\ with the rapidly advancing frontier of knowledge, and they have to devote their whole time to it and specialize\ in a small area. The rest of the population has little idea of the advances that are being made or the ex\ citement they are generating. Seventy years ago, if Eddington is to be believed, only two people understood th\ e general theory of relativity. Nowadays tens of thousands of university graduates do, and many millions of peopl\ e are at least familiar with the idea. If a complete unified theory was discovered, it would only be a matter of tim\ e before it was digested and simplified in the same way and taught in schools, at least in outline. We would the\ n all be able to have some understanding of the laws that govern the universe and are responsible f\ or our existence.

Even if we do discover a complete unified theory, it would not mean that\ we would be able to predict events in general, for two reasons. The first is the limitation that the uncertain\ ty principle of quantum mechanics sets on our powers of prediction. There is nothing we can do to get around that.\ In practice, however, this first limitation is less restrictive than the second one. It arises from the fact that we\ could not solve the equations of the theory exactly, except in very simple situations. (We cannot even solve exactl\ y for the motion of three bodies in Newton’s theory of gravity, and the difficulty increases with the num\ ber of bodies and the complexity of the theory.) We already know the laws that govern the behavior of matter un\ der all but the most extreme conditions. In particular, we know the basic laws that underlie all of c\ hemistry and biology. Yet we have certainly not reduced these subjects to the status of solved problems: w\ e have, as yet, had little success in predicting human behavior from mathematical equations! So even if we do \ find a complete set of basic laws, there will still be in the years ahead the intellectually challenging ta\ sk of developing better approximation methods, so that we can make useful predictions of the probable outcomes\ in complicated and realistic situations. A complete, consistent, unified theory is only the first ste\ p: our goal is a complete understanding of the events around us, and of our own existence.

  PREVIOUS NEXT   A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (10 of 11) [2/20/2001 3:15:56 AM]   A Brief History of Time - Stephen Hawking... Chapter 11 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/j.html (11 of 11) [2/20/2001 3:15:56 AM] CHAPTER 12 CONCLUSION   We find ourselves in a bewildering world. We want to make sense of what \ we see around us and to ask: What is the nature of the universe? What is our place in it and where did it \ and we come from? Why is it the way it is?

To try to answer these questions we adopt some “world picture.” Ju\ st as an infinite tower of tortoises supporting the fiat earth is such a picture, so is the theory of superstrings. Both\ are theories of the universe, though the latter is much more mathematical and precise than the former. Both theor\ ies lack observational evidence: no one has ever seen a giant tortoise with the earth on its back, but then,\ no one has seen a superstring either.

However, the tortoise theory fails to be a good scientific theory becaus\ e it predicts that people should be able to fall off the edge of the world. This has not been found to agree with ex\ perience, unless that turns out to be the explanation for the people who are supposed to have disappeared in the B\ ermuda Triangle!

The earliest theoretical attempts to describe and explain the universe i\ nvolved the idea that events and natural phenomena were controlled by spirits with human emotions who acted in a \ very humanlike and unpredictable manner. These spirits inhabited natural objects, like rivers and mountai\ ns, including celestial bodies, like the sun and moon. They had to be placated and their favor sought in order to\ ensure the fertility of the soil and the rotation of the seasons. Gradually, however, it must have been noticed t\ hat there were certain regularities: the sun always rose in the east and set in the west, whether or not a sacrif\ ice had been made to the sun god.

Further, the sun, the moon, and the planets followed precise paths acros\ s the sky that could be predicted in advance with considerable accuracy. The sun and the moon might still be \ gods, but they were gods who obeyed strict laws, apparently without any exceptions, if one discounts \ stories like that of the sun stopping for Joshua.

At first, these regularities and laws were obvious only in astronomy and\ a few other situations. However, as civilization developed, and particularly in the last 300 years, more and\ more regularities and laws were discovered. The success of these laws led Laplace at the beginning of th\ e nineteenth century to postulate scientific determinism; that is, he suggested that there would be a set \ of laws that would determine the evolution of the universe precisely, given its configuration at one time\ .

Laplace’s determinism was incomplete in two ways. It did not say how \ the laws should be chosen and it did not specify the initial configuration of the universe. These were left to Go\ d. God would choose how the universe began and what laws it obeyed, but he would not intervene in the univers\ e once it had started. In effect, God was confined to the areas that nineteenth-century science did not unders\ tand.

We now know that Laplace’s hopes of determinism cannot be realized, a\ t least in the terms he had in mind.

The uncertainty principle of quantum mechanics implies that certain pair\ s of quantities, such as the position and velocity of a particle, cannot both be predicted with complete accuracy.\ Quantum mechanics deals with this situation via a class of quantum theories in which particles don’t ha\ ve well-defined positions and velocities but are represented by a wave. These quantum theories are deterministic in t\ he sense that they give laws for the evolution of the wave with time. Thus if one knows the wave at one time,\ one can calculate it at any other time.

The unpredictable, random element comes in only when we try to interpret\ the wave in terms of the positions and velocities of particles. But maybe that is our mistake: maybe there \ are no particle positions and velocities, but only waves. It is just that we try to fit the waves to our preconcei\ ved ideas of positions and velocities. The resulting mismatch is the cause of the apparent unpredictability.

In effect, we have redefined the task of science to be the discovery of \ laws that will enable us to predict events up to the limits set by the uncertainty principle. The question remains,\ however: how or why were the laws and the initial state of the universe chosen?

In this book I have given special prominence to the laws that govern gra\ vity, because it is gravity that shapes the large-scale structure of the universe, even though it is the weakest\ of the four categories of forces. The laws of gravity were incompatible with the view held until quite recentl\ y that the universe is unchanging in time:

A Brief History of Time - Stephen Hawking... Chapter 12 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/k.html (1 of 4) [2/20/2001 3:16:08 AM] the fact that gravity is always attractive implies that the universe mus\ t be either expanding or contracting.

According to the general theory of relativity, there must have been a st\ ate of infinite density in the past, the big bang, which would have been an effective beginning of time. Similarly, i\ f the whole universe recollapsed, there must be another state of infinite density in the future, the big crunch,\ which would be an end of time. Even if the whole universe did not recollapse, there would be singularities in any l\ ocalized regions that collapsed to form black holes. These singularities would be an end of time for anyone who \ fell into the black hole. At the big bang and other singularities, all the laws would have broken down, so God wou\ ld still have had complete freedom to choose what happened and how the universe began.

When we combine quantum mechanics with general relativity, there seems t\ o be a new possibility that did not arise before: that space and time together might form a finite, four-dim\ ensional space without singularities or boundaries, like the surface of the earth but with more dimensions. It s\ eems that this idea could explain many of the observed features of the universe, such as its large-scale unifor\ mity and also the smaller-scale departures from homogeneity, like galaxies, stars, and even human beings\ . It could even account for the arrow of time that we observe. But if the universe is completely self-containe\ d, with no singularities or boundaries, and completely described by a unified theory, that has profound implicat\ ions for the role of God as Creator.

Einstein once asked the question: “How much choice did God have in co\ nstructing the universe?” If the no boundary proposal is correct, he had no freedom at all to choose initial\ conditions. He would, of course, still have had the freedom to choose the laws that the universe obeyed. This, \ however, may not really have been all that much of a choice; there may well be only one, or a small number, of\ complete unified theories, such as the heterotic string theory, that are self-consistent and allow the existenc\ e of structures as complicated as human beings who can investigate the laws of the universe and ask about the na\ ture of God.

Even if there is only one possible unified theory, it is just a set of r\ ules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The u\ sual approach of science of constructing a mathematical model cannot answer the questions of why the\ re should be a universe for the model to describe. Why does the universe go to all the bother of existin\ g? Is the unified theory so compelling that it brings about its own existence? Or does it need a creator, and, \ if so, does he have any other effect on the universe? And who created him?

Up to now, most scientists have been too occupied with the development o\ f new theories that describe what the universe is to ask the question why. On the other hand, the people whose business it is to ask why, the philosophers, have not been able to keep up with the advance of scientif\ ic theories. In the eighteenth century, philosophers considered the whole of human knowledge, including science,\ to be their field and discussed questions such as: did the universe have a beginning? However, in the ni\ neteenth and twentieth centuries, science became too technical and mathematical for the philosophers, or a\ nyone else except a few specialists.

Philosophers reduced the scope of their inquiries so much that Wittgenst\ ein, the most famous philosopher of this century, said, “The sole remaining task for philosophy is the an\ alysis of language.” What a comedown from the great tradition of philosophy from Aristotle to Kant!

However, if we do discover a complete theory, it should in time be under\ standable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, sc\ ientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and \ the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason – for then we \ would know the mind of God.

  ALBERT EINSTEIN Einstein’s connection with the politics of the nuclear bomb is well k\ nown: he signed the famous letter to President Franklin Roosevelt that persuaded the United States to take th\ e idea seriously, and he engaged in postwar efforts to prevent nuclear war. But these were not just the isol\ ated actions of a scientist dragged into the world of politics. Einstein’s life was, in fact, to use his own w\ ords, “divided between politics and equations.” Einstein’s earliest political activity came during the First World Wa\ r, when he was a professor in Berlin.

Sickened by what he saw as the waste of human lives, he became involved \ in antiwar demonstrations. His A Brief History of Time - Stephen Hawking... Chapter 12 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/k.html (2 of 4) [2/20/2001 3:16:08 AM] advocacy of civil disobedience and public encouragement of people to ref\ use conscription did little to endear him to his colleagues. Then, following the war, he directed his efforts \ toward reconciliation and improving international relations. This too did not make him popular, and soon his\ politics were making it difficult for him to visit the United States, even to give lectures.

Einstein’s second great cause was Zionism. Although he was Jewish by \ descent, Einstein rejected the biblical idea of God. However, a growing awareness of anti-Semitism, both before \ and during the First World War, led him gradually to identify with the Jewish community, and later to become\ an outspoken supporter of Zionism.

Once more unpopularity did not stop him from speaking his mind. His theo\ ries came under attack; an anti-Einstein organization was even set up. One man was convicted of inc\ iting others to murder Einstein (and fined a mere six dollars). But Einstein was phlegmatic. When a book was\ published entitled 100 Authors Against Einstein, he retorted, “If I were wrong, then one would have been enough!” In 1933, Hitler came to power. Einstein was in America, and declared he \ would not return to Germany. Then, while Nazi militia raided his house and confiscated his bank account, a \ Berlin newspaper displayed the headline “Good News from Einstein – He’s Not Coming Back.” I\ n the face of the Nazi threat, Einstein renounced pacifism, and eventually, fearing that German scientists would\ build a nuclear bomb, proposed that the United States should develop its own. But even before the first atom\ ic bomb had been detonated, he was publicly warning of the dangers of nuclear war and proposing internation\ al control of nuclear weaponry.

Throughout his life, Einstein’s efforts toward peace probably achieve\ d little that would last – and certainly won him few friends. His vocal support of the Zionist cause, however, was du\ ly recognized in 1952, when he was offered the presidency of Israel. He declined, saying he thought he was \ too naive in politics. But perhaps his real reason was different: to quote him again, “Equations are more im\ portant to me, because politics is for the present, but an equation is something for eternity.”   GALILEO GALILEI Galileo, perhaps more than any other single person, was responsible for \ the birth of modern science. His renowned conflict with the Catholic Church was central to his philosophy\ , for Galileo was one of the first to argue that man could hope to understand how the world works, and, moreov\ er, that we could do this by observing the real world.

Galileo had believed Copernican theory (that the planets orbited the su\ n) since early on, but it was only when he found the evidence needed to support the idea that he started to publ\ icly support it. He wrote about Copernicus’s theory in Italian (not the usual academic Latin), and \ soon his views became widely supported outside the universities. This annoyed the Aristotelian professors, who \ united against him seeking to persuade the Catholic Church to ban Copernicanism.

Galileo, worried by this, traveled to Rome to speak to the ecclesiastica\ l authorities. He argued that the Bible was not intended to tell us anything about scientific theories, and that\ it was usual to assume that, where the Bible conflicted with common sense, it was being allegorical. But the Ch\ urch was afraid of a scandal that might undermine its fight against Protestantism, and so took repressive measur\ es. It declared Copernicanism “false and erroneous” in 1616, and commanded Galileo never again to “defe\ nd or hold” the doctrine. Galileo acquiesced.

In 1623, a longtime friend of Galileo’s became the Pope. Immediately \ Galileo tried to get the 1616 decree revoked. He failed, but he did manage to get permission to write a book \ discussing both Aristotelian and Copernican theories, on two conditions: he would not take sides and woul\ d come to the conclusion that man could in any case not determine how the world worked because God could b\ ring about the same effects in ways unimagined by man, who could not place restrictions on God’s omn\ ipotence.

The book, Dialogue Concerning the Two Chief World Systems, was completed and published in 1632, with the full backing of the censors – and was immediately greeted throughout \ Europe as a literary and philosophical masterpiece. Soon the Pope, realizing that people were seeing the book a\ s a convincing argument in favor of A Brief History of Time - Stephen Hawking... Chapter 12 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/k.html (3 of 4) [2/20/2001 3:16:08 AM] Copernicanism, regretted having allowed its publication. The Pope argued\ that although the book had the official blessing of the censors, Galileo had nevertheless contravened t\ he 1616 decree. He brought Galileo before the Inquisition, who sentenced him to house arrest for life and c\ ommanded him to publicly renounce Copernicanism. For a second time, Galileo acquiesced.

Galileo remained a faithful Catholic, but his belief in the independence\ of science had not been crushed. Four years before his death in 1642, while he was still under house arrest, t\ he manuscript of his second major book was smuggled to a publisher in Holland. It was this work, referred to as\ Two New Sciences, even more than his support for Copernicus, that was to be the genesis of modern physics.

  ISAAC NEWTON Isaac Newton was not a pleasant man. His relations with other academics \ were notorious, with most of his later life spent embroiled in heated disputes. Following publication of Principia Mathematica – surely the most influential book ever written in physics – Newton had risen rapidly i\ nto public prominence. He was appointed president of the Royal Society and became the first scientist ever to be\ knighted.

Newton soon clashed with the Astronomer Royal, John Flamsteed, who had e\ arlier provided Newton with much-needed data for Principia, but was now withholding information that Newton wanted. Newton would not\ take no for an answer: he had himself appointed to the governing body of\ the Royal Observatory and then tried to force immediate publication of the data. Eventually he arranged for F\ lamsteed’s work to be seized and prepared for publication by Flamsteed’s mortal enemy, Edmond Halley. \ But Flamsteed took the case to court and, in the nick of time, won a court order preventing distribution of t\ he stolen work. Newton was incensed and sought his revenge by systematically deleting all references to Flamstee\ d in later editions of Principia.

A more serious dispute arose with the German philosopher Gottfried Leibn\ iz. Both Leibniz and Newton had independently developed a branch of mathematics called calculus, which u\ nderlies most of modern physics.

Although we now know that Newton discovered calculus years before Leibni\ z, he published his work much later. A major row ensued over who had been first, with scientists vigor\ ously defending both contenders. It is remarkable, however, that most of the articles appearing in defense of N\ ewton were originally written by his own hand – and only published in the name of friends! As the row grew\ , Leibniz made the mistake of appealing to the Royal Society to resolve the dispute. Newton, as president, appoi\ nted an “impartial” committee to investigate, coincidentally consisting entirely of Newton’s friends! \ But that was not all: Newton then wrote the committee’s report himself and had the Royal Society publish it, offi\ cially accusing Leibniz of plagiarism. Still unsatisfied, he then wrote an anonymous review of the report in the Roya\ l Society’s own periodical. Following the death of Leibniz, Newton is reported to have declared that he had ta\ ken great satisfaction in “breaking Leibniz’s heart.” During the period of these two disputes, Newton had already left Cambrid\ ge and academe. He had been active in anti-Catholic politics at Cambridge, and later in Parliament, and was\ rewarded eventually with the lucrative post of Warden of the Royal Mint. Here he used his talents for deviousne\ ss and vitriol in a more socially acceptable way, successfully conducting a major campaign against counter\ feiting, even sending several men to their death on the gallows.

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Chapter 12 file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/k.html (4 of 4) [2/20/2001 3:16:08 AM] GLOSSARY   Absolute zero: The lowest possible temperature, at which substances contain no heat e\ nergy.

Acceleration : The rate at which the speed of an object is changing.

Anthropic principle : We see the universe the way it is because if it were different we woul\ d not be here to observe it.

Antiparticle : Each type of matter particle has a corresponding antiparticle. When a \ particle collides with its antiparticle, they annihilate, leaving only energy.

Atom : The basic unit of ordinary matter, made up of a tiny nucleus (consist\ ing of protons and neutrons) surrounded by orbiting electrons.

Big bang : The singularity at the beginning of the universe.

Big crunch : The singularity at the end of the universe.

Black hole : A region of space-time from which nothing, not even light, can escape,\ because gravity is so strong.

Casimir effect : The attractive pressure between two flat, parallel metal plates placed\ very near to each other in a vacuum. The pressure is due to a reduction in the usual number of virt\ ual particles in the space between the plates.

Chandrasekhar limit : The maximum possible mass of a stable cold star, above which it must c\ ollapse into a black hole.

Conservation of energy : The law of science that states that energy (or its equivalent in mass\ ) can neither be created nor destroyed.

Coordinates : Numbers that specify the position of a point in space and time.

Cosmological constant : A mathematical device used by Einstein to give space-time an inbuilt t\ endency to expand.

Cosmology : The study of the universe as a whole.

Dark matter : Matter in galaxies, clusters, and possibly between clusters, that can \ not be observed directly but can be detected by its gravitational effect. As much as 90 percent of th\ e mass of the universe may be in the form of dark matter.

Duality : A correspondence between apparently different theories that lead to th\ e same physical results.

Einstein-Rosen bridge : A thin tube of space-time linking two black holes. Also see Wormhole.

Electric charge : A property of a particle by which it may repel (or attract) other pa\ rticles that have a charge of similar (or opposite) sign.

Electromagnetic force : The force that arises between particles with electric charge; the seco\ nd strongest of the four fundamental forces.

Electron : A particle with negative electric charge that orbits the nucleus of an\ atom.

Electroweak unification energy : The energy (around 100 GeV) above which the distinction between the electromagnetic force and the weak force disappears.

A Brief History of Time - Stephen Hawking... Glossary file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/l.html (1 of 4) [2/20/2001 3:16:19 AM] Elementary particle: A particle that, it is believed, cannot be subdivided.

Event : A point in space-time, specified by its time and place.

Event horizon : The boundary of a black hole.

Exclusion principle : The idea that two identical spin-1/2 particles cannot have (within th\ e limits set by the uncertainty principle) both the same position and the same velocity.

Field : Something that exists throughout space and time, as opposed to a parti\ cle that exists at only one point at a time.

Frequency : For a wave, the number of complete cycles per second.

Gamma rays : Electromagnetic rays of very short wavelength, produced in radio-activ\ e decay or by collisions of elementary particles.

General relativity : Einstein’s theory based on the idea that the laws of science should\ be the same for all observers, no matter how they are moving. It explains the force of gravi\ ty in terms of the curvature of a four-dimensional space-time.

Geodesic : The shortest (or longest) path between two points.

Grand unification energy : The energy above which, it is believed, the electro-magnetic force, we\ ak force, and strong force become indistinguishable from each other.

Grand unified theory (GUT) : A theory which unifies the electromagnetic, strong, and weak forces.

Imaginary time : Time measured using imaginary numbers.

Light cone : A surface in space-time that marks out the possible directions for lig\ ht rays passing through a given event.

Light-second (light-year) : The distance traveled by light in one second (year).

Magnetic field : The field responsible for magnetic forces, now incorporated along with\ the electric field, into the electromagnetic field.

Mass : The quantity of matter in a body; its inertia, or resistance to accele\ ration.

Microwave background radiation : The radiation from the glowing of the hot early universe, now so great\ ly red-shifted that it appears not as light but as microwaves (radio waves\ with a wavelength of a few centimeters).

Also see COBE, on page 145.

Naked singularity : A space-time singularity not surrounded by a black hole.

Neutrino : An extremely light (possibly massless) particle that is affected onl\ y by the weak force and gravity.

Neutron : An uncharged particle, very similar to the proton, which accounts for \ roughly half the particles in an atomic nucleus.

Neutron star : A cold star, supported by the exclusion principle repulsion between ne\ utrons.

No boundary condition : The idea that the universe is finite but has no boundary (in imaginar\ y time).

Nuclear fusion : The process by which two nuclei collide and coalesce to form a single,\ heavier nucleus.

Nucleus : The central part of an atom, consisting only of protons and neutrons, \ held together by the strong A Brief History of Time - Stephen Hawking... Glossary file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/l.html (2 of 4) [2/20/2001 3:16:19 AM] force.

Particle accelerator: A machine that, using electromagnets, can accelerate moving charged pa\ rticles, giving them more energy.

Phase : For a wave, the position in its cycle at a specified time: a measure o\ f whether it is at a crest, a trough, or somewhere in between.

Photon : A quantum of light.

Planck’s quantum principle : The idea that light (or any other classical waves) can be emitted or\ absorbed only in discrete quanta, whose energy is proportional to their wavelengt\ h.

Positron : The (positively charged) antiparticle of the electron.

Primordial black hole : A black hole created in the very early universe.

Proportional : ‘X is proportional to Y’ means that when Y is multiplied by any \ number, so is X. ‘X is inversely proportional to Y’ means that when Y is multiplied by any number, X i\ s divided by that number.

Proton : A positively charged particle, very similar to the neutron, that accou\ nts for roughly half the particles in the nucleus of most atoms.

Pulsar : A rotating neutron star that emits regular pulses of radio waves.

Quantum : The indivisible unit in which waves may be emitted or absorbed.

Quantum chromodynamics (QCD): The theory that describes the interactions of quarks and gluon\ s.

Quantum mechanics : The theory developed from Planck’s quantum principle and Heisenberg\ ’s uncertainty principle.

Quark : A (charged) elementary particle that feels the strong force. Protons\ and neutrons are each composed of three quarks.

Radar : A system using pulsed radio waves to detect the position of objects by\ measuring the time it takes a single pulse to reach the object and be reflected back.

Radioactivity : The spontaneous breakdown of one type of atomic nucleus into another.

Red shift : The reddening of light from a star that is moving away from us, due to\ the Doppler effect.

Singularity : A point in space-time at which the space-time curvature becomes infini\ te.

Singularity theorem : A theorem that shows that a singularity must exist under certain circu\ mstances – in particular, that the universe must have started with a singularity.

Space-time : The four-dimensional space whose points are events.

Spatial dimension : Any of the three dimensions that are spacelike – that is, any excep\ t the time dimension.

Special relativity : Einstein’s theory based on the idea that the laws of science should\ be the same for all observers, no matter how they are moving, in the absence of gravitationa\ l phenomena.

Spectrum : The component frequencies that make up a wave. The visible part of the\ sun’s spectrum can be seen in a rainbow.

Spin : An internal property of elementary particles, related to, but not iden\ tical to, the everyday concept of spin.

A Brief History of Time - Stephen Hawking... Glossary file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/l.html (3 of 4) [2/20/2001 3:16:19 AM] Stationary state: One that is not changing with time: a sphere spinning at a constant ra\ te is stationary because it looks identical at any given instant.

String theory : A theory of physics in which particles are described as waves on strin\ gs. Strings have length but no other dimension.

Strong force : The strongest of the four fundamental forces, with the shortest range \ of all. It holds the quarks together within protons and neutrons, and holds the protons and neutrons\ together to form atoms.

Uncertainty principle : The principle, formulated by Heisenberg, that one can never be exactly\ sure of both the position and the velocity of a particle; the more accurately one knows t\ he one, the less accurately one can know the other.

Virtual particle : In quantum mechanics, a particle that can never be directly detected, \ but whose existence does have measurable effects.

Wave/particle duality : The concept in quantum mechanics that there is no distinction between \ waves and particles; particles may sometimes behave like waves, and waves like par\ ticles.

Wavelength : For a wave, the distance between two adjacent troughs or two adjacent \ crests.

Weak force : The second weakest of the four fundamental forces, with a very short r\ ange. It affects all matter particles, but not force-carrying particles.

Weight : The force exerted on a body by a gravitational field. It is proportion\ al to, but not the same as, its mass.

White dwarf : A stable cold star, supported by the exclusion principle repulsion bet\ ween electrons.

Wormhole : A thin tube of space-time connecting distant regions of the universe. \ Wormholes might also link to parallel or baby universes and could provide the possibility of time tra\ vel.

  PREVIOUS NEXT     A Brief History of Time - Stephen Hawking... Glossary file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/l.html (4 of 4) [2/20/2001 3:16:19 AM] ACKNOWLEDGMENTS   Many people have helped me in writing this book. My scientific colleague\ s have without exception been inspiring. Over the years my principal associates and collaborators were\ Roger Penrose, Robert Geroch, Brandon Carter, George Ellis, Gary Gibbons, Don Page, and Jim Hartle. I \ owe a lot to them, and to my research students, who have always given me help when needed.

One of my students, Brian Whitt, gave me a lot of help writing the first\ edition of this book. My editor at Bantam Books, Peter Guzzardi, made innumerable comments which improved the book\ considerably. In addition, for this edition, I would like to thank Andrew Dunn, who helped me revise th\ e text.

I could not have written this book without my communication system. The \ software, called Equalizer, was donated by Walt Waltosz of Words Plus Inc., in Lancaster, California. My\ speech synthesizer was donated by Speech Plus, of Sunnyvale, California. The synthesizer and laptop comput\ er were mounted on my wheelchair by David Mason, of Cambridge Adaptive Communication Ltd. With this syste\ m I can communicate better now than before I lost my voice.

I have had a number of secretaries and assistants over the years in whic\ h I wrote and revised this book. On the secretarial side, I’m very grateful to Judy Fella, Ann Ralph, Laura G\ entry, Cheryl Billington, and Sue Masey. My assistants have been Colin Williams, David Thomas, and Raymond Laflamme,\ Nick Phillips, Andrew Dunn, Stuart Jamieson, Jonathan Brenchley, Tim Hunt, Simon Gill, Jon Rogers, a\ nd Tom Kendall. They, my nurses, colleagues, friends, and family have enabled me to live a very full life\ and to pursue my research despite my disability.

Stephen Hawking   ABOUT THE AUTHOR Stephen Hawking, who was born in 1942 on the anniversary of Galileo’s\ death, holds Isaac Newton’s chair as Lucasian Professor of Mathematics at the University of Cambridge. Widely\ regarded as the most brilliant theoretical physicist since Einstein, he is also the author of Black Holes and Baby Universes, published in 1993, as well as numerous scientific papers and books.   PREVIOUS     A Brief History of Time - Stephen Hawking... Acknowledgments file:///C|/WINDOWS/Desktop/blahh/Stephen Hawking - A brief history of ti\ me/m.html [2/20/2001 3:16:30 AM]