Illustrated Theory of Everything: The Origin and Fate of the Universe (3 page)

But which Friedmann model describes our universe? Will the universe eventu-ally stop expanding and start contracting, or will it expand forever? To answerthis question we need to know the present rate of expansion of the universeand its present average density. If the density is less than a certain criticalvalue, determined by the rate of expansion, the gravitational attraction will betoo 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 theuniverse to recollapse.

We can determine the present rate of expansion by measuring the velocities atwhich other galaxies are moving away from us, using the Doppler effect. Thiscan be done very accurately. However, the distances to the galaxies are notvery well known because we can only measure them indirectly. So all we knowis that the universe is expanding by between 5 percent and 10 percent everythousand million years. However, our uncertainty about the present averagedensity of the universe is even greater.

If we add up the masses of all the stars that we can see in our galaxy and othergalaxies, the total is less than one-hundredth of the amount required to haltthe expansion of the universe, even in the lowest estimate of the rate of expan-sion. But we know that our galaxy and other galaxies must contain a largeamount of dark matter which we cannot see directly, but which we know mustbe there because of the influence of its gravitational attraction on the orbits ofstars and gas in the galaxies. Moreover, most galaxies are found in clusters, andwe can similarly infer the presence of yet more dark matter in between thegalaxies in these clusters by its effect on the motion of the galaxies. When weadd up all this dark matter, we still get only about one-tenth of the amountrequired to halt the expansion. However, there might be some other form ofmatter which we have not yet detected and which might still raise the averagedensity of the universe up to the critical value needed to halt the expansion.The present evidence, therefore, suggests that the universe will probablyexpand forever. But don’t bank on it. All we can really be sure of is that evenif the universe is going to recollapse, it won’t do so for at least another tenthousand million years, since it has already been expanding for at least thatlong. This should not unduly worry us since by that time, unless we havecolonies beyond the solar system, mankind will long since have died out,extinguished along with the death of our sun.

THE BIG BANG

All of the Friedmann solutions have the feature that at some time in thepast, between ten and twenty thousand million years ago, the distancebetween neighboring galaxies must have been zero. At that time, which wecall the big bang, the density of the universe and the curvature of space-timewould have been infinite. This means that the general theory of relativity-on which Friedmann’s solutions are based-predicts that there is a singularpoint in the universe.

All our theories of science are formulated on the assumption that space-timeis smooth and nearly flat, so they would all break down at the big bang singu-larity, where the curvature of space-time is infinite. This means that even ifthere were events before the big bang, one could not use them to determinewhat would happen afterward, because predictability would break down at thebig bang. Correspondingly, if we know only what has happened since the bigbang, we could not determine what happened beforehand. As far as we areconcerned, events before the big bang can have no consequences, so theyshould not form part of a scientific model of the universe. We should thereforecut 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 becauseit smacks of divine intervention. (The Catholic church, on the other hand, hadseized on the big bang model and in 1951 officially pronounced it to be inaccordance with the Bible.) There were a number of attempts to avoid the con-clusion that there had been a big bang. The proposal that gained widest supportwas called the steady state theory. It was suggested in 1948 by two refugees fromNazi-occupied Austria, Hermann Bondi and Thomas Gold, together with theBriton Fred Hoyle, who had worked with them on the development of radarduring the war. The idea was that as the galaxies moved away from each other,new galaxies were continually forming in the gaps in between, from newmatter that was being continually created. The universe would therefore lookroughly the same at all times as well as at all points of space.

The steady state theory required a modification of general relativity to allowfor the continual creation of matter, but the rate that was involved was solow-about one particle per cubic kilometer per year-that it was not in con-flict with experiment. The theory was a good scientific theory, in the sensethat it was simple and it made definite predictions that could be tested byobservation. One of these predictions was that the number of galaxies or sim-ilar objects in any given volume of space should be the same wherever andwhenever we look in the universe.

In the late 1950s and early 1960s, a survey of sources of radio waves from outerspace was carried out at Cambridge by a group of astronomers led by MartinRyle. The Cambridge group showed that most of these radio sources must lieoutside our galaxy, and also that there were many more weak sources thanstrong ones. They interpreted the weak sources as being the more distant ones,and the stronger ones as being near. Then there appeared to be fewer sourcesper unit volume of space for the nearby sources than for the distant ones.

This could have meant that we were at the center of a great region in the uni-verse in which the sources were fewer than elsewhere. Alternatively, it couldhave meant that the sources were more numerous in the past, at the time thatthe radio waves left on their journey to us, than they are now. Either explana-tion contradicted the predictions of the steady state theory. Moreover, thediscovery of the microwave radiation by Penzias and Wilson in 1965 also indi-cated that the universe must have been much denser in the past. The steadystate theory therefore had regretfully to be abandoned.

Another attempt to avoid the conclusion that there must have been a big bangand, therefore, a beginning of time, was made by two Russian scientists,Evgenii Lifshitz and Isaac Khalatnikov, in 1963. They suggested that the bigbang might be a peculiarity of Friedmann’s models alone, which after all wereonly approximations to the real universe. Perhaps, of all the models that wereroughly like the real universe, only Friedmann’s would contain a big bang sin-gularity. In Friedmann’s models, the galaxies are all moving directly away fromeach other. So it is not surprising that at some time in the past they were all atthe same place. In the real universe, however, the galaxies are not just movingdirectly away from each other-they also have small sideways velocities. So inreality they need never have been all at exactly the same place, only very closetogether. Perhaps, then, the current expanding universe resulted not from a bigbang singularity, but from an earlier contracting phase; as the universe had col-lapsed, the particles in it might not have all collided, but they might haveflown past and then away from each other, producing the present expansion ofthe universe. How then could we tell whether the real universe should havestarted out with a big bang?

What Lifshitz and Khalatnikov did was to study models of the universe whichwere roughly like Friedmann’s models but which took account of the irregular-ities and random velocities of galaxies in the real universe. They showed thatsuch models could start with a big bang, even though the galaxies were nolonger always moving directly away from each other. But they claimed thatthis was still only possible in certain exceptional models in which the galaxieswere all moving in just the right way. They argued that since there seemed tobe infinitely more Friedmann-like models without a big bang singularity thanthere were with one, we should conclude that it was very unlikely that therehad been a big bang. They later realized, however, that there was a much moregeneral class of Friedmann-like models which did have singularities, and inwhich the galaxies did not have to be moving in any special way. They there-fore withdrew their claim in 1970.

The work of Lifshitz and Khalatnikov was valuable because it showed that theuniverse could have had a singularity-a big bang-if the general theory of rel-ativity was correct. However, it did not resolve the crucial question: Does gen-eral relativity predict that our universe should have the big bang, a beginningof time? The answer to this came out of a completely different approach start-ed by a British physicist, Roger Penrose, in 1965. He used the way light conesbehave in general relativity, and the fact that gravity is always attractive, toshow that a star that collapses under its own gravity is trapped in a region whoseboundary eventually shrinks to zero size. This means that all the matter in thestar will be compressed into a region of zero volume, so the density of matterand the curvature of space-time become infinite. In other words, one has a sin-gularity contained within a region of space-time known as a black hole.

At first sight, Penrose’s result didn’t have anything to say about the questionof whether there was a big bang singularity in the past. However, at the timethat Penrose produced his theorem, I was a research student desperately look-ing for a problem with which to complete my Ph.D. thesis. I realized that if onereversed the direction of time in Penrose’s theorem so that the collapse becamean expansion, the conditions of his theorem would still hold, provided theuniverse were roughly like a Friedmann model on large scales at the presenttime. Penrose’s theorem had shown that any collapsing star must end in asingularity; the time-reversed argument showed that any Friedmann-likeexpanding universe must have begun with a singularity. For technical reasons,Penrose’s theorem required that the universe be infinite in space. So I coulduse it to prove that there should be a singularity only if the universe wasexpanding fast enough to avoid collapsing again, because only that Friedmannmodel was infinite in space.

During the next few years I developed new mathematical techniques toremove this and other technical conditions from the theorems that provedthat singularities must occur. The final result was a joint paper by Penroseand myself in 1970, which proved that there must have been a big bang singu-larity provided only that general relativity is correct and that the universecontains as much matter as we observe.

There was a lot of opposition to our work, partly from the Russians, whofollowed the party line laid down by Lifshitz and Khalatnikov, and partly frompeople who felt that the whole idea of singularities was repugnant and spoiledthe beauty of Einstein’s theory. However, one cannot really argue with themathematical theorem. So it is now generally accepted that the universe musthave a beginning.

The Theory of Everything: The Origin and Fate of the Universe

Chapter 3 - THIRD LECTURE - BLACK HOLES

The term black hole is of very recent origin. It was coined in 1969 by theAmerican scientist John Wheeler as a graphic description of an idea thatgoes back at least two hundred years. At that time there were two theoriesabout light. One was that it was composed of particles; the other was that itwas made of waves. We now know that really both theories are correct. By thewave/particle duality of quantum mechanics, light can be regarded as both awave and a particle. Under the theory that light was made up of waves, it wasnot clear how it would respond to gravity. But if light were composed of parti-cles, one might expect them to be affected by gravity in the same way thatcannonballs, rockets, and planets are.

On this assumption, a Cambridge don, John Michell, wrote a paper in 1783in the Philosophical Transactions of the Royal Society of London. In it, he point-ed out that a star that was sufficiently massive and compact would have sucha strong gravitational field that light could not escape. Any light emittedfrom the surface of the star would be dragged back by the star’s gravitationalattraction before it could get very far. Michell suggested that there might bea large number of stars like this. Although we would not be able to see thembecause the light from them would not reach us, we would still feel their grav-itational attraction. Such objects are what we now call black holes, becausethat is what they are-black voids in space.

A similar suggestion was made a few years later by the French scientist theMarquis de Laplace, apparently independently of Michell. Interestinglyenough, he included it in only the first and second editions of his book, TheSystem of the World, and left it out of later editions; perhaps he decided that itwas a crazy idea. In fact, it is not really consistent to treat light like cannon-balls in Newton’s theory of gravity because the speed of light is fixed. A can-nonball fired upward from the Earth will be slowed down by gravity and willeventually stop and fall back. A photon, however, must continue upward at aconstant speed. How, then, can Newtonian gravity affect light? A consistenttheory of how gravity affects light did not come until Einstein proposed gen-eral relativity in 1915; and even then it was a long time before the implica-tions of the theory for massive stars were worked out.

To understand how a black hole might be formed, we first need an understand-ing of the life cycle of a star. A star is formed when a large amount of gas, most-ly hydrogen, starts to collapse in on itself due to its gravitational attraction. Asit contracts, the atoms of the gas collide with each other more and more fre-quently and at greater and greater speeds-the gas heats up. Eventually the gaswill be so hot that when the hydrogen atoms collide they no longer bounce offeach other but instead merge with each other to form helium atoms. The heatreleased in this reaction, which is like a controlled hydrogen bomb, is whatmakes the stars shine. This additional heat also increases the pressure of thegas until it is sufficient to balance the gravitational attraction, and the gasstops contracting. It is a bit like a balloon where there is a balance between thepressure of the air inside, which is trying to make the balloon expand, and thetension in the rubber, which is trying to make the balloon smaller.

The stars will remain stable like this for a long time, with the heat from thenuclear reactions balancing the gravitational attraction. Eventually, however,the star will run out of its hydrogen and other nuclear fuels. And paradoxical-ly, the more fuel a star starts off with, the sooner it runs out. This is becausethe more massive the star is, the hotter it needs to be to balance its gravita-tional attraction. And the hotter it is, the faster it will use up its fuel. Our sunhas probably got enough fuel for another five thousand million years or so, butmore massive stars can use up their fuel in as little as one hundred millionyears, much less than the age of the universe. When the star runs out of fuel,it will start to cool off and so to contract. What might happen to it then wasonly first understood at the end of the 1920s.

In 1928 an Indian graduate student named Subrahmanyan Chandrasekhar setsail for England to study at Cambridge with the British astronomer Sir ArthurEddington. Eddington was an expert on general relativity. There is a story thata journalist told Eddington in the early 1920s that he had heard there wereonly three people in the world who understood general relativity. Eddingtonreplied, “I am trying to think who the third person is.”During his voyage from India, Chandrasekhar worked out how big a star couldbe and still separate itself against its own gravity after it had used up all itsfuel. The idea was this: When the star becomes small, the matter particles getvery near each other. But the Pauli exclusion principle says that two matterparticles cannot have both the same position and the same velocity. The mat-ter particles must therefore have very different velocities. This makes themmove away from each other, and so tends to make the star expand. A star cantherefore maintain itself at a constant radius by a balance between the attrac-tion of gravity and the repulsion that arises from the exclusion principle, justas earlier in its life the gravity was balanced by the heat.

Chandrasekhar realized, however, that there is a limit to the repulsion that theexclusion principle can provide. The theory of relativity limits the maximumdifference in the velocities of the matter particles in the star to the speed oflight. This meant that when the star got sufficiently dense, the repulsioncaused 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 halftimes the mass of the sun would not be able to support itself against its owngravity. This mass is now known as the Chandrasekhar limit.

This had serious implications for the ultimate fate of massive stars. If a star’smass is less than the Chandrasekhar limit, it can eventually stop contractingand settle down to a possible final state as a white dwarf with a radius of a fewthousand miles and a density of hundreds of tons per cubic inch. A white dwarfis supported by the exclusion principle repulsion between the electrons in itsmatter. We observe a large number of these white dwarf stars. One of the firstto be discovered is the star that is orbiting around Sirius, the brightest star inthe night sky.

It was also realized that there was another possible final state for a star alsowith a limiting mass of about one or two times the mass of the sun, but muchsmaller than even the white dwarf. These stars would be supported by theexclusion principle repulsion between the neutrons and protons, rather thanbetween the electrons. They were therefore called neutron stars. They wouldhave had a radius of only ten miles or so and a density of hundreds of millionsof tons per cubic inch. At the time they were first predicted, there was no waythat neutron stars could have been observed, and they were not detected untilmuch later.

Stars with masses above the Chandrasekhar limit, on the other hand, have abig problem when they come to the end of their fuel. In some cases they mayexplode or manage to throw off enough matter to reduce their mass below thelimit, but it was difficult to believe that this always happened, no matter howbig the star. How would it know that it had to lose weight? And even if everystar managed to lose enough mass, what would happen if you added more massto a white dwarf or neutron star to take it over the limit? Would it collapse toinfinite density?

Eddington was shocked by the implications of this and refused to believeChandrasekhar’s result. He thought it was simply not possible that a star couldcollapse to a point. This was the view of most scientists. Einstein himself wrotea paper in which he claimed that stars would not shrink to zero size.The hos-tility of other scientists, particularly of Eddington, his former teacher and theleading authority on the structure of stars, persuaded Chandrasekhar to aban-don this line of work and turn instead to other problems in astronomy.However, when he was awarded the Nobel Prize in 1983, it was, at least inpart, for his early work on the limiting mass of cold stars.

Chandrasekhar had shown that the exclusion principle could not halt the col-lapse of a star more massive than the Chandrasekhar limit. But the problem ofunderstanding what would happen to such a star, according to general relativ-ity, was not solved until 1939 by a young American, Robert Oppenheimer. Hisresult, however, suggested that there would be no observational consequencesthat could be detected by the telescopes of the day. Then the war intervenedand Oppenheimer himself became closely involved in the atom bomb project.And after the war the problem of gravitational collapse was largely forgottenas most scientists were then interested in what happens on the scale of theatom and its nucleus. In the 1960s, however, interest in the large-scale prob-lems of astronomy and cosmology was revived by a great increase in the num-ber and range of astronomical observations brought about by the applicationof modern technology. Oppenheimer’s work was then rediscovered andextended by a number of people.

The picture that we now have from Oppenheimer’s work is as follows: Thegravitational field of the star changes the paths of light rays in space-time fromwhat 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 emit-ted from their tips, are bent slightly inward near the surface of the star. Thiscan be seen in the bending of light from distant stars that is observed duringan eclipse of the sun. As the star contracts, the gravitational field at its surfacegets stronger and the light cones get bent inward more. This makes it moredifficult for light from the star to escape, and the light appears dimmer andredder to an observer at a distance.

Eventually, when the star has shrunk to a certain critical radius, the gravita-tional field at the surface becomes so strong that the light cones are bentinward so much that the light can no longer escape. According to the theoryof relativity, nothing can travel faster than light. Thus, if light cannot escape,neither can anything else. Everything is dragged back by the gravitationalfield. So one has a set of events, a region of space-time, from which it is notpossible to escape to reach a distant observer. This region is what we now calla black hole. Its boundary is called the event horizon. It coincides with thepaths of the light rays that just fail to escape from the black hole.

In order to understand what you would see if you were watching a star collapseto form a black hole, one has to remember that in the theory of relativity thereis no absolute time. Each observer has his own measure of time. The time forsomeone on a star will be different from that for someone at a distance, becauseof the gravitational field of the star. This effect has been measured in an exper-iment on Earth with clocks at the top and bottom of a water tower. Supposean intrepid astronaut on the surface of the collapsing star sent a signal everysecond, according to his watch, to his spaceship orbiting about the star. Atsome time on his watch, say eleven o’clock, the star would shrink below thecritical radius at which the gravitational field became so strong that the signalswould no longer reach the spaceship.

His companions watching from the spaceship would find the intervals betweensuccessive signals from the astronaut getting longer and longer as eleveno’clock approached. However, the effect would be very small before 10:59:59.They would have to wait only very slightly more than a second between theastronaut’s 10:59:58 signal and the one that he sent when his watch read10:59:59, but they would have to wait forever for the eleven o’clock signal.The light waves emitted from the surface of the star between 10:59:59 andeleven o’clock, by the astronaut’s watch, would be spread out over an infiniteperiod of time, as seen from the spaceship.

The time interval between the arrival of successive waves at the spaceshipwould get longer and longer, and so the light from the star would appearredder and redder and fainter and fainter. Eventually the star would be so dimthat it could no longer be seen from the spaceship. All that would be left wouldbe a black hole in space. The star would, however, continue to exert the samegravitational force on the spaceship. This is because the star is still visible tothe spaceship, at least in principle. It is just that the light from the surface isso red-shifted by the gravitational field of the star that it cannot be seen.However, the red shift does not affect the gravitational field of the star itself.Thus, the spaceship would continue to orbit the black hole.

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 densitywithin the black hole. This is rather like the big bang at the beginning of time,only it would be an end of time for the collapsing body and the astronaut. Atthe singularity, the laws of science and our ability to predict the future wouldbreak down. However, any observer who remained outside the black holewould not be affected by this failure of predictability, because neither light norany other signal can reach them from the singularity.

This remarkable fact led Roger Penrose to propose the cosmic censorshiphypothesis, which might be paraphrased as “God abhors a naked singularity.”In other words, the singularities produced by gravitational collapse occur onlyin places like black holes, where they are decently hidden from outside viewby an event horizon. Strictly, this is what is known as the weak cosmic censor-ship hypothesis: protect obervers who remain outside the black hole from theconsequences of the breakdown of predictability that occurs at the singularity.But it does nothing at all for the poor unfortunate astronaut who falls into thehole. Shouldn’t God protect his modesty as well?

There are some solutions of the equations of general relativity in which it ispossible for our astronaut to see a naked singularity. He may be able to avoidhitting the singularity and instead fall through a “worm hole” and come out inanother region of the universe. This would offer great possibilities for travel inspace and time, but unfortunately it seems that the solutions may all be high-ly unstable. The least disturbance, such as the presence of an astronaut, maychange them so that the astronaut cannot see the singularity until he hits itand his time comes to an end. In other words, the singularity always lies in hisfuture and never in his past.