Illustrated Theory of Everything: The Origin and Fate of the Universe (4 page)

The strong version of the cosmic censorship hypothesis states that in a realis-tic solution, the singularities always lie either entirely in the future, like thesingularities of gravitational collapse, or entirely in the past, like the big bang.It is greatly to be hoped that some version of the censorship hypothesis holds,because close to naked singularities it may be possible to travel into the past.While this would be fine for writers of science fiction, it would mean that noone’s life would ever be safe. Someone might go into the past and kill yourfather or mother before you were conceived.

In a gravitational collapse to form a black hole, the movements would bedammed by the emission of gravitational waves. One would therefore expectthat it would not be too long before the black hole would settle down to a sta-tionary state. It was generally supposed that this final stationary state woulddepend on the details of the body that had collapsed to form the black hole.The black hole might have any shape or size, and its shape might not even befixed, but instead be pulsating.However, in 1967, the study of black holes was revolutionized by a paper writ-ten in Dublin by Werner Israel. Israel showed that any black hole that is notrotating must be perfectly round or spherical. Its size, moreover, would dependonly on its mass. It could, in fact, be described by a particular solution ofEinstein’s equations that had been known since 1917, when it had been foundby Karl Schwarzschild shortly after the discovery of general relativity. At first,Israel’s result was interpreted by many people, including Israel himself, as evi-dence that black holes would form only from the collapse of bodies that wereperfectly round or spherical. As no real body would be perfectly spherical, thismeant that, in general, gravitational collapse would lead to naked singularities.There was, however, a different interpretation of Israel’s result, which wasadvocated by Roger Penrose and John Wheeler in particular. This was that ablack hole should behave like a ball of fluid. Although a body might start offin an unspherical state, as it collapsed to form a black hole it would settle downto a spherical state due to the emission of gravitational waves. Further calcu-lations supported this view and it came to be adopted generally.

Israel’s result had dealt only with the case of black holes formed from nonro-tating bodies. On the analogy with a ball of fluid, one would expect that ablack hole made by the collapse of a rotating body would not be perfectlyround. It would have a bulge round the equator caused by the effect of the rota-tion. We observe a small bulge like this in the sun, caused by its rotation onceevery twenty-five days or so. In 1963, Roy Kerr, a New Zealander, had found aset of black-hole solutions of the equations of general relativity more generalthan the Schwarzschild solutions. These “Kerr” black holes rotate at aconstant rate, their size and shape depending only on their mass and rate ofrotation. If the rotation was zero, the black hole was perfectly round and thesolution was identical to the Schwarzschild solution. But if the rotation wasnonzero, the black hole bulged outward near its equator. It was therefore nat-ural to conjecture that a rotating body collapsing to form a black hole wouldend up in a state described by the Kerr solution.

In 1970, a colleague and fellow research student of mine, Brandon Carter, tookthe first step toward proving this conjecture. He showed that, provided a sta-tionary rotating black hole had an axis of symmetry, like a spinning top, its sizeand shape would depend only on its mass and rate of rotation. Then, in 1971,I proved that any stationary rotating black hole would indeed have such anaxis of symmetry. Finally, in 1973, David Robinson at Kings College, London,used Carter’s and my results to show that the conjecture had been correct:Such a black hole had indeed to be the Kerr solution.

So after gravitational collapse a black hole must settle down into a state inwhich it could be rotating, but not pulsating. Moreover, its size and shapewould depend only on its mass and rate of rotation, and not on the nature ofthe body that had collapsed to form it. This result became known by themaxim “A black hole has no hair.” It means that a very large amount of infor-mation about the body that has collapsed must be lost when a black hole isformed, because afterward all we can possibly measure about the body is itsmass and rate of rotation. The significance of this will be seen in the next lec-ture. The no-hair theorem is also of great practical importance because it sogreatly restricts the possible types of black holes. One can therefore makedetailed models of objects that might contain black holes, and compare thepredictions of the models with observations.

Black holes are one of only a fairly small number of cases in the history of sci-ence where a theory was developed in great detail as a mathematical modelbefore 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 onebelieve in objects for which the only evidence was calculations based on thedubious theory of general relativity?

In 1963, however, Maarten Schmidt, an astronomer at the Mount PalomarObservatory in California, found a faint, starlike object in the direction of thesource of radio waves called 3C273-that is, source number 273 in the thirdCambridge catalog of radio sources. When he measured the red shift of theobject, he found it was too large to be caused by a gravitational field: If it hadbeen a gravitational red shift, the object would have to be so massive and sonear to us that it would disturb the orbits of planets in the solar system. Thissuggested that the red shift was instead caused by the expansion of the uni-verse, which in turn meant that the object was a very long way away. And tobe visible at such a great distance, the object must be very bright and must beemitting a huge amount of energy.

The only mechanism people could think of that would produce such largequantities of energy seemed to be the gravitational collapse not just of a starbut of the whole central region of a galaxy. A number of other similar “quasi-stellar objects,” or quasars, have since been discovered, all with large red shifts.But they are all too far away, and too difficult, to observe to provide conclu-sive evidence of black holes.

Further encouragement for the existence of black holes came in 1967 with thediscovery by a research student at Cambridge, Jocelyn Bell, of some objects inthe sky that were emitting regular pulses of radio waves. At first, Jocelyn andher supervisor, Anthony Hewish, thought that maybe they had made contactwith an alien civilization in the galaxy. Indeed, at the seminar at which theyannounced their discovery, I remember that they called the first four sourcesto be found LGM 1-4, LGM standing for “Little Green Men.”

In the end, however, they and everyone else came to the less romantic conclu-sion that these objects, which were given the name pulsars, were in fact justrotating neutron stars. They were emitting pulses of radio waves because of acomplicated indirection between their magnetic fields and surrounding matter.This was bad news for writers of space westerns, but very hopeful for the smallnumber of us who believed in black holes at that time. It was the first positiveevidence that neutron stars existed. A neutron star has a radius of about tenmiles, 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 was not unreasonable to expectthat other stars could collapse to even smaller size and become black holes.How could we hope to detect a black hole, as by its very definition it does notemit any light? It might seem a bit like looking for a black cat in a coal cellar.Fortunately, there is a way, since as John Michell pointed out in his pioneer-ing paper in 1783, a black hole still exerts a gravitational force on nearbyobjects. Astronomers have observed a number of systems in which two starsorbit around each other, attracted toward each other by gravity. They alsoobserved systems in which there is only one visible star that is orbiting aroundsome unseen companion.

One cannot, of course, immediately conclude that the companion is a blackhole. It might merely be a star that is too faint to be seen. However, some ofthese systems, like the one called Cygnus X-I, are also strong sources of X rays.The best explanation for this phenomenon is that the X rays are generated bymatter that has been blown off the surface of the visible star. As it falls towardthe unseen companion, it develops a spiral motion-rather like water runningout of a bath-and it gets very hot, emitting X rays. For this mechanism towork, the unseen object has to be very small, like a white dwarf, neutron star,or black hole.

Now, from the observed motion of the visible star, one can determine the low-est possible mass of the unseen object. In the case of Cygnus X-I, this is aboutsix times the mass of the sun. According to Chandrasekhar’s result, this is toomuch for the unseen object to be a white dwarf. It is also too large a mass tobe a neutron star. It seems, therefore, that it must be a black hole.There are other models to explain Cygnus X-I that do not include a blackhole, but they are all rather far-fetched. A black hole seems to be the onlyreally natural explanation of the observations. Despite this, I have a bet withKip Thorne of the California Institute of Technology that in fact Cygnus X-Idoes not contain a black hole. This is a form of insurance policy for me. I havedone a lot of work on black holes, and it would all be wasted if it turned outthat black holes do not exist. But in that case, I would have the consolation ofwinning my bet, which would bring me four years of the magazine Private Eye.

If black holes do exist, Kip will get only one year of Penthouse, because whenwe made the bet in 1975, we were 80 percent certain that Cygnus was a blackhole. By now I would say that we are about 95 percent certain, but the bet hasyet to be settled.

There is evidence for black holes in a number of other systems in our galaxy,and for much larger black holes at the centers of other galaxies and quasars.One can also consider the possibility that there might be black holes withmasses much less than that of the sun. Such black holes could not be formedby gravitational collapse, because their masses are below the Chandrasekharmass limit. Stars of this low mass can support themselves against the force ofgravity even when they have exhausted their nuclear fuel. So, low-mass blackholes could form only if matter were compressed to enormous densities by verylarge external pressures. Such conditions could occur in a very big hydrogenbomb. The physicist John Wheeler once calculated that if one took all theheavy water in all the oceans of the world, one could build a hydrogen bombthat would compress matter at the center so much that a black hole would becreated. Unfortunately, however, there would be no one left to observe it.A more practical possibility is that such low-mass black holes might have beenformed in the high temperatures and pressures of the very early universe. Blackholes could have been formed if the early universe had not been perfectlysmooth and uniform, because then a small region that was denser than aver-age could be compressed in this way to form a black hole. But we know thatthere must have been some irregularities, because otherwise the matter in theuniverse would still be perfectly uniformly distributed at the present epoch,instead of being clumped together in stars and galaxies.

Whether or not the irregularities required to account for stars and galaxieswould have led to the formation of a significant number of these primordialblack holes depends on the details of the conditions in the early universe. Soif we could determine how many primordial black holes there are now, wewould learn a lot about the very early stages of the universe. Primordial blackholes with masses more than a thousand million tons-the mass of a largemountain-could be detected only by their gravitational influence on othervisible matter or on the expansion of the universe. However, as we shalllearn in the next lecture, black holes are not really black after all: They glowlike a hot body, and the smaller they are, the more they glow. So, paradoxi-cally, smaller black holes might actually turn out to be easier to detect thanlarge ones.

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

Chapter 4 - FOURTH LECTURE - BLACK HOLES AIN’T SO BLACK

Before 1970, my research on general relativity had concentrated mainly onthe question of whether there had been a big bang singularity. However,one evening in November of that year, shortly after the birth of my daughter,Lucy, I started to think about black holes as I was getting into bed. My disabil-ity made this rather a slow process, so I had plenty of time. At that date therewas no precise definition of which points in space-time lay inside a black holeand which lay outside.

I had already discussed with Roger Penrose the idea of defining a black hole asthe set of events from which it was not possible to escape to a large distance.This is now the generally accepted definition. It means that the boundary ofthe black hole, the event horizon, is formed by rays of light that just fail to getaway from the black hole. Instead, they stay forever, hovering on the edge ofthe black hole. It is like running away from the police and managing to keepone step ahead but not being able to get clear away.

Suddenly I realized that the paths of these light rays could not be approachingone another, because if they were, they must eventually run into each other. Itwould be like someone else running away from the police in the opposite direc-tion. You would both be caught or, in this case, fall into a black hole. But ifthese light rays were swallowed up by the black hole, then they could not havebeen on the boundary of the black hole. So light rays in the event horizon hadto be moving parallel to, or away from, each other.

Another way of seeing this is that the event horizon, the boundary of the blackhole, is like the edge of a shadow. It is the edge of the light of escape to a greatdistance, but, equally, it is the edge of the shadow of impending doom. And ifyou look at the shadow cast by a source at a great distance, such as the sun, youwill see that the rays of light on the edge are not approaching each other. Ifthe rays of light that form the event horizon, the boundary of the black hole,can never approach each other, the area of the event horizon could stay thesame or increase with time. It could never decrease, because that would meanthat at least some of the rays of light in the boundary would have to beapproaching each other. In fact, the area would increase whenever matter orradiation fell into the black hole.

Also, suppose two black holes collided and merged together to form a singleblack hole. Then the area of the event horizon of the final black hole wouldbe greater than the sum of the areas of the event horizons of the original blackholes. This nondecreasing property of the event horizon’s area placed animportant restriction on the possible behavior of black holes. I was so excitedwith my discovery that I did not get much sleep that night.The next day I rang up Roger Penrose. He agreed with me. I think, in fact, thathe had been aware of this property of the area. However, he had been using aslightly different definition of a black hole. He had not realized that theboundaries of the black hole according to the two definitions would be thesame, provided the black hole had settled down to a stationary state.

THE SECOND LAW OFTHERMODYNAMICS

The nondecreasing behavior of a black hole’s area was very reminiscent of thebehavior of a physical quantity called entropy, which measures the degree ofdisorder of a system. It is a matter of common experience that disorder willtend to increase if things are left to themselves; one has only to leave a housewithout repairs to see that. One can create order out of disorder-for example,one can paint the house. However, that requires expenditure of energy, and sodecreases the amount of ordered energy available.

A precise statement of this idea is known as the second law of thermodynam-ics. It states that the entropy of an isolated system never decreases with time.Moreover, when two systems are joined together, the entropy of the combinedsystem is greater than the sum of the entropies of the individual systems. Forexample, consider a system of gas molecules in a box. The molecules can bethought of as little billiard balls continually colliding with each other andbouncing off the walls of the box. Suppose that initially the molecules are allconfined to the left-hand side of the box by a partition. If the partition is thenremoved, the molecules will tend to spread out and occupy both halves of thebox. At some later time they could, by chance, all be in the right half or all beback in the left half. However, it is overwhelmingly more probable that therewill be roughly equal numbers in the two halves. Such a state is less ordered,or more disordered, than the original state in which all the molecules were inone half. One therefore says that the entropy of the gas has gone up.

Similarly, suppose one starts with two boxes, one containing oxygen moleculesand the other containing nitrogen molecules. If one joins the boxes togetherand removes the intervening wall, the oxygen and the nitrogen molecules willstart to mix. At a later time, the most probable state would be to have athoroughly uniform mixture of oxygen and nitrogen molecules throughout thetwo 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 ofother laws of science. Other laws, such as Newton’s law of gravity, forexample, are absolute law-that is, they always hold. On the other hand, thesecond law is a statistical law-that is, it does not hold always, just in the vastmajority of cases. The probability of all the gas molecules in our box beingfound in one half of the box at a later time is many millions of millions to one,but it could happen.

However, if one has a black hole around, there seems to be a rather easier wayof violating the second law: Just throw some matter with a lot of entropy, suchas a box of gas, down the black hole. The total entropy of matter outside theblack 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 sincethere is no way to look inside the black hole, we cannot see how much entropythe matter inside it has. It would be nice, therefore, if there was some featureof the black hole by which observers outside the black hole could tell itsentropy; this should increase whenever matter carrying entropy fell into theblack hole.

Following my discovery that the area of the event horizon increased whenevermatter fell into a black hole, a research student at Princeton named JacobBekenstein suggested that the area of the event horizon was a measure of theentropy of the black hole. As matter carrying entropy fell into the black hole,the area of the event horizon would go up, so that the sum of the entropy ofmatter outside black holes and the area of the horizons would never go down.This suggestion seemed to prevent the second law of thermodynamics frombeing violated in most situations. However, there was one fatal flaw: If a blackhole has entropy, then it ought also to have a temperature. But a body with anonzero temperature must emit radiation at a certain rate. It is a matter ofcommon experience that if one heats up a poker in the fire, it glows red hotand emits radiation. However, bodies at lower temperatures emit radiation,too; one just does not normally notice it because the amount is fairly small.This radiation is required in order to prevent violations of the second law. Soblack holes ought to emit radiation, but by their very definition, black holesare objects that are not supposed to emit anything. It therefore seemed that thearea of the event horizon of a black hole could not be regarded as its entropy.In fact, in 1972 I wrote a paper on this subject with Brandon Carter and anAmerican colleague, Jim Bardeen. We pointed out that, although there weremany similarities between entropy and the area of the event horizon, there wasthis apparently fatal difficulty. I must admit that in writing this paper I wasmotivated partly by irritation with Bekenstein, because I felt he had misusedmy discovery of the increase of the area of the event horizon. However, itturned out in the end that he was basically correct, though in a manner he hadcertainly not expected.

BLACK HOLE RADIATION

In September 1973, while I was visiting Moscow, I discussed black holes withtwo leading Soviet experts, Yakov Zeldovich and Alexander Starobinsky. Theyconvinced me that, according to the quantum mechanical uncertainty princi-ple, rotating black holes should create and emit particles. I believed their argu-ments on physical grounds, but I did not like the mathematical way in whichthey calculated the emission. I therefore set about devising a better mathemat-ical treatment, which I described at an informal seminar in Oxford at the endof November 1973. At that time I had not done the calculations to find outhow much would actually be emitted. I was expecting to discover just the radi-ation that Zeldovich and Starobinsky had predicted from rotating black holes.However, when I did the calculation, I found, to my surprise and annoyance,that even nonrotating black holes should apparently create and emit particlesat a steady rate.

At first I thought that this emission indicated that one of the approximationsI had used was not valid. I was afraid if Bekenstein found out about it, he woulduse it as a further argument to support his ideas about the entropy of blackholes, which I still did not like. However, the more I thought about it, themore it seemed that the approximations really ought to hold. But what finallyconvinced me that the emission was real was that the spectrum of the emittedparticles was exactly that which would be emitted by a hot body.

The black hole was emitting particles at exactly the correct rate to preventviolations of the second law.

Since then, the calculations have been repeated in a number of different formsby other people. They all confirm that a black hole ought to emit particles andradiation as if it were a hot body with a temperature that depends only on theblack hole’s mass: the higher the mass, the lower the temperature. One canunderstand this emission in the following way: What we think of as emptyspace cannot be completely empty because that would mean that all the fields,such as the gravitational field and the electromagnetic field, would have to beexactly zero. However, the value of a field and its rate of change with time arelike the position and velocity of a particle. The uncertainty principle impliesthat the more accurately one knows one of these quantities, the less accuratelyone can know the other.

So in empty space the field cannot be fixed at exactly zero, because then itwould have both a precise value, zero, and a precise rate of change, also zero.Instead, there must be a certain minimum amount of uncertainty, or quantumfluctuations, in the value of a field. One can think of these fluctuations as pairsof particles of light or gravity that appear together at some time, move apart,and then come together again and annihilate each other. These particles arecalled virtual particles. Unlike real particles, they cannot be observed directly

with a particle detector. However, their indirect effects, such as small changesin the energy of electron orbits and atoms, can be measured and agree with thetheoretical predictions to a remarkable degree of accuracy.

By conservation of energy, one of the partners in a virtual particle pair willhave positive energy and the other partner will have negative energy. The onewith negative energy is condemned to be a short-lived virtual particle. This isbecause real particles always have positive energy in normal situations. It musttherefore seek out its partner and annihilate it. However, the gravitationalfield inside a black hole is so strong that even a real particle can have negativeenergy there.

It is therefore possible, if a black hole is present, for the virtual particle withnegative energy to fall into the black hole and become a real particle. In thiscase it no longer has to annihilate its partner; its forsaken partner may fall intothe black hole as well. But because it has positive energy, it is also possible forit to escape to infinity as a real particle. To an observer at a distance, it willappear to have been emitted from the black hole. The smaller the black hole,the less far the particle with negative energy will have to go before it becomesa real particle. Thus, the rate of emission will be greater, and the apparent tem-perature of the black hole will be higher.

The positive energy of the outgoing radiation would be balanced by a flow ofnegative energy particles into the black hole. By Einstein’s famous equationE = mc2, energy is equivalent to mass. A flow of negative energy into the blackhole therefore reduces its mass. As the black hole loses mass, the area of itsevent horizon gets smaller, but this decrease in the entropy of the black holeis more than compensated for by the entropy of the emitted radiation, so thesecond law is never violated.

BLACK HOLE EXPLOSIONS

The lower the mass of the black hole, the higher its temperature is. So as theblack hole loses mass, its temperature and rate of emission increase. It there-fore loses mass more quickly. What happens when the mass of the black holeeventually becomes extremely small is not quite clear. The most reasonableguess is that it would disappear completely in a tremendous final burst of emis-sion, equivalent to the explosion of millions of H-bombs.

A black hole with a mass a few times that of the sun would have a tempera-ture of only one ten-millionth of a degree above absolute zero. This is muchless than the temperature of the microwave radiation that fills the universe,about 2.7 degrees above absolute zero-so such black holes would give off lessthan they absorb, though even that would be very little. If the universe is des-

tined to go on expanding forever, the temperature of the microwave radiationwill eventually decrease to less than that of such a black hole. The hole willthen absorb less than it emits and will begin to lose mass. But, even then, itstemperature is so low that it would take about 1066years to evaporatecompletely. This is much longer than the age of the universe, which is onlyabout 1010 years.