The Theory of Everything: The Origin and Fate of the Universe - Stephen Hawking (2002)

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.

On the other hand, as we learned in the last lecture, there might be primor-dial black holes with a very much smaller mass that were made by the collapseof irregularities in the very early stages of the universe. Such black holes wouldhave a much higher temperature and would be emitting radiation at a muchgreater rate. A primordial black hole with an initial mass of a thousand mil-lion tons would have a lifetime roughly equal to the age of the universe.Primordial black holes with initial masses less than this figure would alreadyhave completely evaporated. However, those with slightly greater masseswould still be emitting radiation in the form of X rays and gamma rays. Theseare like waves of light, but with a much shorter wavelength. Such holeshardly deserve the epithet black. They really are white hot, and are emittingenergy at the rate of about ten thousand megawatts.

One such black hole could run ten large power stations, if only we could har-ness its output. This would be rather difficult, however. The black hole wouldhave the mass of a mountain compressed into the size of the nucleus of anatom. If you had one of these black holes on the surface of the Earth, therewould be no way to stop it falling through the floor to the center of the Earth.It would oscillate through the Earth and back, until eventually it settled downat the center. So the only place to put such a black hole, in which one mightuse the energy that it emitted, would be in orbit around the Earth. And theonly way that one could get it to orbit the Earth would be to attract it thereby towing a large mass in front of it, rather like a carrot in front of a donkey.This does not sound like a very practical proposition, at least not in theimmediate future.

THE SEARCH FOR PRIMORDIALBLACK HOLES

But even if we cannot harness the emission from these primordial black holes,what are our chances of observing them? We could look for the gamma raysthat the primordial black holes emit during most of their lifetime. Althoughthe radiation from most would be very weak because they are far away, thetotal from all of them might be detectable. We do, indeed, observe such abackground of gamma rays. However, this background was probably generatedby processes other than primordial black holes. One can say that the observa-tions of the gamma ray background do not provide any positive evidence forprimordial black holes. But they tell us that, on average, there cannot be morethan three hundred little black holes in every cubic light-year in the universe.This limit means that primordial black holes could make up at most one mil-lionth of the average mass density in the universe.

With primordial black holes being so scarce, it might seem unlikely that therewould be one that was near enough for us to observe on its own. But sincegravity would draw primordial black holes toward any matter, they should bemuch more common in galaxies. If they were, say, a million times more com-mon in galaxies, then the nearest black hole to us would probably be at adistance of about a thousand million kilometers, or about as far as Pluto, thefarthest known planet. At this distance it would still be very difficult to detectthe 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 severalgamma ray quanta coming from the same direction within a reasonable spaceof time, such as a week.

Otherwise, they might simply be part of the background. But Planck’s quan-tum principle tells us that each gamma ray quantum has a very high energy,because gamma rays have a very high frequency. So to radiate even ten thou-sand megawatts would not take many quanta. And to observe these few quan-ta coming from the distance of Pluto would require a larger gamma ray detec-tor than any that have been constructed so far. Moreover, the detector wouldhave 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 its life andblow up, it would be easy to detect the final burst of emission. But if the blackhole has been emitting for the last ten or twenty thousand million years, thechances of it reaching the end of its life within the next few years are reallyrather small. It might equally well be a few million years in the past or future.So in order to have a reasonable chance of seeing an explosion before yourresearch grant ran out, you would have to find a way to detect any explosionswithin a distance of about one light-year. You would still have the problem ofneeding a large gamma ray detector to observe several gamma ray quanta fromthe explosion. However, in this case, it would not be necessary to determinethat all the quanta came from the same direction. It would be enough toobserve that they all arrived within a very short time interval to be reasonablyconfident that they were coming from the same burst.

One gamma ray detector that might be capable of spotting primordial blackholes is the entire Earth’s atmosphere. (We are, in any case, unlikely to be ableto build a larger detector.) When a high-energy gamma ray quantum hits theatoms in our atmosphere, it creates pairs of electrons and positrons. Whenthese hit other atoms, they in turn create more pairs of electrons and positrons.So one gets what is called an electron shower. The result is a form of lightcalled Cerenkov radiation. One can therefore detect gamma ray bursts bylooking for flashes of light in the night sky.

Of course, there are a number of other phenomena, such as lightning, whichcan also give flashes in the sky. However, one could distinguish gamma raybursts from such effects by observing flashes simultaneously at two or morethoroughly widely separated locations. A search like this has been carried outby two scientists from Dublin, Neil Porter and Trevor Weekes, using telescopesin Arizona. They found a number of flashes but none that could be definitelyascribed to gamma ray bursts from primordial black holes.

Even if the search for primordial black holes proves negative, as it seems itmay, it will still give us important information about the very early stages ofthe universe. If the early universe had been chaotic or irregular, or if the pres-sure of matter had been low, one would have expected it to produce manymore primordial black holes than the limit set by our observations of thegamma ray background. It is only if the early universe was very smooth anduniform, and with a high pressure, that one can explain the absence ofobservable numbers of primordial black holes.

GENERAL RELATIVITY ANDQUANTUM MECHANICS

Radiation from black holes was the first example of a prediction that depend-ed on both of the great theories of this century, general relativity and quantummechanics. It aroused a lot of opposition initially because it upset the existingviewpoint: “How can a black hole emit anything?” When I first announced theresults of my calculations at a conference at the Rutherford Laboratory nearOxford, I was greeted with general incredulity. At the end of my talk the chair-man of the session, John G. Taylor from Kings College, London, claimed it wasall nonsense. He even wrote a paper to that effect.

However, in the end most people, including John Taylor, have come to theconclusion that black holes must radiate like hot bodies if our other ideasabout general relativity and quantum mechanics are correct. Thus eventhough we have not yet managed to find a primordial black hole, there isfairly general agreement that if we did, it would have to be emitting a lot ofgamma and X rays. If we do find one, I will get the Nobel Prize.The existence of radiation from black holes seems to imply that gravitationalcollapse is not as final and irreversible as we once thought. If an astronaut fallsthat extra mass will be returned to the universe in the form of radiation. Thus,in a sense, the astronaut will be recycled. It would be a poor sort of immortal-ity, however, because any personal concept of time for the astronaut wouldalmost certainly come to an end as he was crushed out of existence inside theblack hole. Even the types of particle that were eventually emitted by theblack hole would in general be different from those that made up the astro-naut. The only feature of the astronaut that would survive would be his massor energy.

The approximations I used to derive the emission from black holes shouldwork 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 itsmass gets very small. The most likely outcome seems to be that the black holewould just disappear, at least from our region of the universe. It would takewith it the astronaut and any singularity there might be inside the black hole.This was the first indication that quantum mechanics might remove the sin-gularities that were predicted by classical general relativity. However, themethods that I and other people were using in 1974 to study the quantumeffects of gravity were not able to answer questions such as whether singulari-ties would occur in quantum gravity.

From 1975 onward, I therefore started to develop a more powerful approach toquantum gravity based on Feynman’s idea of a sum over histories. The answersthat this approach suggests for the origin and fate of the universe will bedescribed in the next two lectures. We shall see that quantum mechanicsallows the universe to have a beginning that is not a singularity. This meansthat the laws of physics need not break down at the origin of the universe. Thestate of the universe and its contents, like ourselves, are completely deter-mined by the laws of physics, up to the limit set by the uncertainty principle.So much for free will.