Before 1970, my research on general relativity hadconcentrated mainly on the question of whether or not therehad been a big bang singularity. However, one evening inNovember that year, shortly after the birth of my daughter,Lucy, I started to think about black holes as I was getting intobed. My disability makes this rather a slow process, so I hadplenty of time. At that date there was no precise definition ofwhich points in space-time lay inside a black hole and whichlay outside. I had already discussed with Roger Penrose theidea of defining a black hole as the set of events from which itwas not possible to escape to a large distance, which is nowthe generally accepted definition. It means that the boundary ofthe black hole, the event horizon, is formed by the light raysthat just fail to escape from the black hole, hovering foreverjust on the edge (Fig. 7.1). It is a bit like running away fromthe police and just managing to keep one step ahead but notbeing able to get clear away!
Suddenly I realized that the paths of these light rays couldnever approach one another. If they did they must eventuallyrun into one another. It would be like meeting someone elserunning away from the police in the opposite direction - youwould 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 blackhole. So the paths of light rays in the event horizon hadalways to be moving parallel to, or away from, each other.
Another way of seeing this is that the event horizon, theboundary of the black hole, is like the edge of a shadow - theshadow of impending doom. If you look at the shadow cast bya source at a great distance, such as the sun, you will see thatthe rays of light in the edge are not approaching each other.
If the rays of light that form the event horizon, theboundary of the black hole, can never approach each other,the area of the event horizon might stay the same or increasewith time, but it could never decrease because that would meanthat at least some of the rays of light in the boundary wouldhave to be approaching each other. In fact, the area wouldincrease whenever matter or radiation fell into the black hole(Fig. 7.2). Or if two black holes collided and merged togetherto form a single black hole, the area of the event horizon ofthe final black hole would be greater than or equal to the sumof the areas of the event horizons of the original black holes(Fig. 7.3). This nondecreasing property of the event horizon’sarea placed an important restriction on the possible behavior ofblack holes. I was so excited with my discovery that I did notget much sleep that night. The next day I rang up RogerPenrose. He agreed with me. I think, in fact, that he had beenaware of this property of the area. However, he had beenusing a slightly different definition of a black hole. He had notrealized that the boundaries of the black hole according to thetwo definitions would be the same, and hence so would theirareas, provided the black hole had settled down to a state inwhich it was not changing with time.
The nondecreasing behavior of a black hole’s area was veryreminiscent of the behavior of a physical quantity called entropy,which measures the degree of disorder of a system. It is amatter of common experience that disorder will tend to increaseif things are left to themselves. (One has only to stop makingrepairs around the house to see that!) One can create orderout of disorder (for example, one can paint the house), butthat requires expenditure of effort or energy and so decreasesthe amount of ordered energy available.
A precise statement of this idea is known as the second lawof thermodynamics. It states that the entropy of an isolatedsystem always increases, and that when two systems are joinedtogether, the entropy of the combined system is greater thanthe sum of the entropies of the individual systems. Forexample, consider a system of gas molecules in a box. Themolecules can be thought of as little billiard balls continuallycolliding with each other and bouncing off the walls of the box.
The higher the temperature of the gas, the faster the moleculesmove, and so the more frequently and harder they collide withthe walls of the box and the greater the outward pressure theyexert on the walls. Suppose that initially the molecules are allconfined to the left-hand side of the box by a partition. If thepartition is then removed, the molecules will tend to spread outand occupy both halves of the box. At some later time theycould, by chance, all be in the right half or back in the lefthalf, but it is overwhelmingly more probable that there will beroughly equal numbers in the two halves. Such a state is lessordered, or more disordered, than the original state in which allthe molecules were in one half. One therefore says that theentropy of the gas has gone up. Similarly, suppose one startswith two boxes, one containing oxygen molecules and the othercontaining nitrogen molecules. If one joins the boxes togetherand removes the intervening wall, the oxygen and the nitrogenmolecules will start to mix. At a later time the most probablestate would be a fairly uniform mixture of oxygen and nitrogenmolecules throughout the two boxes. This state would be lessordered, and hence have more entropy, than the initial state oftwo separate boxes.
The second law of thermodynamics has a rather differentstatus than that of other laws of science, such as Newton’s lawof gravity, for example, because it does not hold always, just inthe vast majority of cases. The probability of all the gasmolecules in our first boxfound in one half of the box at a later time is many millionsof millions to one, but it can happen. However, if one has ablack hole around there seems to be a rather easier way ofviolating the second law: just throw some matter with a lot ofentropy such as a box of gas, down the black hole. The totalentropy of matter outside the black hole would go down. Onecould, of course, still say that the total entropy, including theentropy inside the black hole, has not gone down - but sincethere is no way to look inside the black hole, we cannot seehow much entropy the matter inside it has. It would be nice,then, if there was some feature of the black hole by whichobservers outside the blackhole could tell its entropy, and which would increasewhenever matter carrying entropy fell into the black hole.
Following the discovery, described above, that the area of theevent horizon increased whenever matter fell into a black hole,a research student at Princeton named Jacob Bekensteinsuggested that the area of the event horizon was a measure ofthe entropy of the black hole. As matter carrying entropy fellinto a black hole, the area of its event horizon would go up,so that the sum of the entropy of matter outside black holesand the area of the horizons would never go down.
This suggestion seemed to prevent the second law ofthermodynamics 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 aparticular temperature must emit radiation at a certain rate. Itis a matter of common experience that if one heats up apoker in a fire it glows red hot and emits radiation, but bodiesat lower temperatures emit radiation too; one just does notnormally notice it because the amount is fairly small. Thisradiation is required in order to prevent violation of the secondlaw. So black holes ought to emit radiation. But by their verydefinition, black holes are objects that are not supposed to emitanything. It therefore seemed that the area of the eventhorizon of a black hole could not be regarded as its entropy.
In 1972 I wrote a paper with Brandon Carter and anAmerican colleague, Jim Bardeen, in which we pointed out thatalthough there were many similarities between entropy and thearea of the event horizon, there was this apparently fataldifficulty. I must admit that in writing this paper I wasmotivated partly by irritation with Bekenstein, who, I felt, hadmisused my discovery of the increase of the area of the eventhorizon. However, it turned out in the end that he wasbasically correct, though in a manner he had certainly notexpected.
In September 1973, while I was visiting Moscow, I discussedblack holes with two leading Soviet experts, Yakov Zeldovichand Alexander Starobinsky. They convinced me that, accordingto the quantum mechanical uncertainty principle, rotating blackholes should create and emit particles. I believed theirarguments on physical grounds, but I did not like themathematical way in which they calculated the emission. Itherefore set about devising a better mathematical treatment,which I described at an informal seminar in Oxford at the endof November 1973. At that time I had not done the calculationsto find out how much would actually be emitted. I wasexpecting to discover just the radiation that Zeldovich andStarobinsky had predicted from rotating black holes. However,when I did the calculation, I found, to my surprise andannoyance, that even non-rotating black holes should apparentlycreate and emit particles at a steady rate. At first I thoughtthat this emission indicated that one of the approximations Ihad used was not valid. I was afraid that if Bekenstein foundout about it, he would use it as a further argument to supporthis ideas about the entropy of black holes, which I still did notlike. However, the more I thought about it, the more it seemedthat the approximations really ought to hold. But what finallyconvinced me that the emission was real was that the spectrumof the emitted particles was exactly that which would be emittedby a hot body, and that the black hole was emitting particlesat exactly the correct rate to prevent violations of the secondlaw. Since then the calculations have been repeated in anumber of different forms by other people. They all confirmthat a black hole ought to emit particles and radiation as if itwere a hot body with a temperature that depends only on theblack hole’s mass: the higher the mass, the lower thetemperature.
How is it possible that a black hole appears to emit particleswhen we know that nothing can escape from within its eventhorizon? The answer, quantum theory tells us, is that theparticles do not come from within the black hole, but from the“empty” space just outside the black hole’s event horizon! Wecan understand this in the following way: what we think of as“empty” space cannot be completely empty because that wouldmean that all the fields, such as the gravitational andelectromagnetic fields, would have to be exactly zero. However,the value of a field and its rate of change with time are likethe position and velocity of a particle: the uncertainty principleimplies that the more accurately one knows one of thesequantities, the less accurately one can know the other. So inempty space the field cannot be fixed at exactly zero, becausethen it would have both a precise value (zero) and a preciserate of change (also zero). There must be a certain minimumamount of uncertainty, or quantum fluctuations, in the value ofthe field. One can think of these fluctuations as pairs ofparticles of light or gravity that appear together at some time,move apart, and then come together again and annihilate eachother. These particles are virtual particles like the particles thatcarry the gravitational force of the sun: unlike real particles,they cannot be observed directly with a particle detector.
However, their indirect effects, such as small changes in theenergy of electron orbits in atoms, can be measured and agreewith the theoretical predictions to a remarkable degree ofaccuracy. The uncertainty principle also predicts that there willbe similar virtual pairs of matter particles, such as electrons orquarks. In this case, however, one member of the pair will bea particle and the other an antiparticle (the antiparticles of lightand gravity are the same as the particles).
Because energy cannot be created out of nothing, one of thepartners in a particle/antiparticle pair will have positive energy,and the other partner negative energy. The one with negativeenergy is condemned to be a short-lived virtual particle becausereal particles always have positive energy in normal situations. Itmust therefore seek out its partner and annihilate with it.
However, a real particle close to a massive body has lessenergy than if it were far away, because it would take energyto lift it far away against the gravitational attraction of the body.
Normally, the energy of the particle is still positive, but thegravitational field inside a black hole is so strong that even areal particle can have negative energy there. It is thereforepossible, if a black hole is present, for the virtual particle withnegative energy to fall into the black hole and become a realparticle or antiparticle. In this case it no longer has toannihilate with its partner. Its forsaken partner may fall into theblack hole as well. Or, having positive energy, it might alsoescape from the vicinity of the black hole as a real particle orantiparticle (Fig. 7.4). To an observer at a distance, it willappear to have been emitted from the black hole. The smallerthe black hole, the shorter the distance the particle withnegative energy will have to go before it becomes a realparticle, and thus the greater the rate of emission, and theapparent temperature, of the black hole.
The positive energy of the outgoing radiation would bebalanced by a flow of negative energy particles into the blackhole. By Einstein’s equation E = mc2 (where E is energy, m ismass, and c is the speed of light), energy is proportional tomass. A flow of negative energy into the black hole thereforereduces its mass. As the black hole loses mass, the area of itsevent horizon gets smaller, but this decrease in the entropy ofthe black hole is more than compensated for by the entropy ofthe emitted radiation, so the second law is never violated.
Moreover, the lower the mass of the black hole, the higherits temperature. So as the black hole loses mass, itstemperature and rate of emission increase, so it loses massmore quickly. What happens when the mass of the black holeeventually becomes extremely small is not quite clear, but themost reasonable guess is that it would disappear completely ina tremendous final burst of emission, equivalent to the explosionof millions of H-bombs.
A black hole with a mass a few times that of the sun wouldhave a temperature of only one ten millionth of a degree aboveabsolute zero. This is much less than the temperature of themicrowave radiation that fills the universe (about 2.7? aboveabsolute zero), so such black holes would emit even less thanthey absorb. If the universe is destined to go on expandingforever, the temperature of the microwave radiation willeventually decrease to less than that of such a black hole,which will then begin to lose mass. But, even then, itstemperature would be so low that it would take about a millionmillion million million million million million million million millionmillion years (1 with sixty-six zeros after it) to evaporatecompletely. This is much longer than the age of the universe,which is only about ten or twenty thousand million years (1 or2 with ten zeros after it). On the other hand, as mentioned inChapter 6, there might be primordial black holes with a verymuch smaller mass that were made by the collapse ofirregularities in the very early stages of the universe. Such blackholes would have a much higher temperature and would beemitting radiation at a much greater rate. A primordial blackhole with an initial mass of a thousand million tons would havea lifetime roughly equal to the age of the universe. Primordialblack holes with initial masses less than this figure wouldalready have completely evaporated, but those with slightlygreater masses would still be emitting radiation in the form ofX rays and gamma rays. These X rays and gamma rays arelike waves of light, but with a much shorter wavelength. Suchholes hardly deserve the epithet black: they really are white hotand are emitting energy at a rate of about ten thousandmegawatts.
One such black hole could run ten large power stations, ifonly we could harness its power. This would be rather difficult,however: the black hole would have the mass of a mountaincompressed into less than a million millionth of an inch, thesize of the nucleus of an atom! If you had one of these blackholes on the surface of the earth, there would be no way tostop it from falling through the floor to the center of the earth.
It would oscillate through the earth and back, until eventually itsettled down at the center. So the only place to put such ablack hole, in which one might use the energy that it emitted,would be in orbit around the earth - and the only way thatone 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 infront of a donkey. This does not sound like a very practicalproposition, at least not in the immediate future.
But even if we cannot harness the emission from theseprimordial black holes, what are our chances of observingthem? We could look for the gamma rays that the primordialblack holes emit during most of their lifetime. Although theradiation from most would be very weak because they are faraway, the total from all of them might be detectable. We doobserve such a background of gamma rays: Fig. 7.5 showshow the observed intensity differs at different frequencies (thenumber of waves per second). However, this background couldhave been, and probably was, generated by processes otherthan primordial black holes. The dotted line in Fig. 7.5 showshow the intensity should vary with frequency for gamma raysgiven off by primordial black holes, if there were on average300 per cubic light-year. One can therefore say that theobservations of the gamma ray background do not provide anypositive evidence for primordial black holes, but they do tell usthat on average there cannot be more than 300 in every cubiclight-year in the universe. This limit means that primordial blackholes could make up at most one millionth of the matter in theuniverse.
With primordial black holes being so scarce, it might seemunlikely that there would be one near enough for us toobserve as an individual source of gamma rays. But sincegravity would draw 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 canbe no more than 300 primordial black holes per cubiclight-year on average, it tells us nothing about how commonthey might be in our own galaxy. If they were, say, a milliontimes more common than this, then the nearest black hole tous would probably be at a distance of about a thousand millionkilometers, or about as far away as Pluto, the farthest knownplanet. At this distance it would still be very difficult to detectthe steady emission of a black hole, even if it was tenthousand megawatts. In order to observe a primordial blackhole one would have to detect several gamma ray quantacoming from the same direction within a reasonable space oftime, such as a week. Otherwise, they might simply be part ofthe background. But Planck’s quantum principle tells us thateach gamma ray quantum has a very high energy, becausegamma rays have a very high frequency, so it would not takemany quanta to radiate even ten thousand megawatts. And toobserve these few coming from the distance of Pluto wouldrequire a larger gamma ray detector than any that have beenconstructed so far. Moreover, the detector would have to be inspace, because gamma rays cannot penetrate the atmosphere.
Of course, if a black hole as close as Pluto were to reachthe end of its life and blow up, it would be easy to detect thefinal burst of emission. But if the black hole has been emittingfor the last ten or twenty thousand million years, the chance ofit reaching the end of its life within the next few years, ratherthan several million years in the past or future, is really rathersmall! So in order to have a reasonable chance of seeing anexplosion before your research grant ran out, you would haveto find a way to detect any explosions within a distance ofabout one light-year. In fact bursts of gamma rays from spacehave been detected by satellites originally constructed to lookfor violations of the Test Ban Treaty. These seem to occurabout sixteen times a month and to be roughly uniformlydistributed in direction across the sky. This indicates that theycome from outside the Solar System since otherwise we wouldexpect them to be concentrated toward the plane of the orbitsof the planets. The uniform distribution also indicates that thesources are either fairly near to us in our galaxy or rightoutside it at cosmological distances because otherwise, again,they would be concentrated toward the plane of the galaxy. Inthe latter case, the energy required to account for the burstswould be far too high to have been produced by tiny blackholes, but if the sources were close in galactic terms, it mightbe possible that they were exploding black holes. I would verymuch like this to be the case but I have to recognize thatthere are other possible explanations for the gamma ray bursts,such as colliding neutron stars. New observations in the nextfew years, particularly by gravitational wave detectors like LIGO,should enable us to discover the origin of the gamma raybursts.
Even if the search for primordial black holes proves negative,as it seems it may, it will still give us important informationabout the very early stages of the universe. If the earlyuniverse had been chaotic or irregular, or if the pressure ofmatter had been low, one would have expected it to producemany more primordial black holes than the limit already set byour observations of the gamma ray background. Only if theearly universe was very smooth and uniform, with a highpressure, can one explain the absence of observable numbersof primordial black holes.
The idea of radiation from black holes was the first exampleof a prediction that depended in an essential way on both thegreat theories of this century, general relativity and quantummechanics. It aroused a lot of opposition initially because itupset the existing viewpoint: “How can a black hole emitanything?” When I first announced the results of mycalculations at a conference at the Rutherford-AppletonLaboratory near 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 endmost people, including John Taylor, have come to theconclusion that black holes must radiate like hot bodies if ourother ideas about general relativity and quantum mechanics arecorrect. Thus, even though we have not yet managed to find aprimordial black hole, there is fairly general agreement that ifwe did, it would have to be emitting a lot of gamma rays andX rays.
The existence of radiation from black holes seems to implythat gravitational collapse is not as final and irreversible as weonce thought. If an astronaut falls into a black hole, its masswill increase, but eventually the energy equivalent of that extramass will be returned to the universe in the form of radiation.
Thus, in a sense, the astronaut will be “recycled.” It would bea poor sort of immortality, however, because any personalconcept of time for the astronaut would almost certainly cometo an end as he was torn apart inside the black hole! Eventhe types of particles that were eventually emitted by the blackhole would in general be different from those that made up theastronaut: the only feature of the astronaut that would survivewould be his mass or energy.
The approximations I used to derive the emission from blackholes should work well when the black hole has a mass greaterthan a fraction of a gram. However, they will break down atthe end of the black hole’s life when its mass gets very small.
The most likely outcome seems to be that the black hole willjust disappear, at least from our region of the universe, takingwith it the astronaut and any singularity there might be insideit, if indeed there is one. This was the first indication thatquantum mechanics might remove the singularities that werepredicted by general relativity. However, the methods that I andother people were using in 1974 were not able to answerquestions such as whether singularities would occur in quantumgravity. From 1975 onward I therefore started to develop amore powerful approach to quantum gravity based on RichardFeynrnan’s idea of a sum over histories. The answers that thisapproach suggests for the origin and fate of the universe andits contents, such as astronauts, will be de-scribed in the nexttwo chapters. We shall see that although the uncertaintyprinciple places limitations on the accuracy of all ourpredictions, it may at the same time remove the fundamentalunpredictability that occurs at a space-time singularity.