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.
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