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, sinc