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THIRD LECTURE - BLACK HOLES

The term black hole is of very recent origin. It was coined in 1969 by the American scientist John Wheeler as a graphic description of an idea that goes back at least two hundred years. At that time there were two theories about light. One was that it was composed of particles; the other was that it was made of waves. We now know that really both theories are correct. By the wave/particle duality of quantum mechanics, light can be regarded as both a wave and a particle. Under the theory that light was made up of waves, it was not clear how it would respond to gravity. But if light were composed of particles, one might expect them to be affected by gravity in the same way that cannonballs, rockets, and planets are.

On this assumption, a Cambridge don, John Michell, wrote a paper in 1783 in the Philosophical Transactions of the Royal Society of London. In it, he pointed out that a star that was sufficiently massive and compact would have such a strong gravitational field that light could not escape. Any light emitted from the surface of the star would be dragged back by the star’s gravitational attraction before it could get very far. Michell suggested that there might be a large number of stars like this. Although we would not be able to see them because the light from them would not reach us, we would still feel their gravitational attraction. Such objects are what we now call black holes, because that is what they are—black voids in space.

A similar suggestion was made a few years later by the French scientist the Marquis de Laplace, apparently independently of Michell. Interestingly enough, he included it in only the first and second editions of his book, The System of the World, and left it out of later editions; perhaps he decided that it was a crazy idea. In fact, it is not really consistent to treat light like cannonballs in Newton’s theory of gravity because the speed of light is fixed. A cannonball fired upward from the Earth will be slowed down by gravity and will eventually stop and fall back. A photon, however, must continue upward at a constant speed. How, then, can Newtonian gravity affect light? A consistent theory of how gravity affects light did not come until Einstein proposed general relativity in 1915; and even then it was a long time before the implications of the theory for massive stars were worked out.

To understand how a black hole might be formed, we first need an understanding of the life cycle of a star. A star is formed when a large amount of gas, mostly hydrogen, starts to collapse in on itself due to its gravitational attraction. As it contracts, the atoms of the gas collide with each other more and more frequently and at greater and greater speeds—the gas heats up. Eventually the gas will be so hot that when the hydrogen atoms collide they no longer bounce off each other but instead merge with each other to form helium atoms. The heat released in this reaction, which is like a controlled hydrogen bomb, is what makes the stars shine. This additional heat also increases the pressure of the gas until it is sufficient to balance the gravitational attraction, and the gas stops contracting. It is a bit like a balloon where there is a balance between the pressure of the air inside, which is trying to make the balloon expand, and the tension 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 the nuclear reactions balancing the gravitational attraction. Eventually, however, the star will run out of its hydrogen and other nuclear fuels. And paradoxically, the more fuel a star starts off with, the sooner it runs out. This is because the more massive the star is, the hotter it needs to be to balance its gravitational attraction. And the hotter it is, the faster it will use up its fuel. Our sun has probably got enough fuel for another five thousand million years or so, but more massive stars can use up their fuel in as little as one hundred million years, 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 was only first understood at the end of the 1920s.

In 1928 an Indian graduate student named Subrahmanyan Chandrasekhar set sail for England to study at Cambridge with the British astronomer Sir Arthur Eddington. Eddington was an expert on general relativity. There is a story that a journalist told Eddington in the early 1920s that he had heard there were only three people in the world who understood general relativity. Eddington replied, “I am trying to think who the third person is.”

During his voyage from India, Chandrasekhar worked out how big a star could be and still separate itself against its own gravity after it had used up all its fuel. The idea was this: When the star becomes small, the matter particles get very near each other. But the Pauli exclusion principle says that two matter particles cannot have both the same position and the same velocity. The matter particles must therefore have very different velocities. This makes them move away from each other, and so tends to make the star expand. A star can therefore maintain itself at a constant radius by a balance between the attraction of gravity and the repulsion that arises from the exclusion principle, just as earlier in its life the gravity was balanced by the heat.

Chandrasekhar realized, however, that there is a limit to the repulsion that the exclusion principle can provide. The theory of relativity limits the maximum difference in the velocities of the matter particles in the star to the speed of light. This meant that when the star got sufficiently dense, the repulsion caused 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 half times the mass of the sun would not be able to support itself against its own gravity. This mass is now known as the Chandrasekhar limit.

This had serious implications for the ultimate fate of massive stars. If a star’s mass is less than the Chandrasekhar limit, it can eventually stop contracting and settle down to a possible final state as a white dwarf with a radius of a few thousand miles and a density of hundreds of tons per cubic inch. A white dwarf is supported by the exclusion principle repulsion between the electrons in its matter. We observe a large number of these white dwarf stars. One of the first to be discovered is the star that is orbiting around Sirius, the brightest star in the night sky.

It was also realized that there was another possible final state for a star also with a limiting mass of about one or two times the mass of the sun, but much smaller than even the white dwarf. These stars would be supported by the exclusion principle repulsion between the neutrons and protons, rather than between the electrons. They were therefore called neutron stars. They would have had a radius of only ten miles or so and a density of hundreds of millions of tons per cubic inch. At the time they were first predicted, there was no way that neutron stars could have been observed, and they were not detected until much later.

Stars with masses above the Chandrasekhar limit, on the other hand, have a big problem when they come to the end of their fuel. In some cases they may explode or manage to throw off enough matter to reduce their mass below the limit, but it was difficult to believe that this always happened, no matter how big the star. How would it know that it had to lose weight? And even if every star managed to lose enough mass, what would happen if you added more mass to a white dwarf or neutron star to take it over the limit? Would it collapse to infinite density?

Eddington was shocked by the implications of this and refused to believe Chandrasekhar’s result. He thought it was simply not possible that a star could collapse to a point. This was the view of most scientists. Einstein himself wrote a paper in which he claimed that stars would not shrink to zero size.The hostility of other scientists, particularly of Eddington, his former teacher and the leading authority on the structure of stars, persuaded Chandrasekhar to abandon 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 in part, for his early work on the limiting mass of cold stars.

Chandrasekhar had shown that the exclusion principle could not halt the collapse of a star more massive than the Chandrasekhar limit. But the problem of understanding what would happen to such a star, according to general relativity, was not solved until 1939 by a young American, Robert Oppenheimer. His result, however, suggested that there would be no observational consequences that could be detected by the telescopes of the day. Then the war intervened and Oppenheimer himself became closely involved in the atom bomb project. And after the war the problem of gravitational collapse was largely forgotten as most scientists were then interested in what happens on the scale of the atom and its nucleus. In the 1960s, however, interest in the large-scale problems of astronomy and cosmology was revived by a great increase in the number and range of astronomical observations brought about by the application of modern technology. Oppenheimer’s work was then rediscovered and extended by a number of people.

The picture that we now have from Oppenheimer’s work is as follows: The gravitational field of the star changes the paths of light rays in space–time from what 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 emitted from their tips, are bent slightly inward near the surface of the star. This can be seen in the bending of light from distant stars that is observed during an eclipse of the sun. As the star contracts, the gravitational field at its surface gets stronger and the light cones get bent inward more. This makes it more difficult for light from the star to escape, and the light appears dimmer and redder to an observer at a distance.

Eventually, when the star has shrunk to a certain critical radius, the gravitational field at the surface becomes so strong that the light cones are bent inward so much that the light can no longer escape. According to the theory of relativity, nothing can travel faster than light. Thus, if light cannot escape, neither can anything else. Everything is dragged back by the gravitational field. So one has a set of events, a region of space–time, from which it is not possible to escape to reach a distant observer. This region is what we now call a black hole. Its boundary is called the event horizon. It coincides with the paths 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 collapse to form a black hole, one has to remember that in the theory of relativity there is no absolute time. Each observer has his own measure of time. The time for someone on a star will be different from that for someone at a distance, because of the gravitational field of the star. This effect has been measured in an experiment on Earth with clocks at the top and bottom of a water tower. Suppose an intrepid astronaut on the surface of the collapsing star sent a signal every second, according to his watch, to his spaceship orbiting about the star. At some time on his watch, say eleven o’clock, the star would shrink below the critical radius at which the gravitational field became so strong that the signals would no longer reach the spaceship.

His companions watching from the spaceship would find the intervals between successive signals from the astronaut getting longer and longer as eleven o’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 the astronaut’s 10:59:58 signal and the one that he sent when his watch read 10:59:59, 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 and eleven o’clock, by the astronaut’s watch, would be spread out over an infinite period of time, as seen from the spaceship.

The time interval between the arrival of successive waves at the spaceship would get longer and longer, and so the light from the star would appear redder and redder and fainter and fainter. Eventually the star would be so dim that it could no longer be seen from the spaceship. All that would be left would be a black hole in space. The star would, however, continue to exert the same gravitational force on the spaceship. This is because the star is still visible to the spaceship, at least in principle. It is just that the light from the surface is so 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 density within 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. At the singularity, the laws of science and our ability to predict the future would break down. However, any observer who remained outside the black hole would not be affected by this failure of predictability, because neither light nor any other signal can reach them from the singularity.

This remarkable fact led Roger Penrose to propose the cosmic censorship hypothesis, which might be paraphrased as “God abhors a naked singularity.” In other words, the singularities produced by gravitational collapse occur only in places like black holes, where they are decently hidden from outside view by an event horizon. Strictly, this is what is known as the weak cosmic censorship hypothesis: protect obervers who remain outside the black hole from the consequences of the breakdown of predictability that occurs at the singularity. But it does nothing at all for the poor unfortunate astronaut who falls into the hole. Shouldn’t God protect his modesty as well?

There are some solutions of the equations of general relativity in which it is possible for our astronaut to see a naked singularity. He may be able to avoid hitting the singularity and instead fall through a “worm hole” and come out in another region of the universe. This would offer great possibilities for travel in space and time, but unfortunately it seems that the solutions may all be highly unstable. The least disturbance, such as the presence of an astronaut, may change them so that the astronaut cannot see the singularity until he hits it and his time comes to an end. In other words, the singularity always lies in his future and never in his past.

The strong version of the cosmic censorship hypothesis states that in a realistic solution, the singularities always lie either entirely in the future, like the singularities 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 no one’s life would ever be safe. Someone might go into the past and kill your father or mother before you were conceived.

In a gravitational collapse to form a black hole, the movements would be dammed by the emission of gravitational waves. One would therefore expect that it would not be too long before the black hole would settle down to a stationary state. It was generally supposed that this final stationary state would depend 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 be fixed, but instead be pulsating.

However, in 1967, the study of black holes was revolutionized by a paper written in Dublin by Werner Israel. Israel showed that any black hole that is not rotating must be perfectly round or spherical. Its size, moreover, would depend only on its mass. It could, in fact, be described by a particular solution of Einstein’s equations that had been known since 1917, when it had been found by Karl Schwarzschild shortly after the discovery of general relativity. At first, Israel’s result was interpreted by many people, including Israel himself, as evidence that black holes would form only from the collapse of bodies that were perfectly round or spherical. As no real body would be perfectly spherical, this meant that, in general, gravitational collapse would lead to naked singularities. There was, however, a different interpretation of Israel’s result, which was advocated by Roger Penrose and John Wheeler in particular. This was that a black hole should behave like a ball of fluid. Although a body might start off in an unspherical state, as it collapsed to form a black hole it would settle down to a spherical state due to the emission of gravitational waves. Further calculations supported this view and it came to be adopted generally.

Israel’s result had dealt only with the case of black holes formed from nonrotating bodies. On the analogy with a ball of fluid, one would expect that a black hole made by the collapse of a rotating body would not be perfectly round. It would have a bulge round the equator caused by the effect of the rotation. We observe a small bulge like this in the sun, caused by its rotation once every twenty-five days or so. In 1963, Roy Kerr, a New Zealander, had found a set of black–hole solutions of the equations of general relativity more general than the Schwarzschild solutions. These “Kerr” black holes rotate at a constant rate, their size and shape depending only on their mass and rate of rotation. If the rotation was zero, the black hole was perfectly round and the solution was identical to the Schwarzschild solution. But if the rotation was nonzero, the black hole bulged outward near its equator. It was therefore natural to conjecture that a rotating body collapsing to form a black hole would end up in a state described by the Kerr solution.

In 1970, a colleague and fellow research student of mine, Brandon Carter, took the first step toward proving this conjecture. He showed that, provided a stationary rotating black hole had an axis of symmetry, like a spinning top, its size and 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 an axis 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 in which it could be rotating, but not pulsating. Moreover, its size and shape would depend only on its mass and rate of rotation, and not on the nature of the body that had collapsed to form it. This result became known by the maxim “A black hole has no hair.” It means that a very large amount of information about the body that has collapsed must be lost when a black hole is formed, because afterward all we can possibly measure about the body is its mass and rate of rotation. The significance of this will be seen in the next lecture. The no-hair theorem is also of great practical importance because it so greatly restricts the possible types of black holes. One can therefore make detailed models of objects that might contain black holes, and compare the predictions of the models with observations.

Black holes are one of only a fairly small number of cases in the history of science where a theory was developed in great detail as a mathematical model before 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 one believe in objects for which the only evidence was calculations based on the dubious theory of general relativity?

In 1963, however, Maarten Schmidt, an astronomer at the Mount Palomar Observatory in California, found a faint, starlike object in the direction of the source of radio waves called 3C273—that is, source number 273 in the third Cambridge catalog of radio sources. When he measured the red shift of the object, he found it was too large to be caused by a gravitational field: If it had been a gravitational red shift, the object would have to be so massive and so near to us that it would disturb the orbits of planets in the solar system. This suggested that the red shift was instead caused by the expansion of the universe, which in turn meant that the object was a very long way away. And to be visible at such a great distance, the object must be very bright and must be emitting a huge amount of energy.

The only mechanism people could think of that would produce such large quantities of energy seemed to be the gravitational collapse not just of a star but of the whole central region of a galaxy. A number of other similar “quasistellar 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 conclusive evidence of black holes.

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

In the end, however, they and everyone else came to the less romantic conclusion that these objects, which were given the name pulsars, were in fact just rotating neutron stars. They were emitting pulses of radio waves because of a complicated indirection between their magnetic fields and surrounding matter. This was bad news for writers of space westerns, but very hopeful for the small number of us who believed in black holes at that time. It was the first positive evidence that neutron stars existed. A neutron star has a radius of about ten miles, 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 expect that 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 not emit 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 pioneering paper in 1783, a black hole still exerts a gravitational force on nearby objects. Astronomers have observed a number of systems in which two stars orbit around each other, attracted toward each other by gravity. They also observed systems in which there is only one visible star that is orbiting around some unseen companion.

One cannot, of course, immediately conclude that the companion is a black hole. It might merely be a star that is too faint to be seen. However, some of these 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 by matter that has been blown off the surface of the visible star. As it falls toward the unseen companion, it develops a spiral motion—rather like water running out of a bath—and it gets very hot, emitting X rays. For this mechanism to work, 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 lowest possible mass of the unseen object. In the case of Cygnus X-I, this is about six times the mass of the sun. According to Chandrasekhar’s result, this is too much for the unseen object to be a white dwarf. It is also too large a mass to be 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 black hole, but they are all rather far-fetched. A black hole seems to be the only really natural explanation of the observations. Despite this, I have a bet with Kip Thorne of the California Institute of Technology that in fact Cygnus X–I does not contain a black hole. This is a form of insurance policy for me. I have done a lot of work on black holes, and it would all be wasted if it turned out that black holes do not exist. But in that case, I would have the consolation of winning 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 when we made the bet in 1975, we were 80 percent certain that Cygnus was a black hole. By now I would say that we are about 95 percent certain, but the bet has yet 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 with masses much less than that of the sun. Such black holes could not be formed by gravitational collapse, because their masses are below the Chandrasekhar mass limit. Stars of this low mass can support themselves against the force of gravity even when they have exhausted their nuclear fuel. So, low-mass black holes could form only if matter were compressed to enormous densities by very large external pressures. Such conditions could occur in a very big hydrogen bomb. The physicist John Wheeler once calculated that if one took all the heavy water in all the oceans of the world, one could build a hydrogen bomb that would compress matter at the center so much that a black hole would be created. Unfortunately, however, there would be no one left to observe it.

A more practical possibility is that such low–mass black holes might have been formed in the high temperatures and pressures of the very early universe. Black holes could have been formed if the early universe had not been perfectly smooth and uniform, because then a small region that was denser than average could be compressed in this way to form a black hole. But we know that there must have been some irregularities, because otherwise the matter in the universe 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 galaxies would have led to the formation of a significant number of these primordial black holes depends on the details of the conditions in the early universe. So if we could determine how many primordial black holes there are now, we would learn a lot about the very early stages of the universe. Primordial black holes with masses more than a thousand million tons—the mass of a large mountain—could be detected only by their gravitational influence on other visible matter or on the expansion of the universe. However, as we shall learn in the next lecture, black holes are not really black after all: They glow like a hot body, and the smaller they are, the more they glow. So, paradoxically, smaller black holes might actually turn out to be easier to detect than large ones.