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FOURTH LECTURE - BLACK HOLES AIN’T SO BLACK

Before 1970, my research on general relativity had concentrated mainly on the question of whether there had been a big bang singularity. However, one evening in November of that year, shortly after the birth of my daughter, Lucy, I started to think about black holes as I was getting into bed. My disability made this rather a slow process, so I had plenty of time. At that date there was no precise definition of which points in space-time lay inside a black hole and which lay outside.

I had already discussed with Roger Penrose the idea of defining a black hole as the set of events from which it was not possible to escape to a large distance. This is now the generally accepted definition. It means that the boundary of the black hole, the event horizon, is formed by rays of light that just fail to get away from the black hole. Instead, they stay forever, hovering on the edge of the black hole. It is like running away from the police and managing to keep one step ahead but not being able to get clear away.

Suddenly I realized that the paths of these light rays could not be approaching one another, because if they were, they must eventually run into each other. It would be like someone else running away from the police in the opposite direction. You would 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 black hole. So light rays in the event horizon had to be moving parallel to, or away from, each other.

Another way of seeing this is that the event horizon, the boundary of the black hole, is like the edge of a shadow. It is the edge of the light of escape to a great distance, but, equally, it is the edge of the shadow of impending doom. And if you look at the shadow cast by a source at a great distance, such as the sun, you will see that the rays of light on the edge are not approaching each other. If the rays of light that form the event horizon, the boundary of the black hole, can never approach each other, the area of the event horizon could stay the same or increase with time. It could never decrease, because that would mean that at least some of the rays of light in the boundary would have to be approaching each other. In fact, the area would increase whenever matter or radiation fell into the black hole.

Also, suppose two black holes collided and merged together to form a single black hole. Then the area of the event horizon of the final black hole would be greater than the sum of the areas of the event horizons of the original black holes. This nondecreasing property of the event horizon’s area placed an important restriction on the possible behavior of black holes. I was so excited with my discovery that I did not get much sleep that night.

The next day I rang up Roger Penrose. He agreed with me. I think, in fact, that he had been aware of this property of the area. However, he had been using a slightly different definition of a black hole. He had not realized that the boundaries of the black hole according to the two definitions would be the same, provided the black hole had settled down to a stationary state.

THE SECOND LAW OF

THERMODYNAMICS

The nondecreasing behavior of a black hole’s area was very reminiscent of the behavior of a physical quantity called entropy, which measures the degree of disorder of a system. It is a matter of common experience that disorder will tend to increase if things are left to themselves; one has only to leave a house without repairs to see that. One can create order out of disorder—for example, one can paint the house. However, that requires expenditure of energy, and so decreases the amount of ordered energy available.

A precise statement of this idea is known as the second law of thermodynamics. It states that the entropy of an isolated system never decreases with time. Moreover, when two systems are joined together, the entropy of the combined system is greater than the sum of the entropies of the individual systems. For example, consider a system of gas molecules in a box. The molecules can be thought of as little billiard balls continually colliding with each other and bouncing off the walls of the box. Suppose that initially the molecules are all confined to the left-hand side of the box by a partition. If the partition is then removed, the molecules will tend to spread out and occupy both halves of the box. At some later time they could, by chance, all be in the right half or all be back in the left half. However, it is overwhelmingly more probable that there will be roughly equal numbers in the two halves. Such a state is less ordered, or more disordered, than the original state in which all the molecules were in one half. One therefore says that the entropy of the gas has gone up.

Similarly, suppose one starts with two boxes, one containing oxygen molecules and the other containing nitrogen molecules. If one joins the boxes together and removes the intervening wall, the oxygen and the nitrogen molecules will start to mix. At a later time, the most probable state would be to have a thoroughly uniform mixture of oxygen and nitrogen molecules throughout the two boxes. This state would be less ordered, and hence have more entropy, than the initial state of two separate boxes.

The second law of thermodynamics has a rather different status than that of other laws of science. Other laws, such as Newton’s law of gravity, for example, are absolute law—that is, they always hold. On the other hand, the second law is a statistical law—that is, it does not hold always, just in the vast majority of cases. The probability of all the gas molecules in our box being found in one half of the box at a later time is many millions of millions to one, but it could happen.

However, if one has a black hole around, there seems to be a rather easier way of violating the second law: Just throw some matter with a lot of entropy, such as a box of gas, down the black hole. The total entropy of matter outside the black hole would go down. One could, of course, still say that the total entropy, including the entropy inside the black hole, has not gone down. But since there is no way to look inside the black hole, we cannot see how much entropy the matter inside it has. It would be nice, therefore, if there was some feature of the black hole by which observers outside the black hole could tell its entropy; this should increase whenever matter carrying entropy fell into the black hole.

Following my discovery that the area of the event horizon increased whenever matter fell into a black hole, a research student at Princeton named Jacob Bekenstein suggested that the area of the event horizon was a measure of the entropy of the black hole. As matter carrying entropy fell into the black hole, the area of the event horizon would go up, so that the sum of the entropy of matter outside black holes and the area of the horizons would never go down.

This suggestion seemed to prevent the second law of thermodynamics from being violated in most situations. However, there was one fatal flaw: If a black hole has entropy, then it ought also to have a temperature. But a body with a nonzero temperature must emit radiation at a certain rate. It is a matter of common experience that if one heats up a poker in the fire, it glows red hot and emits radiation. However, bodies at lower temperatures emit radiation, too; one just does not normally notice it because the amount is fairly small. This radiation is required in order to prevent violations of the second law. So black holes ought to emit radiation, but by their very definition, black holes are objects that are not supposed to emit anything. It therefore seemed that the area of the event horizon of a black hole could not be regarded as its entropy.

In fact, in 1972 I wrote a paper on this subject with Brandon Carter and an American colleague, Jim Bardeen. We pointed out that, although there were many similarities between entropy and the area of the event horizon, there was this apparently fatal difficulty. I must admit that in writing this paper I was motivated partly by irritation with Bekenstein, because I felt he had misused my discovery of the increase of the area of the event horizon. However, it turned out in the end that he was basically correct, though in a manner he had certainly not expected.

BLACK HOLE RADIATION

In September 1973, while I was visiting Moscow, I discussed black holes with two leading Soviet experts, Yakov Zeldovich and Alexander Starobinsky. They convinced me that, according to the quantum mechanical uncertainty principle, rotating black holes should create and emit particles. I believed their arguments on physical grounds, but I did not like the mathematical way in which they calculated the emission. I therefore set about devising a better mathematical treatment, which I described at an informal seminar in Oxford at the end of November 1973. At that time I had not done the calculations to find out how much would actually be emitted. I was expecting to discover just the radiation that Zeldovich and Starobinsky had predicted from rotating black holes. However, when I did the calculation, I found, to my surprise and annoyance, that even nonrotating black holes should apparently create and emit particles at a steady rate.

At first I thought that this emission indicated that one of the approximations I had used was not valid. I was afraid if Bekenstein found out about it, he would use it as a further argument to support his ideas about the entropy of black holes, which I still did not like. However, the more I thought about it, the more it seemed that the approximations really ought to hold. But what finally convinced me that the emission was real was that the spectrum of the emitted particles was exactly that which would be emitted by a hot body.

The black hole was emitting particles at exactly the correct rate to prevent violations of the second law.

Since then, the calculations have been repeated in a number of different forms by other people. They all confirm that a black hole ought to emit particles and radiation as if it were a hot body with a temperature that depends only on the black hole’s mass: the higher the mass, the lower the temperature. One can understand this emission in the following way: What we think of as empty space cannot be completely empty because that would mean that all the fields, such as the gravitational field and the electromagnetic field, would have to be exactly zero. However, the value of a field and its rate of change with time are like the position and velocity of a particle. The uncertainty principle implies that the more accurately one knows one of these quantities, the less accurately one can know the other.

So in empty space the field cannot be fixed at exactly zero, because then it would have both a precise value, zero, and a precise rate of change, also zero. Instead, there must be a certain minimum amount of uncertainty, or quantum fluctuations, in the value of a field. One can think of these fluctuations as pairs of particles of light or gravity that appear together at some time, move apart, and then come together again and annihilate each other. These particles are called virtual particles. Unlike real particles, they cannot be observed directly with a particle detector. However, their indirect effects, such as small changes in the energy of electron orbits and atoms, can be measured and agree with the theoretical predictions to a remarkable degree of accuracy.

By conservation of energy, one of the partners in a virtual particle pair will have positive energy and the other partner will have negative energy. The one with negative energy is condemned to be a short-lived virtual particle. This is because real particles always have positive energy in normal situations. It must therefore seek out its partner and annihilate it. However, the gravitational field inside a black hole is so strong that even a real particle can have negative energy there.

It is therefore possible, if a black hole is present, for the virtual particle with negative energy to fall into the black hole and become a real particle. In this case it no longer has to annihilate its partner; its forsaken partner may fall into the black hole as well. But because it has positive energy, it is also possible for it to escape to infinity as a real particle. To an observer at a distance, it will appear to have been emitted from the black hole. The smaller the black hole, the less far the particle with negative energy will have to go before it becomes a real particle. Thus, the rate of emission will be greater, and the apparent temperature of the black hole will be higher.

The positive energy of the outgoing radiation would be balanced by a flow of negative energy particles into the black hole. By Einstein’s famous equation E = mc2, energy is equivalent to mass. A flow of negative energy into the black hole therefore reduces its mass. As the black hole loses mass, the area of its event horizon gets smaller, but this decrease in the entropy of the black hole is more than compensated for by the entropy of the emitted radiation, so the second law is never violated.

BLACK HOLE EXPLOSIONS

The lower the mass of the black hole, the higher its temperature is. So as the black hole loses mass, its temperature and rate of emission increase. It therefore loses mass more quickly. What happens when the mass of the black hole eventually becomes extremely small is not quite clear. The most reasonable guess is that it would disappear completely in a tremendous final burst of emission, equivalent to the explosion of millions of H-bombs.

A black hole with a mass a few times that of the sun would have a temperature of only one ten-millionth of a degree above absolute zero. This is much less than the temperature of the microwave radiation that fills the universe, about 2.7 degrees above absolute zero—so such black holes would give off less than they absorb, though even that would be very little. If the universe is destined to go on expanding forever, the temperature of the microwave radiation will eventually decrease to less than that of such a black hole. The hole will then absorb less than it emits and will begin to lose mass. But, even then, its temperature is so low that it would take about 1066years to evaporate completely. This is much longer than the age of the universe, which is only about 1010 years.

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

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

THE SEARCH FOR PRIMORDIAL

BLACK HOLES

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

With primordial black holes being so scarce, it might seem unlikely that there would be one that was near enough for us to observe on its own. But since gravity would draw primordial black holes toward any matter, they should be much more common in galaxies. If they were, say, a million times more common in galaxies, then the nearest black hole to us would probably be at a distance of about a thousand million kilometers, or about as far as Pluto, the farthest known planet. At this distance it would still be very difficult to detect the steady emission of a black hole even if it was ten thousand megawatts.

In order to observe a primordial black hole, one would have to detect several gamma ray quanta coming from the same direction within a reasonable space of time, such as a week.

Otherwise, they might simply be part of the background. But Planck’s quantum principle tells us that each gamma ray quantum has a very high energy, because gamma rays have a very high frequency. So to radiate even ten thousand megawatts would not take many quanta. And to observe these few quanta coming from the distance of Pluto would require a larger gamma ray detector than any that have been constructed so far. Moreover, the detector would have to be in space, because gamma rays cannot penetrate the atmosphere.

Of course, if a black hole as close as Pluto were to reach the end of its life and blow up, it would be easy to detect the final burst of emission. But if the black hole has been emitting for the last ten or twenty thousand million years, the chances of it reaching the end of its life within the next few years are really rather small. It might equally well be a few million years in the past or future. So in order to have a reasonable chance of seeing an explosion before your research grant ran out, you would have to find a way to detect any explosions within a distance of about one light-year. You would still have the problem of needing a large gamma ray detector to observe several gamma ray quanta from the explosion. However, in this case, it would not be necessary to determine that all the quanta came from the same direction. It would be enough to observe that they all arrived within a very short time interval to be reasonably confident that they were coming from the same burst.

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

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

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

GENERAL RELATIVITY AND

QUANTUM MECHANICS

Radiation from black holes was the first example of a prediction that depended on both of the great theories of this century, general relativity and quantum mechanics. It aroused a lot of opposition initially because it upset the existing viewpoint: “How can a black hole emit anything?” When I first announced the results of my calculations at a conference at the Rutherford Laboratory 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 end most people, including John Taylor, have come to the conclusion that black holes must radiate like hot bodies if our other ideas about general relativity and quantum mechanics are correct. Thus even though we have not yet managed to find a primordial black hole, there is fairly general agreement that if we did, it would have to be emitting a lot of gamma and X rays. If we do find one, I will get the Nobel Prize.

The existence of radiation from black holes seems to imply that gravitational collapse is not as final and irreversible as we once thought. If an astronaut falls into a black hole, its mass will increase. Eventually, the energy equivalent of that extra mass will be returned to the universe in the form of radiation. Thus, in a sense, the astronaut will be recycled. It would be a poor sort of immortality, however, because any personal concept of time for the astronaut would almost certainly come to an end as he was crushed out of existence inside the black hole. Even the types of particle that were eventually emitted by the black hole would in general be different from those that made up the astronaut. The only feature of the astronaut that would survive would be his mass or energy.

The approximations I used to derive the emission from black holes should work well when the black hole has a mass greater than a fraction of a gram. However, they will break down at the end of the black hole’s life, when its mass gets very small. The most likely outcome seems to be that the black hole would just disappear, at least from our region of the universe. It would take with it the astronaut and any singularity there might be inside the black hole. This was the first indication that quantum mechanics might remove the singularities that were predicted by classical general relativity. However, the methods that I and other people were using in 1974 to study the quantum effects of gravity were not able to answer questions such as whether singularities would occur in quantum gravity.

From 1975 onward, I therefore started to develop a more powerful approach to quantum gravity based on Feynman’s idea of a sum over histories. The answers that this approach suggests for the origin and fate of the universe will be described in the next two lectures. We shall see that quantum mechanics allows the universe to have a beginning that is not a singularity. This means that the laws of physics need not break down at the origin of the universe. The state of the universe and its contents, like ourselves, are completely determined by the laws of physics, up to the limit set by the uncertainty principle.

So much for free will.