Three Roads to Quantum Gravity (13 page)

T
he reason why we have been considering an accelerating observer is that her situation is very similar to that of an observer hovering just above the horizon of a black hole. So the two laws we found at the end of the last chapter, Unruh’s law and Bekenstein’s law, can be applied to tell us what we see as we hover over a black hole. Applying the analogy, we can predict that an observer outside the black hole will see themself as embedded in a gas of hot photons. Their temperature must be related to the acceleration the engines need to deliver to keep the spacecraft hovering a fixed distance above the horizon. Furthermore, the photons that this observer detects will be randomized because a complete description of them will require information that is beyond the horizon, coded in correlations between the photons she sees and photons that remain beyond the horizon (
Figure 18
). To measure this missing information she will attribute an entropy to the black hole. And this entropy will turn out to be proportional to the area of the horizon of the black hole.

Although the analogy is very useful, there is an important difference between the two situations. The temperature and entropy measured by the accelerating observer are consequences of her motion alone. If she turns off her engines, the photons making up her horizon will catch up with her. She can then see into her hidden region. She no longer sees a hot gas of photons, so she measures no temperature. There is no missing information as she sees only empty space, which is
consistent with the fact that there is no hidden region, so no horizon. But with the black hole there are an infinite number of observers who agree that there is a horizon, beyond which they cannot see. And this is not just a consequence of their motion, for all observers who do not fall through the horizon will agree that the black hole and its horizon are there. This means that all observers who are far from a black hole will agree that it has a temperature and an entropy.

Radiation from black holes, as discovered by Stephen Hawking. The photon that travels away from the black hole has random properties and motion because it is correlated, as in one of the photons in
Figure 17
, with the one lost behind the horizon. Because observers outside the horizon cannot recover the information that the infalling photon carries, the outmoving photon appears to have a thermal motion, like a molecule in a hot gas. The result is that the radiation leaving the black hole has a non-zero temperature. It also has an entropy, which is a measure of the missing information.

For simple black holes, which do not rotate and have no electric charge, the values of the temperature and entropy can be expressed very simply. The area of the horizon of a simple black hole is proportional to the square of its mass, in Planck units. The entropy S is proportional to this quantity. In terms of Planck units, we have the simple formula where A is the area of the horizon, and G is the gravitational constant.

There is a very simple way to interpret this equation which is due to Gerard ’t Hooft, who did important work in elementary particle physics - for which he won the 1999 Nobel Prize for Physics - before turning his attention to the problem of quantum gravity. He suggests that the horizon of a black hole is like a computer screen, with one pixel for every four Planck areas. Each pixel can be on or off, which means that it codes one bit of information. The total number of bits of information contained within a black hole is then equal to the total number of such pixels that it would take to cover the horizon. The Planck units are very, very tiny. It would take 10
⁶⁶
Planck area pixels to cover a single square centimetre. So an astrophysical black hole whose horizon has a diameter of several kilometres can contain a stupendous amount of information.

Entropy has another meaning besides being a measure of information. If a system has entropy, it will act in ways that are irreversible in time. This is because of the second law of thermodynamics, which says that entropy can only be created, not destroyed. If you shatter a teapot by dropping it on the floor, you have greatly increased its entropy - it will be very difficult to put it back together. In thermodynamics the irreversibility of a process is measured by an increase of entropy, because that measures the amount of information lost to random motion. But such information, once lost, can never be recovered, so the entropy cannot normally decrease. This is one way of expressing the second law of thermodynamics.

Black holes also behave in a way that is not reversible in
time, because things can fall into a black hole but nothing can come out of one. This turns out to have a consequence, first discovered by Stephen Hawking, for the area of the horizon of a black hole. He showed by a very elegant proof that the area of the horizon of a black hole can never decrease in time. So it was natural to suggest that the area of the horizon of a black hole is analogous to entropy, in that it is a quantity that can only increase in time. The great insight of Bekenstein was that this was not just an analogy. He argued that a black hole has real entropy, which he conjectured is proportional to the area of its horizon and measures the amount of information trapped beyond that horizon.

You may wonder why fifteen thousand other physicists were not able to take this step if it was based on a simple analogy, apparent to anyone who looked at the problem. The reason is that the analogy is not quite complete. For if nothing can come out of a black hole, then it has zero temperature. This is because temperature measures the energy in random motion, and if there is nothing in a box, there can be no motion of any kind, random or otherwise. But ordinarily a system cannot have entropy without it being hot. This is because the missing information results in random motion, which means there is heat. So, if black holes have entropy, there is a violation of the laws of thermodynamics.

So this seemed to be not a brilliant move, but the kind of misuse of analogy that characterizes the thinking of novices in any field. But a few people did take Bekenstein seriously, including Stephen Hawking, Paul Davies and Bill Unruh. The mystery was solved first by Hawking, who realized that if black holes were hot there was no contradiction with the laws of thermodynamics. By following a chain of reasoning roughly like the story above, he was able to show that an observer outside a black hole would see it to be at a finite temperature. Expressed in Planck units, the temperature T of a black hole is inversely proportional to its mass, m. This is a third law, Hawking’s law:

The constant k is very small in normal units. As a result,
astrophysical black holes have temperatures of a very small fraction of a degree. They are therefore much colder than the 2.7 degree microwave background. But a black hole of much smaller mass would be correspondingly hotter, even if it were smaller in size. A black hole the mass of Mount Everest would be no larger than a single atomic nucleus, but it would glow with a temperature greater than the centre of a star.

The radiation emitted by a black hole, called Hawking radiation, carries away energy. By Einstein’s famous relationship between mass and energy, E = mc
²
, this means that the radiation carries away mass as well. This implies that a black hole in empty space must lose mass, for there is no other source of energy to power the radiation it emits. The process by which a black hole radiates away its mass is called black hole evaporation. As a black hole evaporates, its mass decreases. But since its temperature is inversely proportional to its mass, as it loses mass it gets hotter. This will go on at least until the temperature becomes so hot that each photon emitted has roughly the Planck energy. At this point the mass of the black hole is itself roughly equal to the Planck mass, and its horizon is a few Planck lengths across. We have got down to the regime where quantum gravity holds sway. What happens to the black hole next could only be decided by a full quantum theory of gravity.

The evaporation of an astrophysical black hole is a very slow process. The evaporation rate, which depends on the temperature, is very low because the temperature itself is so low, initially. It would take a black hole the mass of the Sun about 10
⁵⁷
times the present age of the universe to evaporate. So this is not something we are going to observe soon. But the question of what happens at the end of black hole evaporation is one that fascinates those of us who think about quantum gravity. It is a subject in which it is easy to find paradoxes to mull over. For example, what happens to the information trapped inside a black hole? We have said that the amount of trapped information is proportional to the area of the horizon of the black hole. When the black hole evaporates, the area of its horizon decreases. Does this mean that the amount of trapped information decreases as well? If not, then there
seems to be a contradiction, but if so we must explain how the information gets out, as it is coded in photons that are trapped behind the horizon.

Similarly, we can also ask whether the entropy of the black hole decreases as the area of the horizon shrinks. It seems that it must, as the two quantities are related. But surely this violates the second law of thermodynamics, which states that entropy can never decrease? One answer is that it need not, because the radiation emitted by the black hole has lots of entropy, which makes up for that lost by the black hole. The second law of thermodynamics requires only that the total entropy of the world never increase. If we include in this total the entropy of the black hole, then all the evidence we have is that the second law of thermodynamics still holds. When something falls into a black hole the outside world may lose some entropy, but the increase in the entropy of the black hole will more than make up for it. On the other hand, if the black hole radiates it loses surface area and hence entropy, but the entropy of the outside world will increase to make up for it.

The result of all this is at the same time very satisfying and deeply puzzling. It is satisfying because the study of black holes has led to a beautiful extension of the laws of thermodynamics. It seemed at first that black holes would violate the laws of thermodynamics. But eventually we realized that if black holes themselves have entropy and temperature, then the laws of thermodynamics would remain true. What is puzzling is that in most circumstances entropy is a measure of missing information. In classical general relativity a black hole is not something complicated: it is described by a few numbers such as its mass and electric charge. But if it has entropy there must be some missing information. The classical theory of black holes gives us no clue as to what that information is about. Nor do the calculations by Bekenstein, Hawking and Unruh give us any hint about what it might be.

But if there is no clue from the classical theory as to the nature of the missing information, there is only one possibility, which is that we need the quantum theory of a black hole to reveal it to us. If we could understand a black hole as a
purely quantum system, then its entropy would have to turn out to include some information about itself that is evident only at the quantum level. So we may now pose a question which could be answered only if we have a quantum theory of gravity. What is the nature of the information trapped in a quantum black hole? Keep this in mind as we go ahead and explore the different approaches to quantum gravity, for a good test of a theory of quantum gravity is how well it is able to answer this question.

A
t the beginning of the twentieth century, few physicists believed in atoms. Now there are few educated people who do not believe in them. But what about space? If we take a bit of space, say a cube 1 centimetre on each side, we can divide each side in two to give eight smaller pieces of space. We can divide each of these again, and so on. With matter there is a limit to how small we can divide something, for at some point we are left with individual atoms. Is the same true of space? If we continue dividing, do we eventually come to a smallest unit of space, some smallest possible volume? Or can we go on for ever, dividing space into smaller and smaller bits, without ever having to stop? All three of the roads I described in the Prologue favour the same answer to this question: that there is indeed a smallest unit of space. It is much smaller than an atom of matter, but nevertheless, as I shall describe in this chapter and the next three, there are good reasons to believe that the continuous appearance of space is as much an illusion as the smooth appearance of matter. When we look on a small enough scale, we see that space is made of things that we can count.