by Lee Smolin
But black holes offer us a way around this problem. Whatever is happening on very small scales near the horizon of the black hole will be enlarged by the effect whereby the wavelengths of light are stretched as the light climbs up to us. This means that if we can observe light coming from very close to the horizon of a black hole, we may be able to see the quantum structure of space itself.
Unfortunately, it has so far proved impractical to make a black hole, so no one has been able to do this experiment. But since the early 1970s several remarkable predictions have been made about what we would see if we could detect light coming from the region just outside a black hole. These predictions constitute the first set of lessons to have come from combining relativity and quantum mechanics. The next three chapters are devoted to them.
CHAPTER 6
ACCELERATION AND HEAT
To really understand what a black hole is like, we must imagine ourselves looking at one up close. What would we see if we were to hover just outside the horizon of a black hole (Figure 15)? A black hole has a gravitational field, like a planet or a star. So to hover just above its surface we must keep our rocket engines on. If we turn off our engines we shall go into a free fall that will quickly take us through the horizon and into the interior of the black hole. To avoid this we must continually accelerate to keep ourselves from being pulled down by the black hole’s gravitational field. Our situation is similar to that of an astronaut in a lunar lander hovering over the surface of the Moon; the main difference is that we do not see a surface below us. Anything that falls towards the black hole accelerates past us as it falls towards the horizon, just below us. But we do not see the horizon because it is made up of photons that cannot reach us, even though they are moving in our direction. They are held in place by the black hole’s gravitational field. So we see light coming from things between us and the horizon, but we see no light from the horizon itself.
You may well think there is something wrong with this. Are we really able to hover over a surface made of photons which never reach us, even though they are moving in our direction? Surely this contradicts relativity, which says that nothing can outrun light? This is true, but there is some fine print. If you are an inertial observer (that is, if you are moving at constant speed, without accelerating) light will always catch up with you. But if you continually accelerate, then light, if it starts out from a point sufficiently far behind you, will never be able to catch you up. In fact this has nothing to do with a black hole. Any observer who continually accelerates, anywhere in the universe, will find themself in a situation rather like that of someone hovering just above the horizon of a black hole. We can see this from Figure 16: given enough of a head start, an accelerating observer can outrun photons. So an accelerating observer has a hidden region simply by virtue of the fact that photons cannot catch up with her. And she has a horizon, which is the boundary of her hidden region. The boundary separates those photons that will catch up with her from those that will not. It is made up of photons which, in spite of their moving at the speed of light, never come any closer to her. Of course, this horizon is due entirely to the acceleration. As soon as the observer turns off her engines and moves inertially, the light from the horizon and beyond will catch her up.
FIGURE 15
A rocket hovering just outside the horizon of a black hole. By keeping its engines on, the rocket can hover a fixed distance over the horizon.
FIGURE 16
We see in bold the worldline of an observer who is constantly accelerating. She approaches but never passes the path of a light ray, which is her horizon since she can see nothing beyond it provided she continues to accelerate. Behind the horizon we see the path of a light ray that never catches up with her. We also see what her trajectory will be if she stops accelerating: she will then pass through her horizon and be able to see what lies on the other side.
This may seem confusing. How can an observer continually accelerate if it is not possible to travel faster than light? Rest assured that what I am saying in no way contradicts relativity. The reason is that while the continually accelerating observer never goes faster than light, she approaches ever closer to that limit. In each interval of time the same acceleration results in smaller and smaller increases in velocity. She comes ever closer to the speed of light, but never reaches it. This is because her mass increases as she approaches the speed of light. Were her speed to match that of light, her mass would become infinite. But one cannot accelerate an object that has infinite mass, hence one cannot accelerate an object to the speed of light or beyond. At the same time, relative to our clocks, her time seems to run slower and slower as her speed approaches, but never reaches, that of light. This goes on for as long as she keeps her engines on and continues to accelerate.
What we are describing here is a metaphor which is very useful for thinking about black holes. An observer hovering just above the surface of a black hole is in many ways just like an observer who is continually accelerating in a region far from any star or black hole. In both cases there is an invisible region whose boundary is a horizon. The horizon is made of light that travels in the same direction as the observer, but never comes any closer to her. To fall through the horizon, the observer has only to turn off her engines. When she does, the light that forms the horizon catches her up and she passes into the hidden region behind it.
But while the situation of an accelerating observer is analogous to that of an observer just outside a black hole, in some ways her situation is simpler. So in this chapter we shall take a small detour and consider the world as seen by an observer who constantly accelerates. This will teach us the concepts we need to understand the quantum properties of a black hole.
Of course, the two situations are not completely analogous. They differ in that the black hole’s horizon is an objective property of the black hole, which is seen by many other observers. However, the invisible region and horizon of an accelerating observer are consequences only of her acceleration, and are seen only by her. Still, the metaphor is very useful. To see why, let us ask a simple question: what does our continually accelerating observer see when she looks around her?
Assume that the region she accelerates through is completely empty. There is no matter or radiation anywhere nearby - there is nothing but the vacuum of empty space. Let us equip our accelerating observer with a suite of scientific instruments, like the ones carried by space probes: particle detectors, thermometers, and so on. Before she turns on her engines she sees nothing, for she is in a region where space is truly empty. Surely turning on her engines does not change this?
In fact it does. First she will experience the normal effect of acceleration, which is to make her feel heavy, just as though she were all of a sudden in a gravitational field. The equivalence between the effects of acceleration and gravity is familiar from the experiences of life and from the science fiction fantasies of artificial gravity in rotating space stations. It is also the most basic principle of Einstein’s general theory of relativity. Einstein called this the equivalence principle. It states that if one is in a windowless room, and has no contact with the outside, it is impossible to tell if one’s room is sitting on the surface of the Earth, or is far away in empty space but accelerating at a rate equal to that by which we see objects fall towards the Earth.
But one of the most remarkable advances of modern theoretical physics has been the discovery that acceleration has another effect which seems at first to have nothing at all to do with gravity. This new effect is very simple: as soon as she accelerates, our observer’s particle detectors will begin to register, in spite of the fact that, according to a normal observer who is not accelerating, the space through which she is travelling is empty. In other words, she will not agree with her non-accelerating friends on the very simple question of whether the space through which they are travelling is empty. The observers who do not accelerate see a completely empty space - a vacuum. Our accelerating observer sees herself as travelling th
rough a region filled with particles. These effects have nothing to do with her engines - they would still be appar-ent if she was being accelerated by being pulled by a rope. They are a universal consequence of her acceleration through space.
Even more remarkable is what she will see if she looks at her thermometer. Before she began accelerating it read zero, because temperature is a measure of the energy in random motion, and in empty space there is nothing to give a non-zero temperature. Now the thermometer registers a temperature, even though all that has changed is her acceleration. If she experiments, she will find that the temperature is proportional to her acceleration. Indeed, all her instruments will behave exactly as if she were all of a sudden surrounded by a gas of photons and other particles, all at a temperature which increases in proportion to her acceleration.
I must stress that what I am describing has never been observed. It is a prediction that was first made in the early 1970s by a brilliant young Canadian physicist, Bill Unruh, who was then barely out of graduate school. What he found was that, as a result of quantum theory and relativity, there must be a new effect, never observed but still universal, whereby anything which is accelerated must experience itself to be embedded in a hot gas of photons, the temperature of which is proportional to the acceleration. The exact relation between temperature T and acceleration a is known, and is given by a famous formula first derived by Unruh. This formula is so simple we can quote it here:
The factor /2πc, where is Planck’s constant and c is the speed of light, is small in ordinary units, which means that the effect has so far escaped experimental confirmation. But it is not inaccessible, and there are proposals to measure it by accelerating electrons with huge lasers. In a world without quantum theory, Planck’s constant would be zero and there would be no effect. The effect also goes away when the speed of light goes to infinity, so it would also vanish in Newtonian physics.
This effect implies that there is a kind of addendum to Einstein’s famous equivalence principle. According to Einstein, a constantly accelerating observer should be in a situation just like an observer sitting on the surface of a planet. Unruh told us that this is true only if the planet has been heated to a temperature that is proportional to the acceleration.
What is the origin of the heat detected by an accelerating observer? Heat is energy, which we know cannot be created nor destroyed. Thus if the observer’s thermometer heats up there must be a source of the energy. So where does it come from? The energy comes from the observer’s own rocket engines. This makes sense, for the effect is present only as long as the observer is accelerating, and this requires a constant input of energy. Heat is not only energy, it is energy in random motion. So we must ask how the radiation measured by an accelerating particle detector becomes randomized. To understand this we have to delve into the mysteries of the quantum theoretic description of empty space.
According to quantum theory, no particle can sit exactly still for this would violate Heisenberg’s uncertainty principle. A particle that remains at rest has a precise position, for it never moves. But for the same reason it has also a precise momentum, namely zero. This also violates the uncertainty principle: we cannot know both position and momentum to arbitrary precision. The principle tells us that if we know the position of a particle with absolute precision we must be completely ignorant of the value of its momentum, and vice versa. As a consequence, even if we could remove all the energy from a particle, there would remain some intrinsic random motion. This motion is called the zero point motion.
What is less well known is that this principle also applies to the fields that permeate space, such as the electric and magnetic fields that carry the forces originating in magnets and electric currents. In this case the roles of position and momentum are played by the electric and magnetic fields. If one measures the precise value of the electric field in some region, one must be completely ignorant of the magnetic field, and so on. This means that if we measure both the electric and magnetic fields in a region we cannot find that both are zero. Thus, even if we could cool a region of space down to zero temperature, so that it contained no energy, there would still be randomly fluctuating electric and magnetic fields. These are called the quantum fluctuations of the vacuum. These quantum fluctuations cannot be detected by any ordinary instrument, sitting at rest, because they carry no energy, and only energy can register its presence in a detector. But the amazing thing is that they can be detected by an accelerating detector, because the acceleration of the detector provides a source of energy. It is exactly these random quantum fluctuations that raise the temperature of the thermometers carried by our accelerating observer.
This still does not completely explain where the randomness comes from. It turns out to have to do with another central concept in quantum theory, which is that there are non-local correlations between quantum systems. These correlations can be observed in certain special situations such as the Einstein-Podolsky-Rosen experiment. In this experiment two photons are created together, but travel apart at the speed of light. But when they are measured it is found that their properties are correlated in such a way that a complete description of either one of them involves the other. This is true no matter how far apart they travel (Figure 17). The photons that make up the vacuum electric and magnetic fields come in pairs that are correlated in exactly this way. What is more, each photon detected by our accelerating observer’s thermometer is correlated with one that is beyond her horizon. This means that part of the information she would need if she wanted to give a complete description of each photon she sees is inaccessible to her, because it resides in a photon that is in her hidden region. As a result, what she observes is intrinsically random. As with the atoms in a gas, there is no way for her to predict exactly how the photons she observes are moving. The result is that the motion she sees is random. But random motion is, by definition, heat. So the photons she sees are hot!
Let us follow this story a bit further. Physicists have a measure of how much randomness is present in any hot system. It is called entropy, and is a measure of exactly how much disorder or randomness there is in the motion of the atoms in any hot system. This measure can be applied also to photons. For example, we can say that the photons coming from the test pattern on my television, being random, have more entropy than the photons that convey The X Files to my eyes. The photons detected by the accelerating detector are random, and so do have a finite amount of entropy.
Entropy is closely related to the concept of information. Physicists and engineers have a measure of how much information is available in any signal or pattern. The information carried by a signal is defined to be equal to the number of yes/no questions whose answers could be coded in that signal. In our digital world, most signals are transmitted as a sequence of bits. These are sequences of ones and zeroes, which may also be thought of as sequences of yeses and noes. The information content of a signal is thus equal to the number of bits, as each bit may be coding the answer to a yes/no question. A megabyte is then precisely a measure of information in this sense, and a computer with a memory of, say, 100 megabytes can store 100 million bytes of information. As each byte contains 8 bits, and each corresponds to the answer to a single yes/no question, this means that the 100 megabyte memory can store the answers to 800 million yes/no questions.
FIGURE 17
The Einstein-Podolsky-Rosen (EPR) experiment. Two photons are created by the decay of an atom. They travel in opposite directions, and are then measured at two events which are outside each other’s light cones. This means that no information can flow to the left event about which measurement the right observer chooses to make. Nevertheless, there are correlations between what the left observer sees and what the right observer chooses to measure. These correlations do not transmit information faster than light because they can be detected only when the statistics from the measurements on each side are compared.
In a random system such as a gas at some non-zero temperature, a larg
e amount of information is coded in the random motion of the molecules. This is information about the positions and motions of the molecules that does not get specified when one describes the gas in terms of quantities such as density and temperature. These quantities are averaged over all the atoms in the gas, so when one talks about a gas in this way most of the information about the actual positions and motions of the molecules is thrown away. The entropy of a gas is a measure of this information - it is equal to the number of yes/no questions that would have to be answered to give a precise quantum theoretic description of all the atoms in the gas.
Information about the exact states of the hot photons seen by the accelerating observer is missing because it is coded in the states of the photons in her hidden region. Because the randomness is a result of the presence of the hidden region, the entropy should incorporate some measure of how much of the world cannot be seen by the accelerating observer. It should have something to do with the size of her hidden region. This is almost right; it is actually a measure of the size of the boundary that separates her from her hidden region. The entropy of the hot radiation she observes as a result of her acceleration turns out to be exactly proportional to the area of her horizon! This relationship between the area of a horizon and entropy was discovered by a Ph.D. student named Jacob Bekenstein, who was working at Princeton at about the time that Bill Unruh made his great discovery. Both were students of John Wheeler, who a few years before had given the black hole its name. Bekenstein and Unruh were in a long line of remarkable students Wheeler trained, which included Richard Feynman.