Three Roads to Quantum Gravity (11 page)

This has the following interesting consequence. Imagine that we are floating some distance from a black hole. We drop a clock into the black hole, which sends us a pulse of light every thousandth of a second. We receive the signal and convert it to sound. At first we hear the signal as a high-pitched tone, as we receive the signals at a frequency of a thousand times a second. But as the clock nears the horizon of the black hole, each signal is delayed more and more by the fact that it takes a little more time for each successive pulse to arrive. So the tone we hear falls in pitch as the clock nears the horizon. Just as the clock crosses the horizon, the pitch falls to zero, and after that we hear nothing.

This means that the frequency of light is decreased by its having to climb out from the region near the horizon. This can also be understood from quantum theory, as the frequency of light is proportional to its energy, and, just as it takes us energy to climb a flight of stairs, it takes a certain amount of energy for the photon to climb up to us from its starting point just outside the black hole. The closer to the horizon the photon begins its flight, the more energy it must give up as it travels to us. So the closer to the horizon it starts, the more its frequency will have decreased by the time it reaches us. Another consequence is that the wavelength of the light is
lengthened as the frequency is decreased. This is because the wavelength of light is always inversely proportional to its frequency. As a result, if the frequency is diminished, the wavelength must be increased by the same factor.

But this means that the black hole is acting as a kind of microscope. It is not an ordinary microscope, as it does not act by enlarging images of objects. Rather, it acts by stretching wavelengths of light. But nevertheless, this is very useful to us. For suppose that at very short distances space has a different nature than the space we see looking around at ordinary scales. Space would then look very different from the simple three-dimensional Euclidean geometry that seems to suffice to describe the immediately perceptible world. There are various possibilities, and we shall be discussing these in later chapters. Space may be discrete, which means that geometry comes in bits of a certain absolute size. Or there may be quantum uncertainty in the very geometry of space. Just as electrons cannot be localized at precise points in the atom, but are forever dancing around the nucleus, the geometry of space may itself be dancing and fluctuating.

Ordinarily we cannot see what is happening on very small length scales. The reason is that we cannot use light to look at something which is smaller than the wavelength of that light. If we use ordinary light, even the best microscope will not resolve any object smaller than a few thousand times the diameter of an atom, which is the wavelength of the visible part of the spectrum of light. To see smaller objects we can use ultraviolet light, but no microscope in existence, not even one that uses electrons or protons in place of light, can come anywhere near the resolution required to see the quantum structure of space.

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.

T
o 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.

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.

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 through 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.