Three Roads to Quantum Gravity (12 page)

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.

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 large 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.

What those two young physicists did remains the most important step yet made in the search for quantum gravity. They gave us two general and simple laws, which were the first physical predictions to come from the study of quantum gravity. They are:

These two laws are the basis for our understanding of quantum black holes, as we shall see in the next chapter.