Stephen Hawking (21 page)

So it was established by the early 1970s that a black hole could rotate, but it could not pulsate (Hawking played a small part in this work, too). The size and shape of a black hole depend only on its mass and the speed at which it rotates; the horizon, all that we can see from the outside Universe, carries no identifying features that can tell us what the hole was made of. Physicists call this lack of identifying features the “no hair” theorem. A black hole has no “hair” in the sense that it has no identifying features, and because all we can ever know about it is its mass and its rate of rotation, this makes the mathematical study of black holes much simpler than scientists had feared it would be.

As nothing can get out of a black hole, its mass can never decrease. So the discovery that the surface area of the horizon can never decrease may not seem that dramatic to ordinary mortals. But Stephen Hawking tells how the moment this hit him was so dramatic that it has stuck in his memory for more than forty years. It happened, as we mentioned in the last chapter, one evening in November 1970, not long after the birth of his daughter, Lucy, as he was getting ready for bed. The idea was so exciting that he spent most of the night thinking about the implications.

He was so excited largely because he and Penrose had only just, at that time, come up with a practical mathematical
definition of a black-hole horizon in terms of the tracks of light rays through spacetime. With this definition, he realized, the surface area of the black hole would always increase if matter or radiation fell into the hole, and even if two black holes collided with one another and merged, the area of the new black hole would always be greater than (or, just possibly, the same as) the areas of the two original black holes put together.

This discovery may have made Hawking so excited that he could not sleep, and it may have impressed Roger Penrose when Hawking telephoned him the next day to discuss the idea, but initially it made very little impression on other astronomers and physicists, who regarded such notions as rather esoteric. After all, the X-ray observations (the ones that led to the identification of Cygnus X-1 with a visible star) were made the next year, in 1971, and it was not until the end of 1972 that the consensus that the X-rays come from a black hole orbiting that star was reached. What really began to make other physicists sit up and take notice of Hawking's ideas about the increasing area of a black hole was the seemingly outrageous suggestion that this might be connected with the branch of physics known as thermodynamics.

Thermodynamics is simply the study of heat and motion, as the name implies. It was developed as a branch of science during the nineteenth century and was of great immediate practical value in the age of steam engines. It rests upon some simple, basic rules, such as the fact that heat cannot flow from a cold object to a hot one (immortalized by the musical duo Flanders and Swann in the memorable couplet “Heat won't flow from a colder to a hotter/You can try it if you like but
you'd far better notter”). But thermodynamics goes far beyond the day-to-day practicalities of making steam engines work more effectively and leads on to fundamental truths about the nature of time and the fate of the Universe. One especially important concept, closely linked to the inability of heat to flow “from a colder to a hotter,” is known as entropy.

In everyday language, entropy is the law that tells us that things wear out. Hot things cool off as time passes, and heat flows out of them. Buildings fall down and crumble away; living things grow old and die. These changes are linked to the passage of time, marking a distinction between the past and the future. They correspond to an increase in the amount of
disorder
in the Universe. This disorder is measured in terms of entropy. The flow of time from the past to the future means that the entropy of the Universe must always increase. The same applies to any closed system—the amount of entropy can only increase (or, at best, stay the same); it can never decrease. Now, obviously, the presence of living things on Earth seems to run counter to this rule: we create order out of disorder by building houses and so on. But the point is that the Earth is not a closed system. It “feeds” off the energy flowing from the Sun, dumping entropy as a result. If you take the whole Solar System and treat it as a closed system, the entropy does increase, just as the laws of thermodynamics require.

So Hawking's dramatic realization, coming with such force that evening in November 1970, was to lead to the idea that the law which says that the area of a black hole can only stay the same or increase is equivalent to the law which says that the entropy of a closed system can only stay the same or
increase. But even Hawking didn't make that connection at first.

This is the kind of step that is quite often made in science by a junior researcher, not yet hidebound by tradition. The thought of trying to make a connection between the gravitational physics of black holes and the thermodynamic physics of Victorian steam engines would have daunted even the genius of a Hawking. But to a research student, just setting out on a scientific career and faced with two pieces of information that seem to say the same kind of thing in different ways, the similarity seemed worth remarking on.

Of course, research students very often remark on odd similarities and coincidences in science, and most of the time it turns out that there is nothing significant in the “discovery” at all. But when a student at Princeton University, Jacob Bekenstein, suggested that the size of the horizon around the singularity might literally be a measure of the entropy of a black hole, he started an avalanche of investigation that led Hawking to the discovery that black holes are not necessarily black after all—they explode.

Just as research students are expected to come up with wild ideas (most of which prove fruitless), so it is a common theme in science that some of the most important developments are a result of somebody trying to prove that somebody else's theory is wrong. This happened to good effect in the 1950s and early 1960s, when Fred Hoyle backed a rival model to the Big Bang, the steady-state hypothesis, and became its most vocal proponent. Astronomers determined to prove Hoyle wrong worked much harder at establishing the accuracy of the Big Bang model than they might have
done had there been no rival on the scene. But sometimes the effort can rebound.

Hawking was annoyed by Bekenstein's suggestion. Even a research student ought to have realized that there is a direct connection between entropy and temperature, so that if the area of a black hole were indeed a measure of entropy it would also be a measure of temperature. And if a black hole had a temperature, then heat would flow out of it, into the cold (–270°C) of the Universe. It would radiate energy, contradicting the most basic fact known about black holes, that nothing at all—not even electromagnetic radiation—can escape from them. Together with Brandon Carter and Jim Bardeen, Hawking wrote a paper, published in
Communications in Mathematical Physics
, pointing out this seemingly fatal flaw in Bekenstein's suggestion. It gave the formula for working out the temperature of a black hole according to this ridiculous notion and was published in 1973. But far from agreeing with Bekenstein, the team commented, “In fact the effective temperature of a black hole is absolute zero. . . . No radiation could be emitted from the hole.”
¹

Within a year, however, Hawking had changed his mind. The reasons why he had second thoughts were related to another line of research on black holes he had been pursuing: the possibility, first aired in 1971, that very small “miniholes,” smaller even than the nucleus of an atom, might have been produced in the Big Bang and could still be at large in the Universe today.

The critical mass needed to make a black hole simply by an object collapsing under its own weight is, as we have mentioned, about three times the mass of the Sun, and the Earth
itself would become a black hole if it were squeezed down to about a centimeter. But absolutely anything will make a black hole if it is squeezed hard enough—a bag of sugar, a coin, the book you are reading,
anything
. The difficulty is that, the lighter the object you want to make into a black hole, the harder you would have to squeeze it.

Hawking reasoned that as we look back in time toward the beginning, we look back to higher and higher densities and pressures. So if we look back far enough, we come to a time when the pressure was great enough to squeeze any amount of matter you fancy, even a few grams, into a black hole.

The one snag with this argument is that, if the Universe had been perfectly smooth and uniform back then, no miniholes could form; the only black hole would be the entire Universe itself. But provided there were some irregularities, some variations in density from place to place in the early Universe, then at the appropriate stage of the Big Bang a few grams of matter, any region that just happened to be a little denser than the average, could indeed get pinched off from the rest of spacetime, forming tiny black holes that would last forever (or so Hawking thought in 1971) and still be around today.

We know that the Universe cannot have been perfectly smooth and uniform in the Big Bang because, if it had been, there would be no way that irregularities such as galaxies could have formed as the Universe expanded. There must have been “seeds” in the form of tiny irregularities on which galaxies could grow by gravitational attraction. So Hawking's notion of primordial black miniholes seemed plausible, even if there was no obvious way to test the idea.

In fact, although lightweight by the standards of conventional black holes, even a minihole may have rather a lot of mass by everyday standards. A black hole weighing about a billion tons, for example (the mass of a mountain here on Earth), would have a radius roughly the same as that of a proton. Less massive miniholes would be correspondingly smaller. And if you are dealing with objects as small as that, physicists knew, you have to use the quantum description of reality in order to understand what is going on.

Now the plot began to thicken. In 1969, Roger Penrose had shown that it is possible for a
rotating
black hole to lose energy and slow down as it does so. The way this happens is rather like the way in which space scientists sometimes use the gravitational pull of the planets to speed up spacecraft moving around the Solar System. For example, between its launch in 1989 and its arrival at Jupiter in 1995, a probe named Galileo underwent a “slingshot” maneuver around the Earth and Venus, and eventually ended up in orbit around Jupiter. But in order to get there, it followed a circuitous route.

After its launch, Galileo was sent not outward through the Solar System toward Jupiter, but inward to fly by Venus. By diving around Venus on a carefully calculated orbit, the spacecraft gained energy and speed and was deflected toward the Earth. Venus lost a corresponding amount of energy but, being vastly more massive than the space probe, slowed down in its orbit by only a minuscule amount. At the end of 1990, the speeding Galileo carried out another slingshot maneuver, this time involving the Earth, and entered an orbit that brought it back for a second slingshot past the Earth some two years later. Only then was it moving fast enough to reach Jupiter in
a reasonable time—and it is a sign of how much the probe's speed was increased that it reached Jupiter sooner, even after years of delicate maneuvering to take advantage of the three slingshots, than if it had gone straight out through the Solar System when it was launched.

Penrose showed that similar gravitational effects could boost the energy of electromagnetic radiation near a rotating black hole. The radiation gains energy; the rotation of the hole slows down. In 1973, two Soviet researchers, Yakov Zel'dovich and Alex Starobinsky, extended this idea to show that a rotating black hole should also throw off particles. Their argument had to do with the uncertainty principle of quantum physics, and we shall explain it shortly. They persuaded Hawking that the effect would be real, and he set about trying to find a precise mathematical treatment to describe the phenomenon. He was surprised, and at first annoyed, to discover that the equations said that the same process should be at work even for a nonrotating black hole.

“I was afraid,” Hawking wrote in
A Brief History of Time
, “that 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.”
²
In 1977, he wrote in the January issue of
Scientific American
that he “put quite a lot of effort into trying to get rid of this embarrassing effect,”
³
but to no avail. In the end, Hawking had to accept the mathematical evidence rather than his prejudices. He had found that all black holes emit energetic particles and that therefore every black hole has a temperature. The temperature exactly matches the thermodynamic predictions related to the surface
area of the black hole. We shall now describe how it works (leaving out the detailed mathematics).

Quantum uncertainty doesn't just mean that human instruments are incapable of measuring any quantity precisely. It means that the Universe itself does not “know” any quantity with absolute precision. This applies to energy as much as to anything else. Although we are used to thinking of empty space as containing nothing at all and therefore having zero energy, the quantum rules say that there is some uncertainty about this.
Perhaps
each tiny bit of the vacuum actually contains rather a lot of energy.

If the vacuum contained enough energy, it could convert this into particles, in line with
E = mc
²
. But things are not as simple as this. If the hypothetical energy of uncertainty in the vacuum were converted into particles and the particles became permanent features of the Universe, the rules of uncertainty would be violated—both human observers and the Universe would now be certain that there was something, in the form of a particle or two, where previously there had been nothing. Uncertainty works two ways: it is just as forbidden to be certain that the energy is nonzero, in these circumstances, as it is to be certain that the energy is zero.