by John Gribbin
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 th
icken. 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 = mc2. 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.
In fact, the precise version of the uncertainty rule says that energy can only be “borrowed” from the vacuum for a very short time, a time determined by Planck’s constant. This is related to the uncertainty inherent in the measurement of time itself. The only way in which this energy can then be converted into particles is if particles are always created in pairs, which then interact with one another and annihilate themselves before the Universe has time to “notice” that the energy has been borrowed. This means that the particles created out of the vacuum are matched in a special way.
Every variety of particle, such as an electron, has a counterpart known as an antiparticle (in the electron’s case, a positron). Antiparticles have been manufactured in experiments using particle accelerators, and they are also found in cosmic rays (energetic particles reaching the Earth from space), as well as being predicted by quantum theory, so there is no doubt that they exist. In many ways, an antiparticle is a mirror image of its particle equivalent: the positron, for example, carries positive charge, whereas the electron carries negative charge. And whenever a particle meets its antiparticle counterpart, the two annihilate each other.
So according to quantum theory, the vacuum is a seething sea of “virtual” particles. Pairs such as electron-positron are constantly being created, interacting with one another, and disappearing in accordance with the quantum rules. Overall, no energy is released, but virtual pairs flicker in and out of existence all the time, below the threshold of reality.
What Hawking showed was that, even for a nonrotating black hole, this process can drain off energy from a hole and release it into the Universe at large. What happens is that a pair of virtual particles is created just outside the horizon of the hole. In the tiny fraction of a second allowed by quantum uncertainty, one of the particles is captured by the hole. So the other particle has nothing to annihilate with, and escapes, carrying energy with it.
Where has the energy come from? In effect, it is the gravitational energy of the hole. The energy of the hole creates two particles, but it captures only one of them, so only half the energy debt is repaid and the net effect is that the hole loses mass. Other things being equal—if the hole does not gain mass from somewhere else—it will steadily shrink away as a result, evaporating like a puddle in the sunshine. This process is slow but sure, taking billions of years to shrivel even a proton-sized minihole to the point where it explodes. Hawking had contradicted his own earlier conclusion that the surface area of a black hole cannot decrease. Having established a link between black holes and thermodynamics by showing that—according to general relativity alone—black holes cannot shrink, he had now found that if you add quantum theory to the brew, the link with thermodynamics is strengthened, but now black holes must shrink.
For ordinary black holes, made out of dead stars, this effect would be of no real importance. A black hole with three or four times the mass of our Sun and a horizon roughly as big as the surface of a neutron star will be constantly swallowing traces of gas and dust from its surroundings, even in the depths of space, and it is simple to show that the mass lost by Hawking Radiation is much less than the mass gained by this accretion. If nobody had thought of the notion of miniholes, nobody would have been very interested in Hawking Radiation. But since Hawking had already come up with the notion of miniholes, the idea of quantum evaporation of black holes made an immediate impact.
A hole smaller than a proton will not eat up much material from its surroundings, even if it happens to be inside a planet. To a hole that small, even solid matter is mostly empty space! So the Hawking Radiation from the surface of a minihole will actually dominate its behavior. Hawking showed that the radiation produced in this way gives the hole a temperature, exactly the temperature suggested by the work of Bekenstein. For a black hole with the mass of our Sun, this temperature is about one ten-millionth of a degree K (with the resulting ultra-feeble Hawking Radiation easily overwhelmed by infalling matter); but for a minihole with a mass of a billion tons and the size of a proton, the temperature is about 120 billion K. As these examples indicate, the temperature depends on one over the mass of the hole, so a
s it loses mass and gets smaller, such a hole gets hotter and radiates energy faster, until it finally explodes in a burst of X-rays and gamma rays.
Science fiction fans may be intrigued to know that if we could find a proton-sized minihole today, it would be a more than useful energy source. The output from such a hole would be about 6,000 megawatts and could make a substantial contribution to the energy requirements of even a large country. Unfortunately, though, holding on to such a hole if you found it would be tricky—remember that it would weigh a billion tons and gravity would tend to pull it down toward the center of the Earth.
The lifetime of such a minihole depends on the exact mass it starts out with, but roughly proton-sized black holes born in the Big Bang should be exploding here and there in the Universe today.
Intriguingly, detectors flown on satellites have reported occasional bursts of gamma radiation coming from the depths of space, and there is no universally accepted explanation for this phenomenon. It is just possible that the Hawking Radiation from exploding black holes has actually been discovered, although it will be almost impossible ever to prove this.
Hawking had achieved something that even he had thought to be almost impossible, using a combination of general relativity and quantum physics (plus a smattering of thermodynamics) in one package to describe a physical phenomenon. It was this work that made his name outside the specialist circles of mathematicians and astronomers, and any physicist today can tell you what Hawking Radiation is and why it is important. But in a quirky gesture which is in some ways typical of Hawking’s attitude toward established conventions, the astonishing discovery that “black holes are not black” was announced first not in the pages of a scientific journal such as Nature but in an essay that Hawking entered for a somewhat obscure competition organized by the Gravity Research Foundation in America.
