in which a black-hole horizon
is clothed in an atmosphere
of radiation and hot particles
that slowly evaporate,
and the hole shrinks
and then explodes
Black Holes Grow
The Idea hit Stephen Hawking one evening in November 1970, as he was preparing for bed. It hit with such force that he was left almost gasping for air. Never before or since has an idea come to him so quickly.
Preparing for bed was not easy. Hawking’s body is afflicted with amyotrophic lateral sclerosis (ALS), a disease that gradually destroys the nerves which control the body’s muscles and leaves the muscles, one after another, to waste away in disuse. He moved slowly, with legs wobbling and at least one hand always firmly grasping a countertop or bedpost, as he brushed his teeth, disrobed, struggled into his pajamas, and climbed into bed. That evening he moved even more slowly than usual, since his mind was preoccupied with the Idea. The Idea excited him. He was ecstatic, but he didn’t tell his wife, Jane; that would have made him most unpopular, since he was supposed to be concentrating on getting to bed.
He lay awake for many hours that night. He couldn’t sleep. His mind kept roaming over the Idea’s ramifications, its connections to other things.
The Idea had been triggered by a simple question. How much gravitational radiation (ripples of spacetime curvature) can two black holes produce, when they collide and coalesce to form a single hole? Hawking had been vaguely aware for some time that the single final hole would have to be larger, in some sense, than the “sum” of the two original holes, but in what sense, and what could that tell him about the amount of gravitational radiation produced?
Then, as he was preparing for bed, it had hit him. Suddenly, a series of mental pictures and diagrams had coalesced in his mind to produce the Idea: It was the area of the hole’s horizon that would be larger. He was sure of it; the pictures and diagrams had coalesced into an unequivocal, mathematical proof. No matter what the masses of the two original holes might be (the same or very different), and no matter how the holes might spin (in the same direction or opposite or not at all), and no matter how the holes might collide (head-on or at a glancing angle), the area of the final hole’s horizon must always be larger than the sum of the areas of the original holes’ horizons. So what? So a lot, Hawking realized as his mind roamed over the ramifications of this area-increase theorem
First of all, in order for the final hole’s horizon to have a large area, the final hole must have a large mass (or equivalently a large energy), which means that not too much energy could have been ejected, as gravitational radiation. But “not too much” was still quite a bit. By combining his new area-increase theorem with an equation that describes the mass of a black hole in terms of its surface area and spin, Hawking deduced that as much as 50 percent of the mass of the two original holes could be converted to gravitational-wave energy, leaving as little as 50 percent behind in the mass of the final hole.1
There were other ramifications Hawking realized in the months that followed his sleepless November night. Most important, perhaps, was a new answer to the question of how to define the concept of a hole’s horizon when the hole is “dynamical,” that is, when it is vibrating wildly (as it must during collisions), or when it is growing rapidly (as it will when it is first being created by an imploding star).
Precise and fruitful definitions are essential to physics research. Only after Hermann Minkowski had defined the absolute interval between two events (Box 2.1) could he deduce that, although space and time are “relative,” they are unified into an “absolute” spacetime. Only after Einstein had defined the trajectories of freely falling particles to be straight lines (Figure 2.2) could he deduce that spacetime is curved (Figure 2.5), and thereby develop his laws of general relativity. And only after Hawking had defined the concept of a dynamical hole’s horizon could he and others explore in detail how black holes change when pummeled by collisions or by infalling debris.
Before November 1970, most physicists, following Roger Penrose’s lead, had thought of a hole’s horizon as “the outermost location where photons trying to escape the hole get pulled inward by gravity.” This old definition of the horizon was an intellectual blind alley, Hawking realized in the ensuing months, and to brand it as such he gave it a new, slightly contemptuous name, a name that would stick. He called it the apparent horizon.2
Hawking’s contempt had several roots. First, the apparent horizon is a relative concept, not an absolute one. Its location depends on the observers’ reference frame; observers falling into the hole might see it at a different location from observers at rest outside the hole. Second, when matter falls into the hole, the apparent horizon can jump suddenly, without warning, from one location to another—a rather bizarre behavior, one not conducive to easy insights. Third and most important, the apparent horizon had no connection at all to the flash of congealing mental pictures and diagrams that had produced Hawking’s New Idea.
Hawking’s new definition of the horizon, by contrast, was absolute (the same in all reference frames), not relative, so he called it the absolute horizon. This absolute horizon is beautiful, Hawking thought. It has a beautiful definition: It is “the boundary in spacetime between events (outside the horizon) that can send signals to the distant Universe and those (inside the horizon) that cannot.” And it has a beautiful evolution: When a hole eats matter or collides with another hole or does anything at all, its absolute horizon changes shape and size in a smooth, continuous way, instead of a sudden, jumping way (Box 12.1). Most important, the absolute horizon meshed perfectly with Hawking’s New Idea:
Absolute and Apparent Horizons for a Newborn Black Hole
The spacetime diagrams shown below describe the implosion of a spherical star to form a spherical black hole; compare with Figure 6.7. The dotted curves are outgoing light rays; in other words, they are the world lines (trajectories through spacetime) of photons—the fastest signals that can be sent radially outward, toward the distant Universe. For optimal escape, the photons are idealized as not being absorbed or scattered at all by the star’s matter.
The apparent horizon (left diagram) is the outermost location where outgoing light rays, trying to escape the hole, get pulled inward toward the singularity (for example, the outgoing rays QQ′ and RR′). The apparent horizon is created suddenly, full-sized, at E, where the star’s surface shrinks through the critical circumference. The absolute horizon (right diagram) is the boundary between events that can send signals to the distant Universe (for example, events P and S which send signals along the light rays pp′ and SS′) and events that cannot send signals to the distant Universe (for example, Q and R). The absolute horizon is created at the star’s center, at the event labeled C, well before the star’s surface shrinks through the critical circumference. The absolute horizon is just a point when created, but it then expands smoothly, like a balloon being blown up, and emerges through the star’s surface precisely when the surface shrinks through the critical circumference (the circle labeled E). It then stops expanding, and thereafter coincides with the suddenly created apparent horizon.
Hawking could see, in his congealed mental pictures and diagrams, that the areas of absolute horizons (but not necessarily apparent horizons) will increase not only when black holes collide and coalesce, but also when they are being born, when matter or gravitational waves fall into them, when the gravity of other objects in the Universe raises tides on them, and when rotational energy is being extracted from the swirl of space just outside their horizons. Indeed, the areas of absolute horizons will almost always increase, and can never decrease. The physical reason is simple: Everything that a hole encounters sends energy inward through its absolute horizon, and there is no way that any energy can come back out. Since all forms of energy produce gravity, this means that the hole’s gravity is continually being strengthened, and correspondingly, its surface area is continually growing.
Hawking’s conclusion, stated more precisely, was this: In any region of space, and at any moment of time (as measured in anyone’s reference frame), measure the areas of all the absolute horizons of all black holes and add the areas together to get a total area. Then wait however long you might wish, and again measure the areas of all the absolute horizons and add them If no black holes have moved out through the “walls” of your region of space between the measurements, then the total horizon area cannot have decreased, and it almost always will have increased, at least a little bit
Hawking was well aware that the choice of definition of horizon, absolute or apparent, could not influence in any way any predictions for the outcomes of experiments that humans or other beings might perform; for example, it could not influence predictions of the waveforms of gravitational radiation produced in black-hole collisions (Chapter 10), nor could it influence predictions of the number of X-rays emitted by hot gas falling into and through a black hole’s horizon (Chapter 8). However, the choice of definition could strongly influence the ease with which theoretical physicists deduce, from Einstein’s general relativistic equations, the properties and behaviors of black holes. The chosen definition would become a central tool in the paradigm by which theorists guide their research; it would influence their mental pictures, their diagrams, the words they say when communicating with each other, and their intuitive leaps of insight. And for this purpose, Hawking believed, the new, absolute horizon, with its smoothly increasing area, would be superior to the old, apparent horizon, with its discontinuous jumps in size.
Stephen Hawking was not the first physicist to think about absolute horizons and discover their area increase. Roger Penrose at Oxford had already done so, before Hawking’s sleepless November night. In fact, Hawking’s insights were based largely on foundations laid by Penrose (Chapter 13) and a recent conversation with him. However, Penrose had not recognized the power of the area-increase theorem, and so had not perfected nor published it. Why had he missed its power? Because he had not developed any clear mental picture of the absolute horizon’s location. He had missed what Hawking saw so clearly that sleepless night: After black holes collide, their merged absolute horizon should soon settle down into a quiescent location whose surface area can be computed from the standard equations for quiescent black holes.
Werner Israel at the University of Alberta, Canada, also caught a glimpse of the area increase theorem before Hawking; but, failing to recognize its importance, he, too, let it slip by unpublished. Moreover, by contrast with Hawking and Penrose, Israel was still using—indeed, was hypnotized by—the old concept of the apparent horizon, as were all the rest of us relativity theorists. The apparent horizon had played a central role in Penrose’s amazing 1964 discovery that Einstein’s laws force every black hole to have a singularity at its center (Chapter 13). The apparent horizon thereby had proved its power. Blinded by that power, we relativists could not conceive of replacing the apparent horizon, as the definition of a black hole’s surface, by the absolute horizon.
We also were paying little attention to the absolute horizon because it violates our cherished notion that an effect should not precede its cause. When matter falls into a black hole, the absolute horizon starts to grow (“effect”) before the matter reaches it (“cause”). The horizon grows in anticipation that the matter will soon be swallowed and will increase the hole’s gravitational pull (Box 12.2). This seeming paradox has a simple origin. The very definition of the absolute horizon depends on what will happen in the future: on whether or not signals will ultimately escape to the distant Universe. In the terminology of philosophers, it is a teleological definition (a definition that relies on “final causes”), and it forces the horizon’s evolution to be teleological. Since teleological viewpoints have rarely if ever been useful in modern physics, the absolute horizon seemed hardly worth exploring.
Evolution of an Accreting Hole’s Apparent and Absolute Horizons
The spacetime diagram below illustrates the jerky evolution of the apparent horizon and the teleological evolution of the absolute horizon. At some initial moment of time (on a horizontal slice near the bottom of the diagram), an old, nonspinning black hole is surrounded by a thin, spherical shell of matter. The shell is like the rubber of a balloon, and the hole is like a pit at the balloon’s center. The hole’s gravity pulls on the shell (the balloon’s rubber), forcing it to shrink and ultimately be swallowed by the hole (the pit). The apparent horizon (the outermost location at which outgoing light rays—shown dotted—are being pulled inward) jumps outward suddenly, and discontinuously, at the moment when the shrinking shell reaches the location of the final hole’s critical circumference. The absolute horizon (the boundary between events that can and cannot send outgoing light rays to the distant Universe) starts to expand before the hole swallows the shell. It expands in anticipation of swallowing, and then, just as the hole swallows, it comes to rest at the same location as the jumping apparent horizon.
Hawking is a bold thinker. He is far more willing than most physicists to take off in radical new directions, if those directions “smell” right. The absolute horizon smelled right to him, so despite its radical nature, he embraced it, and his embrace paid off. Within a few months, Hawking and James Hartle were able to derive, from Einstein’s general relativity laws, a set of elegant equations that describe how the absolute horizon continuously and smoothly expands and changes its shape, in anticipation of swallowing infalling debris or gravitational waves, or in anticipation of being pulled on by the gravity of other bodies.
In November 1970, Stephen Hawking was just beginning to reach full stride as a physicist. He had made several important discoveries already, but he was not yet a dominant figure. As we move on through this chapter, we shall watch him become dominant.
How, with his severe disability, has Hawking been able to out-think and out-intuit his leading colleague-competitors, people like Roger Penrose, Werner Israel, and (as we shall see) Yakov Borisovich Zel’-dovich? They had the use of their hands; they could draw pictures and perform many-page-Iong calculations on paper—calculations in which one records many complex intermediate results along the way, and then goes back, picks them up one by one, and combines them to get a final result; calculations that I cannot conceive of anyone doing in his head. By the early 1970s, Hawking’s hands were largely paralyzed; he could neither draw pictures nor write down equations. His research had to be done entirely in his head.
Because the loss of control over his hands was so gradual, Hawking has had plenty of time to adapt. He has gradually trained his mind to think in a manner different from the minds of other physicists: He thinks in new types of intuitive mental pictures and mental equations that, for him, have replaced paper-and-pen drawings and written equations. Hawking’s mental pictures and mental equations have turned out to be more powerful, for some kinds of problems, than the old paper-and-pen ones, and less powerful for others, and he has gradually learned to concentrate on problems for which his new methods give greater power, a power that nobody else can begin to match.
Hawking’s disability has helped him in other ways. As he himself has often commented, it has freed him from the responsibility of lecturing to university students, and he thus has had far more free time for research than his more healthy colleagues. More important, perhaps, his disease in some ways has improved his attitude toward life.
Stephen Hawking with his wife Jane and their son Timothy in Cambridge, England, in 1980. [Photo by Kip Thorne.]
Hawking contracted ALS in 1963, soon after he began graduate school at Cambridge University. ALS is a catch-all name for a variety of motor neuron diseases, most of which kill fairly quickly. Thinking he had only a few years to live, Hawking at first lost his enthusiasm for life and physics. However, by the winter of 1964–65, it became apparent that his was a rare variant of ALS, a variant that saps the central nervous system’s control of muscles over many years’ time, not just a few. Suddenly life seemed wonderful. He returned to physics with greater vigor and enthusiasm than he had ever had as a healthy, devil-may-care undergraduate student; and with his new lease on life, he married Jane Wilde, whom he had met shortly after contracting ALS and with whom he had fallen in love during the early phases of his disease.
Stephen’s marriage to Jane was essential to his success and happiness in the 1960s and 1970s and into the 1980s. She made for them a normal home and a normal life in the midst of physical adversity.
The happiest smile I ever saw in my life was Stephen’s the evening in August 1972 in the French Alps when Jane, I, and the Hawkings’ two oldest children, Robert and Lucy, returned from a day’s excursion into the mountains. Through foolishness we had missed the last ski lift down the mountain, and had been forced to descend about 1000 meters on foot. Stephen, who had fretted about our tardiness, broke out into an enormous smile, and tears came to his eyes, as he saw Jane, Robert, and Lucy enter the dining room where he was poking at his evening meal, unable to eat.
Hawking lost the use of his limbs and then his voice very gradually. In June 1965, when we first met, he walked with a cane and his voice was only slightly shaky. By 1970 he required a four-legged walker. By 1972 he was confined to a motorized wheelchair and had largely lost the ability to write, but he could still feed himself with some ease, and most native English speakers could still understand his speech, though with difficulty. By 1975 he could no longer feed himself, and only people accustomed to his speech could understand it. By 1981 even I was having severe difficulty understanding him unless we were in an absolutely quiet room; only people who were with him a lot could understand with ease. By 1985 his lungs would not remain clear of fluid of their own accord, and he had to have a tracheostomy so they could be cleared regularly by suctioning. The price was high: He completely lost his voice. To compensate, he acquired a computer-driven voice synthesizer with an American accent for which he would apologize sheepishly. He controls the computer by a simple switch clutched in one hand, which he squeezes as a menu of words scrolls by on the computer screen. Grabbing one word after another from the scrolling menu with his switch, he builds up his sentences. It is painfully slow, but effective; he can produce no more than one short sentence per minute, but his sentences are enunciated clearly by the synthesizer, and are often pearls.
As his speech deteriorated, Hawking learned to make every sentence count. He found ways to express his ideas that were clearer and more succinct than the ways he had used in the early years of his disease. With clarity and succinctness of expression came improved clarity of thought, and greater impact on his colleagues—but also a tendency to seem oracular: When he issues a pronouncement on some deep question, we, his colleagues, sometimes cannot be sure, until after much thought and calculation of our own, whether he is just speculating or has strong evidence. He sometimes doesn’t tell us, and we occasionally wonder whether he, with his absolutely unique insights, is playing games with us. He does, after all, still retain a streak of the impishness that made him popular in his undergraduate days at Oxford, and a sense of humor that rarely deserts him, even in times of trial. (Before his tracheostomy, when I began to have trouble understanding his speech, I sometimes found myself saying over and over again, as many as ten times, “Stephen, I still don’t understand; please say it again.” Showing a bit of frustration, he would continue to repeat himself until I suddenly understood: He was telling me a wonderfully funny, off-the-wall, one-line joke. When I finally caught it, he would grin with pleasure.)
Entropy
Having extolled Hawking’s ability to out-think and out-intuit all his colleague-competitors, I must now confess that he has not managed to do so all the time, just most. Among his defeats, perhaps the most spectacular was at the hands of one of John Wheeler’s graduate students, Jacob Bekenstein. But in the midst of that defeat, as we shall see, Hawking produced a far greater triumph: his discovery that black holes can evaporate. The tortuous route to that discovery will occupy much of the rest of this chapter.
The playing field on which Hawking was defeated was that of black-hole thermodynamics. Thermodynamics is the set of physical laws that govern the random, statistical behavior of large numbers of atoms, for example, the atoms that make up the air in a room or those that make up the entire Sun. The atoms’ statistical behavior includes, among other things, their random jiggling caused by heat; and correspondingly, the laws of thermodynamics include, among other things, the laws that govern heat. Hence the name thermodynamics.
A year before Hawking discovered his area theorem, Demetrios Christodoulou, a nineteen-year-old graduate student in Wheeler’s Princeton group, noticed that the equations that describe slow changes in the properties of black holes (for example, when they slowly accrete gas) resemble some of the equations of thermodynamics. The resemblance was remarkable, but there was no reason to think it anything more than a coincidence.
This resemblance was strengthened by Hawking’s area theorem: The area theorem closely resembled the second law of thermodynamics. In fact, the area theorem, as expressed earlier in this chapter, becomes the second law of thermodynamics if we merely replace the phrase “horizon areas” by the word entropy: In any region of space, and at any moment of time (as measured in anyone’s reference frame), measure the total entropy of everything there. Then wait however long you might wish, and again measure the total entropy. If nothing has moved out through the “walls” of your region of space between the measurements, then the total entropy cannot have decreased, and it almost always will have increased, at least a little bit
What is this thing called “entropy” that increases? It is the amount of “randomness” in the chosen region of space, and the increase of entropy means that things are continually becoming more and more random.
Stated more precisely (see Box 12.3), entropy is the logarithm of the number of ways that all the atoms and molecules in our chosen region can be distributed, without changing that region’s macroscopic appearance.3 When there are many possible ways for the atoms and molecules to be distributed, there is a huge amount of microscopic randomness and the entropy is huge.
The law of entropy increase (the second law of thermodynamics) has great power. As an example, suppose that we have a room containing air and a few crumpled-up newspapers. The air and paper together contain less entropy than they would have if the paper were burned in the air to form carbon dioxide, water vapor, and a bit of ash. In other words, when the room contains the original air and paper, there are fewer ways that its molecules can be randomly distributed than when it contains the final air, carbon dioxide, water vapor, and ash. That is why the paper burns naturally and easily if a spark ignites it, and why the burning cannot easily and naturally be reversed to create paper from carbon dioxide, water, ash, and air. Entropy increases during burning; entropy would decrease during unburning; thus, burning occurs and unburning does not.
Stephen Hawking noticed immediately, in November 1970, the remarkable similarity between the second law of thermodynamics and his law of area increase, but it was obvious to him that the similarity was a mere coincidence. One would have to be crazy, or at least a little dim-witted, to claim that the area of a hole’s horizon in some sense is the hole’s entropy, Hawking thought. After all, there is nothing at all random about a black hole. A black hole is just the opposite of random; it is simplicity incarnate. Once a black hole has settled down into a quiescent state (by emitting gravitational waves; Figure 7.4), it is left totally “hairless”: All of its properties are precisely determined by just three numbers, its mass, its angular momentum, and its electric charge. The hole has no randomness whatsoever.
Entropy in a Child’s Playroom
Imagine a square playroom containing 20 toys. The floor of the room is made of 100 large tiles (with 10 tiles running along each side), and a father has cleaned the room, throwing all the toys onto the northernmost row of tiles. The father cared not one whit which toys landed on which tiles, so they are all randomly scrambled. One measure of their randomness is the number of ways that they could have landed (each of which the father considers as equally satisfactory), that is, the number of ways that the 20 toys can be distributed over the 10 tiles of the northern row. This number turns out to be 10 × 10 × 10 × ... × 10, with one factor of 10 for each toy; that is, 1020.
This number, 1020, is one description of the amount of randomness in the toys. However, it is a rather unwieldly description, since 1020 is such a big number. More easy to manipulate is the logarithm of 1020, that is, the number of factors of 10 that must be multiplied together to get 1020. The logarithm is 20; and this logarithm of the number of ways the toys could be scattered over the tiles is the toys’ entropy.
Now suppose that a child comes into the room and plays with the toys, throwing them around with abandon, and then leaves. The father returns and sees a mess. The toys are now far more randomly distributed than before. Their entropy has increased. The father doesn’t care just where each toy is; all he cares is that they have been scattered randomly throughout the room. How many different ways might they have been scattered? How many ways could the 20 toys be distributed over the 100 tiles? 100 × 100 × 100 × ... × 100, with one factor of 100 for each toy; that is, 10020 = 1040 ways. The logarithm of this number is 40, so the child increased the toys’ entropy from 20 to 40.
“Aha, but then the father cleans up the room and thereby reduces the toys’ entropy back to 20,” you might say. “Doesn’t this violate the second law of thermodynamics?” No, not at all. The toys’ entropy may be reduced by the father’s cleaning, but the entropy in the father’s body and in the room’s air has increased: It took a lot of energy to throw the toys back onto the northernmost tiles, energy that the father got by “burning up” some of his body’s fat. The burning converted neatly organized fat molecules into disorganized waste products, for example, the carbon dioxide that he exhaled randomly into the room; and the resulting increase in the father’s and the room’s entropy (the increase in the number of ways their atoms and molecules can be distributed) far more than made up for the decrease in the toys’ entropy.
Jacob Bekenstein was not persuaded. It seemed likely to him that a black hole’s area in some deep sense is its entropy—or, more precisely, its entropy multiplied by some constant. If not, Bekenstein reasoned, if black holes have vanishing entropy (no randomness at all) as Hawking claimed, then black holes could be used to decrease the entropy of the Universe and thereby violate the second law of thermodynamics. All one need do is bundle all the air molecules from some room into a small package and drop them into a black hole. The air molecules and all the entropy they carry will disappear from our Universe when the package enters the hole, and if the hole’s entropy does not increase to compensate for this loss, then the total entropy of the Universe will have been reduced. This violation of the second law of thermodynamics would be highly unsatisfactory, Bekenstein argued. To preserve the second law, a black hole must possess an entropy that goes up when the package falls through its horizon, and the most promising candidate for that entropy, it seemed to Bekenstein, was the hole’s surface area.
Not at all, Hawking responded. You can lose air molecules by throwing them down a black hole, and you can also lose entropy. That is just the nature of black holes. We will just have to accept this violation of the second law of thermodynamics, Hawking argued; the properties of black holes require it—and besides, it has no serious consequences at all. For example, although under ordinary circumstances a violation of the second law of thermodynamics might permit one to make a perpetual motion machine, when it is a black hole that causes the violation, no perpetual motion machine is possible. The violation is just a tiny peculiarity in the laws of physics, one that the laws presumably live with quite happily.
Bekenstein was not convinced.
All the world’s black-hole experts lined up on Hawking’s side—all, that is, except Bekenstein’s mentor, John Wheeler. “Your idea is just crazy enough that it might be right,” Wheeler told Bekenstein. With this encouragement, Bekenstein plowed forward and tightened up his conjecture. He estimated just how much a hole’s entropy would have to grow, when a package of air is dropped into it, to preserve the second law of thermodynamics, and he estimated how much the plunging package would increase the horizon’s area; and from these rough estimates, he deduced a relationship between entropy and area which, he thought, might always preserve the second law of thermodynamics: The entropy, he concluded, is approximately the horizon’s area divided by a famous area associated with the (as yet ill-understood) laws of quantum gravity, the Planck–Wheeler area, 2.61 × 10−66 square centimeter.4 (We shall learn the significance of the Planck–Wheeler area in the next two chapters.) For a 10-solar-mass hole, this entropy would be the hole’s area, 11,000 square kilometers, divided by the Planck—Wheeler area, 2.61 × 10−66 square centimeter, which is roughly 1079.
This is an enormous amount of entropy. It represents a huge amount of randomness. Where does this randomness reside? Inside the hole, Bekenstein conjectured. The hole’s interior must contain a huge number of atoms or molecules or something, all randomly distributed, and the total number of ways they could be distributed must be5 101079
Nonsense, responded most of the leading black-hole physicists, including Hawking and me. The hole’s interior contains a singularity, not atoms or molecules.
Nevertheless, the similarity between the laws of thermodynamics and the properties of black holes was impressive.
In August 1972, with the golden age of black-hole research in full swing, the world’s leading black-hole experts and about fifty students congregated in the French Alps for an intense month of lectures and joint research. The site was the same Les Houches summer school, on the same green hillside opposite Mont Blanc, at which nine years earlier (1963) I had been taught the intricacies of general relativity (Chapter 10). In 1963 I had been a student. Now, in 1972, I was supposed to be an expert. In the mornings we “experts” lectured to each other and the students about the discoveries we had made during the past five years and about our current struggles toward new insights. During most afternoons we continued our current struggles: Igor Novikov and I closeted ourselves in a small log cabin and struggled to discover the laws that govern gas as it accretes into black holes and emits X-rays (Chapter 8), while on couches in the school’s lounge my students Bill Press and Saul Teukolsky sought ways to discover whether a spinning black hole is stable against small perturbations (Chapter 7), and fifty meters above me on the hillside, James Bardeen, Brandon Carter, and Stephen Hawking joined forces to try to deduce from Einstein’s general relativity equations the full set of laws that govern the evolution of black holes. The setting was idyllic, the physics delicious.
By the end of the month, Bardeen, Carter, and Hawking had consolidated their insights into a set of laws of black-hole mechanics that bore an amazing resemblance to the laws of thermodynamics. Each black-hole law, in fact, turned out to be identical to a thermodynamics law, if one only replaced the phrase “horizon area” by “entropy,” and the phrase “horizon surface gravity” by “temperature.” (The surface gravity, roughly speaking, is the strength of gravity’s pull as felt by somebody at rest just above the horizon.)
When Bekenstein (who was one of the fifty students at the school) saw this perfect fit between the two sets of laws, he became more convinced than ever that the horizon area is the hole’s entropy. Bardeen, Carter, Hawking, I, and the other experts, by contrast, saw in this fit a firm proof that the horizon area cannot be the hole’s entropy in disguise. If it were, then similarly the surface gravity would have to be the hole’s temperature in disguise, and that temperature would not be zero. However, the laws of thermodynamics insist that any and every object with a nonzero temperature must emit radiation, at least a little bit (that is how the radiators that warm some homes work), and everybody knew that black holes cannot emit anything. Radiation can fall into a black hole, but none can ever come out.
If Bekenstein had followed his intuition to its logical conclusion, he would have asserted that somehow a black hole must have a finite temperature and must emit radiation, and we today would look back on him as an astounding prophet. But Bekenstein waffled. He conceded that it was obvious a black hole cannot radiate, but he clung tenaciously to his faith in black-hole entropy.
The first hint that black holes, in fact, can radiate came from Yakov Borisovich Zel’dovich, in June 1971, fourteen months before the Les Houches summer school. However, nobody was paying any attention, and for this I bear the brunt of the shame since I was Zel’dovich’s confidant and foil as he groped toward a radical new insight.
Zel’dovich had brought me to Moscow for my second several-week stint as a member of his research group. On my first stint, two years earlier, he had commandeered for me, in the midst of Moscow’s housing crunch, a spacious private apartment on Shabolovka Street, near October Square. While some of my Russian friends shared one-room apartments with their spouses, children, and a set of parents—one room, not one bedroom—I had had all to myself an apartment with bedroom, living room, kitchen, television, and elegant china. On this second stint I lived more modestly, in a single room at a hotel owned by the Soviet Academy of Sciences, down the street from my old apartment.
At 6:30 one morning, I was roused from my sleep by a phone call from Zel’dovich. “Come to my flat, Kip! I have a new idea about spinning black holes!” Knowing that coffee, tea, and pirozhki (pastries containing ground beef, fish, cabbage, jam, or eggs) would be waiting, I sloshed cold water on my face, threw on my clothes, grabbed my briefcase, dashed down five flights of stairs into the street, grabbed a crowded trolley, transferred to a trolley bus, and alighted at Number 2B Vorobyevskoye Shosse in the Lenin Hills, 10 kilometers south of the Kremlin. Number 4, next door, was the residence of Alexei Kosygin, the Premier of the U.S.S.R.6
I walked through an open gate in the eight-foot-high iron fence and entered a four-acre, forested yard surrounding the massive, squat apartment house Number 2B and its twin Number 2A, with their peeling yellow paint. As one reward for his contributions to Soviet nuclear might (Chapter 6), Zel’dovich had been given one of 2B’s eight apartments: the southwest quarter of the second floor. The apartment was enormous by Moscow standards, 1500 square feet; he shared it with his wife, Varvara Pavlova, one daughter, and a son-in-law.
Zel’dovich met me at the apartment door, with a warm grin on his face and the sounds of his bustling family emerging from back rooms. I removed my shoes, put on slippers from the pile beside the door, and followed him into the shabby but comfortable living/dining room, with its overstuffed couch and chairs. On one wall was a map of the world, with colored pins identifying all the places to which Zel’dovich had been invited (London, Princeton, Beijing, Bombay, Tokyo, and many more), and which the Soviet state, in its paranoid fear of losing nuclear secrets, had forbade him to visit.
Zel’dovich, his eyes dancing, sat me down at the long dining table dominating the room’s center, and announced, “A spinning black hole must radiate. The departing radiation will kick back at the hole and gradually slow its spin, and then halt it. With the spin gone, the radiation will stop, and the hole will live forever thereafter in a perfectly spherical, nonspinning state.”
“That’s one of the craziest things I’ve ever heard,” I asserted. (Open confrontation is not my style, but Zel’dovich thrived on it. He wanted it, he expected it, and he had brought me to Moscow in part to serve as a sparring partner, an opponent against whom to test ideas.) “How can you make such a crazy claim?” I asked. “Everyone knows that radiation can flow into a hole, but nothing, not even radiation, can come out.”
Zel’dovich explained his reasoning: “A spinning metal sphere emits electromagnetic radiation, and so, similarly, a spinning black hole should emit gravitational waves.”
A typical Zel’dovich proof, I thought to myself. Pure physical intuition, based on nothing more than analogy. Zel’dovich doesn’t understand general relativity well enough to compute what a black hole should do, so instead he computes the behavior of a spinning metal sphere, he then asserts that a black hole will behave analogously, and he wakes me up at 6:30 A.M. to test his assertion.
However, I had already seen Zel’dovich make discoveries with little more basis than this; for example, his 1965 claim that when a mountainous star implodes, it produces a perfectly spherical black hole (Chapter 7), a claim that turned out to be right and that foretold the hairlessness of holes. I thus proceeded cautiously. “I had no idea that a spinning metal sphere emits electromagnetic radiation. How?”
“The radiation is so weak,” Zel’dovich explained, “that nobody has ever observed it, nor predicted it before. However, it must occur. The metal sphere will radiate when electromagnetic vacuum fluctuations tickle it. Similarly, a black hole will radiate when gravitational vacuum fluctuations graze its horizon.”
I was too dumb in 1971 to realize the deep significance of this remark, but several years later it would become clear. All previous theoretical studies of black holes had been based on Einstein’s general relativistic laws, and those studies were unequivocal: A black hole cannot radiate. However, we theorists knew that general relativity is only an approximation to the true laws of gravity—an approximation that should be excellent when dealing with black holes, we thought, but an approximation nonetheless.7 The true laws, we were sure, must be quantum mechanical, so we called them the laws of quantum gravity. Although those quantum gravity laws were only vaguely understood at best, John Wheeler had deduced in the 1950s that they must entail gravitational vacuum fluctuations, tiny, unpredictable fluctuations in the curvature of spacetime, fluctuations that remain even when spacetime is completely empty of all matter and one tries to remove all gravitational waves from it, that is, when it is a perfect vacuum (Box 12.4). Zel’dovich was claiming to foresee, from his electromagnetic analogy, that these gravitational vacuum fluctuations would cause spinning black holes to radiate. “But how?” I asked, puzzled.
Zel’dovich bounded to his feet, strode to a one-meter-square blackboard on the wall opposite his map, and began drawing a sketch and talking at the same time. His sketch (Figure 12.1) showed a wave flowing toward a spinning object, skimming around its surface for a while, and then flowing away. The wave might be electromagnetic and the spinning body a metal sphere, Zel’dovich explained, or the wave might be gravitational and the body a black hole.
Box 12.4
Vacuum Fluctuations
Vacuum fluctuations are, for electromagnetic and gravitational waves, what “claustrophobic degeneracy motions” are for electrons.
Recall (Chapter 4) that if one confines an electron to a small region of space, then no matter how hard one tries to slow it to a stop, the laws of quantum mechanics force the electron to continue moving randomly, unpredictably. This is the claustrophobic degeneracy motion that produces the pressure by which white-dwarf stars support themselves against their own gravitational squeeze.
Similarly, if one tries to remove all electromagnetic or gravitational oscillations from some region of space, one will never succeed. The laws of quantum mechanics insist that there always remain some random, unpredictable oscillations, that is, some random, unpredictable electromagnetic and gravitational waves. These are the vacuum fluctuations that (according to Zel’dovich) will “tickle” a spinning metal sphere or black hole and cause it to radiate.
These vacuum fluctuations cannot be stopped by removing their energy, because they contain, on average, no energy at all. At some locations and some moments of time they have positive energy that has been “borrowed” from other locations, and those other locations, as a result, have negative energy. Just as banks will not let customers maintain negative bank balances for long, so the laws of physics force the regions of negative energy to quickly suck energy out of their positive-energy neighbors, thereby restoring themselves to a zero or positive balance. This continual, random, borrowing and returning of energy is what drives the vacuum fluctuations.
Just as an electron’s degeneracy motions become more vigorous when one confines the electron to a smaller and smaller region (Chapter 4), so also the vacuum fluctuations of electromagnetic and gravitational waves are more vigorous in small regions than in large, that is, more vigorous for small wavelengths than for large. This, as we shall see in Chapter 13, has profound consequences for the nature of the singularities at the centers of black holes.
Electromagnetic vacuum fluctuations are well understood and are a common feature of everyday physics. For example, they play a key role in the operation of a fluorescent light tube. An electrical discharge excites mercury vapor atoms in the tube, and then random electromagnetic vacuum fluctuations tickle each excited atom, causing it, at some random time, to emit some of its excitation energy as an electromagnetic wave (a photon).* This emission is called spontaneous because, when it was first identified as a physical effect, physicists did not realize it was being triggered by vacuum fluctuations. As another example, inside a laser, random electromagnetic vacuum fluctuations interfere with the coherent laser light (interference in the sense of Box 10.3), thereby modulating the laser light in unpredictable ways. This causes the photons emerging from the laser to come out at random, unpredictable times, instead of uniformly one after another—a phenomenon called photon shot noise.
Gravitational vacuum fluctuations, by contrast with electromagnetic, have never yet been seen experimentally. Technology of the 1990s, with great effort, should be able to detect highly energetic gravitational waves from black-hole collisions (Chapter 10), but not the waves’ far weaker vacuum fluctuations.
*This “primary” photon gets absorbed by a phosphor coating on the tube’s walls, which in turn emits “secondary” photons that we see as light.
12.1 Zel’dovich’s mechanism by which vacuum fluctuations cause a spinning body to radiate.
The incoming wave is not a “real” wave, Zel’dovich explained, but rather a vacuum fluctuation. As this fluctuational wave sweeps around the spinning body, it behaves like a line of ice skaters making a turn: The outer skaters must whip around at high speed while the inner ones move much more slowly; similarly, the wave’s outer parts move at a very high speed, the speed of light, while its inner parts move much more slowly than light and, in fact, more slowly than the body’s surface is spinning.8 In such a situation, Zel’dovich asserted, the rapidly spinning body will grab hold of the fluctuational wave and accelerate it, much like a small boy accelerating a slingshot as he swings it faster and faster. The acceleration feeds some of the body’s spin energy into the wave, amplifying it. The new, amplified portion of the wave is a “real wave” with positive total energy, while the original, unamplified portion remains a vacuum fluctuation with zero total energy (Box 12.4). The spinning body has thus used the vacuum fluctuation as a sort of catalyst for creating a real wave, and as a template for the shape of the real wave. This is similar, Zel’dovich pointed out, to the manner in which vacuum fluctuations cause a vibrating molecule to “spontaneously” emit light (Box 12.4).
Zel’dovich told me he had proved that a spinning metal sphere radiates in this way; his proof was based on the laws of quantum electrodynamics—that is, the well-known laws that arise from a marriage of quantum mechanics with Maxwell’s laws of electromagnetism. Though he did not have a similar proof that a spinning black hole will radiate, he was quite sure by analogy that it must. In fact, he asserted, a spinning hole will radiate not only gravitational waves, but also electromagnetic waves (photons9), neutrinos, and all other forms of radiation that can exist in nature.
I was quite sure that Zel’dovich was wrong. Several hours later, with no agreement in sight, Zel’dovich offered me a wager. In the novels of Ernest Hemingway, Zel’dovich had read of White Horse scotch, an elegant and esoteric brand of whisky. If detailed calculations with the laws of physics showed that a spinning black hole radiates, then I was to bring Zel’dovich a bottle of White Horse scotch from America. If the calculations showed that there is no such radiation, Zel’dovich would give me a bottle of fine Georgian cognac.
I accepted the wager, but I knew it would not be settled quickly. To settle it would require understanding the marriage of general relativity and quantum mechanics far more deeply than anyone did in 1971.
Having made the wager, I soon forgot it. I have a lousy memory, and my own research was concentrated elsewhere. Zel’dovich, however, did not forget; several weeks after arguing with me, he wrote down his argument and submitted it for publication. The referee probably would have rejected his manuscript had it come from somebody else; his argument was too heuristic for acceptance. But Zel’dovich’s reputation carried the day; his paper was published—and hardly anyone paid any attention. Black-hole radiation just seemed horribly implausible.
A year later, at the Les. Houches summer school, we “experts” were still ignoring Zel’dovich’s idea. I don’t recall it being mentioned even onece.10
We stayed at the Hotel Rossiya, just off Red Square near the Kremlin Although we ventured out nearly every day to give lectures at one institute or another, or to visit a museum or the opera or ballet, our interactions with Soviet physicists occurred for the most part in the Hawkings’ two-room hotel suite, with its view of St. Basil’s Cathedral. One after another, the Soviet Union’s leading theoretical physicists came to the hotel to pay homage to Hawking and to converse.
Among the physicists who made repeated trips to Hawking’s hotel room were Zel’dovich and his graduate student Alexi Starobinsky. Hawking found Zel’dovich and Starobinsky as fascinating as they did him. On one visit, Starobinsky described Zel’dovich’s conjecture that a spinning black hole should radiate, described a partial marriage of quantum mechanics with general relativity that he and Zel’dovich had developed (based on earlier, pioneering work by Bryce DeWitt, Leonard Parker, and others), and then described a proof, using this partial marriage, that a spinning hole does, indeed, radiate. Zel’dovich was well on his way toward winning his bet with me.
Left: Stephen Hawking listening to a lecture at the Les Houches summer school in summer 1972. Right: Yakov Borisovich Zel’dovich at the blackboard in his apartment in Moscow in summer 1971. [Photos by Kip Thorne.]
Of all the things Hawking learned from his conversations in Moscow, this one intrigued him most. However, he was skeptical of the manner in which Zel’dovich and Starobinsky had combined the laws of general relativity with the laws of quantum mechanics, so, after returning to Cambridge, he began to develop his own partial marriage of quantum mechanics and general relativity and use it to test Zel’-dovich’s claim that spinning holes should radiate.
In the meantime, several other physicists in America were doing the same thing, among them William Unruh (a recent student of Wheeler’s) and Don Page (a student of mine). By early 1974 Unruh and Page, each in his own way, had tentatively confirmed Zel’dovich’s prediction: A spinning hole should emit radiation until all of its spin energy has been used up and its emission stops. I would have to concede my bet.
Black Holes Shrink and Explode
Then came a bombshell. Stephen Hawking, first at a conference in England and then in a brief technical article in the journal Nature, announced an outrageous prediction, a prediction that conflicted with Zel’dovich, Starobinsky, Page, and Unruh. Hawking’s calculations confirmed that a spinning black hole must radiate and slow its spin. However, they also predicted that, when the hole stops spinning, its radiation does not stop. With no spin left, and no spin energy left, the hole keeps on emitting radiation of all sorts (gravitational, electromagnetic, neutrino), and as it emits, it keeps on losing energy. Whereas the spin energy was stored in the swirl of space outside the horizon, the energy now being lost could come from only one place: from the hole’s interior!
Equally amazing, Hawking’s calculations predicted that the spectrum of the radiation (that is, the amount of energy radiated at each wavelength) is precisely like the spectrum of thermal radiation from a hot body. In other words, a black hole behaves precisely as though its horizon has a finite temperature, and that temperature, Hawking concluded, is proportional to the hole’s surface gravity. This (if Hawking was right) was incontrovertible proof that the Bardeen—Carter—Hawking laws of black-hole mechanics are the laws of thermodynamics in disguise, and that, as Bekenstein had claimed two years earlier, a black hole has an entropy proportional to its surface area.
Hawking’s calculations said more. Once the hole’s spin has slowed, its entropy and the area of its horizon are proportional to its mass squared, while its temperature and surface gravity are proportional to its mass divided by its area, which means inversely proportional to its mass. Therefore, as the hole continues to emit radiation, converting mass into outflowing energy, its mass goes down, its entropy and area go down, and its temperature and surface gravity go up. The hole shrinks and becomes hotter. In effect, the hole is evaporating.
A hole that has recently formed by stellar implosion (and that thus has a mass larger than about 2 Suns) has a very low temperature: less than 3 × 10−8 degree above absolute zero (0.03 microkelvin). Therefore, the evaporation at first is very slow; so slow that the hole will require longer than 1067 years (1057 times the present age of the Universe) to shrink appreciably. However, as the hole shrinks and heats up, it will radiate more strongly and its evaporation will quicken. Finally, when the hole’s mass has been reduced to somewhere between a thousand tons and 100 million tons (we are not sure where), and its horizon has shrunk to a fraction the size of an atomic nucleus, the hole will be so extremely hot (between a trillion and 100,000 trillion degrees) that it will explode violently, in a fraction of a second.
The world’s dozen experts on the partial marriage of general relativity with quantum theory were quite sure that Hawking had made a mistake. His conclusion violated everything then known about black holes. Perhaps his partial marriage, which differed from other people’s, was wrong; or perhaps he had the right marriage, but had made a mistake in his calculations.
For the next several years the experts minutely examined Hawking’s version of the partial marriage and their own versions, Hawking’s calculations of the waves from black holes and their own calculations. Gradually one expert after another came to agree with Hawking, and in the process they firmed up the partial marriage, producing a new set of physical laws. The new laws are called the laws of quantum fields in curved spacetime because they come from a partial marriage in which the black hole is regarded as a non—quantum mechanical, general relativistic, curved spacetime object, while the gravitational waves, electromagnetic waves, and other types of radiation are regarded as quantum fields—in other words, as waves that are subject to the laws of quantum mechanics and that therefore behave sometimes like waves and sometimes like particles (see Box 4.1). [A full marriage of general relativity and quantum theory, that is, the fully correct laws of quantum gravity, would treat everything, including the hole’s curved space-time, as quantum mechanical, that is, as subject to the uncertainty principle (Box 10.2), to wave/particle duality (Box 4.1), and to vacuum fluctuations (Box 12.4). We shall meet this full marriage and some of its implications in the next chapter.]
How was it possible to reach agreement on the fundamental laws of quantum fields in curved spacetime without any experiments to guide the choice of the laws? How could the experts claim near certainty that Hawking was right without experiments to check their claims? Their near certainty came from the requirement that the laws of quantum fields and the laws of curved spacetime be meshed in a totally consistent way. (If the meshing were not totally consistent, then the laws of physics, when manipulated in one manner, might make one prediction, for example, that black holes never radiate, and when manipulated in another manner, might make a different prediction, for example, that black holes must always radiate. The poor physicists, not knowing what to believe, might be put out of business.)
The new, meshed laws had to be consistent with general relativity’s laws of curved spacetime in the absence of quantum fields and with the laws of quantum fields in the absence of spacetime curvature. This and the demand for a perfect mesh, analogous to the demand that the rows and columns of a crossword puzzle mesh perfectly, turned out to determine the form of the new laws almost11 completely. If the laws could be meshed consistently at all (and they must be, if the physicists’ approach to understanding the Universe makes any sense), then they could be meshed only in the manner described by the new, agreed-upon laws of quantum fields in curved spacetime.
The requirement that the laws of physics mesh consistently is often used as a tool in the search for new laws. However, this consistency requirement has rarely exhibited such great power as here, in the arena of quantum fields in curved spacetime. For example, when Einstein was developing his laws of general relativity (Chapter 2), considerations of consistency could not and did not tell him his starting premise, that gravity is due to a curvature of spacetime; this starting premise came largely from Einstein’s intuition. However, with this premise in hand, the requirement that the new general relativistic laws mesh consistently with Newton’s laws of gravity when gravity is weak, and with the laws of special relativity when there is no gravity at all, determined the forms of the new laws almost uniquely; for example, it was the key to Einstein’s discovery of his field equation.
In September 1975,I returned to Moscow for my fifth visit, bearing a bottle of White Horse scotch for Zel’dovich. To my surprise, I discovered that, although all the Western experts by now had agreed that Hawking was right and black holes can evaporate, nobody in Moscow believed Hawking’s calculations or conclusions. Although several confirmations of Hawking’s claims, derived by new, completely different methods, had been published during 1974 and 1975, those confirmations had had little impact in the U.S.S.R. Why? Because Zel’dovich and Starobinsky, the greatest Soviet experts, were disbelievers: They continued to maintain that, after a radiating black hole has lost all its spin, it must stop radiating, and it therefore cannot evaporate completely. I argued with Zel’dovich and Starobinsky, to no avail; they knew so much more about quantum fields in curved spacetime than I that although (as usual) I was quite sure I had truth on my side, I could not counter their arguments.
My return flight to America was scheduled for Tuesday, 23 September. On Monday evening, as I was packing my bags in my tiny room at the University Hotel, the telephone rang. It was Zel’dovich: “Come to my flat, Kip! I want to talk about black-hole evaporation!” Tight for time, I sought a taxi in front of the hotel. None was in sight, so in standard Muscovite fashion I flagged down a passing motorist and offered him five rubles to take me to Number 2B Vorobyevskoye Shosse. He nodded agreement and we were off, down back streets I had never traveled. My fear of being lost abated when we swung onto Vorobyevskoye Shosse. With a grateful “Spasibo!” I alighted in front of 2B, jogged through the gate and forested grounds, into the building, and up the stairs to the second floor, southwest corner.
Zel’dovich and Starobinsky greeted me at the door, grins on their faces and their hands above their heads. “We give up; Hawking is right; we were wrong!” For the next hour they described to me how their version of the laws of quantum fields in a black hole’s curved spacetime, while seemingly different from Hawking’s, was really completely equivalent. They had concluded black holes cannot evaporate because of an error in their calculations, not because of wrong laws. With the error corrected, they now agreed. There is no escape. The laws require that black holes evaporate.
There are several different ways to picture black-hole evaporation, corresponding to the several different ways to formulate the laws of quantum fields in a black hole’s curved spacetime. However, all the ways acknowledge vacuum fluctuations as the ultimate source of the outflowing radiation. Perhaps the simplest pictorial description is one based on particles rather than waves:
Vacuum fluctuations, like “real,” positive-energy waves, are subject to the laws of wave/particle duality (Box 4.1); that is, they have both wave aspects and particle aspects. The wave aspects we have met already (Box 12.4): The waves fluctuate randomly and unpredictably, with positive energy momentarily here, negative energy momentarily there, and zero energy on average. The particle aspect is embodied in the concept of virtual particles, that is, particles that flash into existence in pairs (two particles at a time), living momentarily on fluctuational energy borrowed from neighboring regions of space, and that then annihilate and disappear, giving their energy back to the neighboring regions. For electromagnetic vacuum fluctuations, the virtual particles are virtual photons; for gravitational vacuum fluctuations, they are virtual gravitons.12
12.2 The mechanism of black-hole evaporation, as viewed by someone who is falling into the hole. Left: A black hole’s tidal gravity pulls a pair of virtual photons apart, thereby feeding energy into them. Right: The virtual photons have acquired enough energy from tidal gravity to materialize, permanently, into real photons, one of which escapes from the hole while the other falls toward the hole’s center.
Tidal gravity near the horizon is very strong; it pulls the virtual photons apart with a huge force, thereby feeding great energy into them, as seen by the infalling observer who is halfway between the photons. The increase in photon energy is sufficient, by the time the photons are a quarter of a horizon circumference apart, to convert them into real long-lived photons (right half of Figure 12.2), and have enough energy left over to give back to the neighboring, ‘negative-energy regions of space. The photons, now real, are liberated from each other. One is inside the horizon and lost forever from the external Universe. The other escapes from the hole, carrying away the energy (that is, the mass13) that the hole’s tidal gravity gave to it. The hole, with its mass reduced, shrinks a bit.
This mechanism of emitting particles does not depend at all on the fact that the particles were photons, and their associated waves were electromagnetic. The mechanism will work equally well for all other forms of particle/wave (that is, for all other types of radiation—gravitational, neutrino, and so forth), and therefore a black hole radiates all types of radiation.
Before the virtual particles have materialized into real particles, they must stay closer together than roughly the wavelength of their waves. To acquire enough energy from the hole’s tidal gravity to materialize, however, they must get as far apart as about a quarter of the circumference of the hole. This means that the wavelengths of the particle/waves that the hole emits will be about one-fourth the hole’s circumference in size, and larger.
A black hole with mass twice as large as the Sun has a circumference of about 35 kilometers, and thus the particle/waves that it emits have wavelengths of about 9 kilometers and larger. These are enormous wavelengths compared to light or ordinary radio waves, but not much different from the lengths of the gravitational waves that the hole would emit if it were to collide with another hole.
During the early years of his career, Hawking tried to be very careful and rigorous in his research. He never asserted things to be true unless he could give a nearly airtight proof of them. However, by 1974 he had changed his attitude: “I would rather be right than rigorous,” he told me firmly. Achieving high rigor requires much time. By 1974 Hawking had set for himself goals of understanding the full marriage of general relativity with quantum mechanics, and understanding the origin of the Universe—goals that to achieve would require enormous amounts of time and concentration. Perhaps feeling more finite than other people feel because of his life-shortening disease, Hawking felt he could not afford to dally with his discoveries long enough to achieve high rigor, nor could he afford to explore all the important features of his discoveries. He must push on at high speed.
Thus it was that Hawking, in 1974, having proved firmly that a black hole radiates as though it had a temperature proportional to its surface gravity, went on to assert, without real proof, that all of the other similarities between the laws of black-hole mechanics and the laws of thermodynamics were more than a coincidence: The black-hole laws are the same thing as the thermodynamic laws, but in disguise. From this assertion and his firmly proved relationship between temperature and surface gravity, Hawking inferred a precise relationship between the hole’s entropy and its surface area: The entropy is 0.10857 ... times14 the surface area, divided by the Planck–Wheeler area. In other words, a 10-solar-mass, nonspinning hole has an entropy of 4.6 × 1078, which is approximately the same as Bekenstein’s conjecture.
Bekenstein, of course, was sure Hawking was right, and he glowed with pleasure. By the end of 1975, Zel’dovich, Starobinsky, I, and Hawking’s other colleagues were also strongly inclined to agree. However, we would not feel fully satisfied until we understood the precise nature of a black hole’s enormous randomness. There must be 104.6 × 1078 ways to distribute something inside the black hole, without changing its external appearance (its mass, angular momentum, and charge), but what was that something? And how, in simple physical terms, could one understand the thermal behavior of a black hole—the fact that the hole behaves just like an ordinary body with temperature? As Hawking moved on to research on quantum gravity and the origin of the Universe, Paul Davies, Bill Unruh, Robert Wald, James York, I, and many others of his colleagues zeroed in on these issues. Gradually over the next ten years we arrived at the new understanding embodied in Figure 12.3.
Figure 12.3a depicts a black hole’s vacuum fluctuations, as viewed by observers falling inward through the horizon. The vacuum fluctuations consist of pairs of virtual particles. Occasionally tidal gravity manages to give one of the plethora of pairs sufficient energy for its two virtual particles to become real, and for one of them to escape from the hole. This was the viewpoint on vacuum fluctuations and black-hole evaporation discussed in Figure 12.2.
Figure 12.3b depicts a different viewpoint on the hole’s vacuum fluctuations, the viewpoint of observers who reside just above the hole’s horizon and are forever at rest relative to the horizon. To prevent themselves from being swallowed by the hole, such observers must accelerate hard, relative to falling observers—using a rocket engine or hanging by a rope. For this reason, these observers’ viewpoint is called the “accelerated viewpoint.” It is also the viewpoint of the “membrane paradigm” (Chapter 11).
Surprisingly, from the accelerated viewpoint, the vacuum fluctuations consist not of virtual particles flashing in and out of existence, but rather of real particles with positive energies and long lives; see Box 12.5. The real particles form a hot atmosphere around the hole, much like the atmosphere of the Sun. Associated with these real particles are real waves. As a particle moves upward through the atmosphere, gravity pulls on it, reducing its energy of motion; correspondingly, as a wave moves upward, it becomes gravitationally redshifted to longer and longer wavelengths (Figure 12.3b).
12.3 (a) Observers falling into a black hole (the two little men in space suits) see vacuum fluctuations near the hole’s horizon to consist of pairs of virtual particles, (b) As viewed by observers just above the horizon and at rest relative to the horizon (the little man hanging by a rope and the little man blasting his rocket engine), the vacuum fluctuations consist of a hot atmosphere of real particles; this is the “accelerated viewpoint.” (c) The atmosphere’s particles, in the accelerated viewpoint, appear to be emitted by a hot, membrane-like horizon. They fly upward short distances, and most are then pulled back into the horizon. However, a few of the particles manage to escape the hole’s grip and evaporate into outer space.
Acceleration Radiation
In 1975, Wheeler’s recent student William Unruh, and independently Paul Davies at King’s College, London, discovered (using the laws of quantum fields in curved spacetime) that accelerated observers just above a black hole’s horizon must see the vacuum fluctuations there not as virtual pairs of particles but rather as an atmosphere of real particles, an atmosphere that Unruh called “acceleration radiation.”
This startling discovery revealed that the concept of a real particle is relative, not absolute; that is, it depends on one’s reference frame. Observers in freely falling frames who plunge through the hole’s horizon see no real particles outside the horizon, only virtual ones. Observers in accelerated frames who, by their acceleration, remain always above the horizon see a plethora of real particles.
How is this possible? How can one observer claim that the horizon is surrounded by an atmosphere of real particles and the other that it is not? The answer lies in the fact that the virtual particles’ vacuum fluctuational waves are not confined solely to the region above the horizon; part of each fluctuational wave is inside the horizon and part is outside.
•The freely falling observers, who plunge through the horizon, can see both parts of the vacuum fluctuational wave, the part inside the horizon and the part outside; so such observers are well aware (by their measurements) that the wave is a mere vacuum fluctuation and correspondingly that its particles are virtual, not real.
•The accelerated observers, who remain always outside the horizon, can see only the outside part of the vacuum fluctuational wave, not the inside part; and correspondingly, by their measurements they are unable to discern that the wave is a mere vacuum fluctuation accompanied by virtual particles. Seeing only a part of the fluctuational wave, they mistake it for “the real thing”—a real wave accompanied by real particles, and as a result their measurements reveal all around the horizon an atmosphere of real particles.
That this atmosphere’s real particles can gradually evaporate and fly off into the external Universe (Figure 12.3c) is an indication that the viewpoint of the accelerated observers is just as correct, that is, just as valid, as that of the freely falling observers: What the freely falling observers see as virtual pairs converted into real particles by tidal gravity, followed by evaporation of one of the real particles, the accelerated observers see simply as the evaporation of one of the particles that was always real and always populated the black hole’s atmosphere. Both viewpoints are correct; they are the same physical situation, seen from two different reference frames.
Figure 12.3c shows the motion of a few of the particles in a black-hole atmosphere, from the accelerated viewpoint. The particles appear to be emitted by the horizon; most fly upward a short distance and are then pulled back down to the horizon by the hole’s strong gravity, but a few manage to escape the hole’s grip. The escaping particles are the same ones as the infalling observers see materialize from virtual pairs (Figure 12.3a). They are Hawking’s evaporating particles.
From the accelerated viewpoint, the horizon behaves like a high-temperature, membrane-like surface; it is the membrane of the “membrane paradigm” described in Chapter 11. Just as the Sun’s hot surface emits particles (for example, the photons that make daylight on Earth), so the horizon’s hot membrane emits particles: the particles that make up the hole’s atmosphere, and the few that evaporate. The gravitational redshift reduces the particles’ energy as they fly upward from the membrane, so although the membrane itself is extremely hot, the evaporating radiation is much cooler.
The accelerated viewpoint not only explains the sense in which a black hole is hot, it also accounts for the hole’s enormous randomness. The following thought experiment (invented by me and my postdoc, Wojciech Zurek) explains how.
Throw into a black hole’s atmosphere a small amount of material containing some small amount of energy (or, equivalently, mass), angular momentum (spin), and electric charge. From the atmosphere this material will continue on down through the horizon and into the hole. Once the material has entered the hole, it is impossible by examining the hole from outside to learn the nature of the injected material (whether it consisted of matter or of antimatter, of photons and heavy atoms, or of electrons and positrons), and it is impossible to learn just where the material was injected. Because a black hole has no “hair,” all one can discover, by examining the hole from outside, are the total amounts of mass, angular momentum, and charge that entered the atmosphere.
Ask how many ways those amounts of mass, angular momentum, and charge could have been injected into the hole’s hot atmosphere. This question is analogous to asking how many ways the child’s toys could have been distributed over the tiles in the playroom of Box 12.3, and correspondingly, the logarithm of the number of ways to inject must be the increase in the atmosphere’s entropy, as described by the standard laws of thermodynamics. By a fairly simple calculation, Zurek and I were able to show that this increase in thermodynamic entropy is precisely equal to ¼ times the increase in the horizon’s area, divided by the Planck–Wheeler area; that is, it is precisely the increase in the horizon’s area in disguise, the same disguise that Hawking inferred, in 1974, from the mathematical similarity of the laws of black-hole mechanics and the laws of thermodynamics.
The outcome of this thought experiment can be expressed succinctly as follows: A black hole’s entropy is the logarithm of the number of ways that the hole could have been made. This means that there are 104.6 × 1078 different ways to make a 10-solar-mass black hole whose entropy is 4.6 × 1078. This explanation of the entropy was originally conjectured by Bekenstein in 1972, and a highly abstract proof was given by Hawking and his former student, Gary Gibbons, in 1977.
The thought experiment also shows the second law of thermodynamics in action. The energy, angular momentum, and charge that one throws into the hole’s atmosphere can have any form at all; for example, they might be the roomful of air wrapped up in a bag, which we met earlier in this chapter while puzzling over the second law. When the bag is thrown into the hole’s atmosphere, the entropy of the external Universe is reduced by the amount of entropy (randomness) in the bag. However, the entropy of the hole’s atmosphere, and thence of the hole, goes up by more than the bag’s entropy, so the total entropy of hole plus external Universe goes up. The second law of thermodynamics is obeyed.
Similarly, it turns out, when the black hole evaporates some particles, its own surface area and entropy typically go down; but the particles get distributed randomly in the external Universe, increasing its entropy by more than the hole’s entropy loss. Again the second law is obeyed.
How long does it take for a black hole to evaporate and disappear? The answer depends on the hole’s mass. The larger the hole, the lower its temperature, and thus the more weakly it emits particles and the more slowly it evaporates. The total lifetime, as worked out by Don Page in 1975 when he was jointly my student and Hawking’s, is 1.2 × 1067 years if the hole’s mass is twice that of the Sun. The lifetime is proportional to the cube of the hole’s mass, so a 20-solar-mass hole has a life of 1.2 × 1070 years. These lifetimes are so enormous compared to the present age of the Universe, about 1 × 1010 years, that the evaporation is totally irrelevant for astrophysics. Nevertheless, the evaporation has been very important for our understanding of the marriage between general relativity and quantum mechanics; the struggle to understand the evaporation taught us the laws of quantum fields in curved spacetime.
Holes far less massive than 2 Suns, if they could exist, would evaporate far more rapidly than 1067 years. Such small holes cannot be formed in the Universe today because degeneracy pressures and nuclear pressures prevent small masses from imploding, even if one squeezes them with all the force the present-day Universe can muster (Chapters 4 and 5). However, such holes might have formed in the big bang, where matter experienced densities and pressures and gravitational squeezes that were enormously higher than in any modern-day star.
Detailed calculations by Hawking, Zel’dovich, Novikov, and others have shown that tiny lumps in the matter emerging from the big bang could have produced tiny black holes, if the lumps’ matter had a rather soft equation of state (that is, had only small increases of pressure when squeezed). Powerful squeezing by other, adjacent matter in the very early Universe, like the squeezing of carbon in the jaws of a powerful anvil to form diamond, could have made the tiny lumps implode to produce tiny holes.
A promising way to search for such tiny primordial black holes is by searching for the particles they produce when they evaporate. Black holes weighing less than about 500 billion kilograms (5 × 1014 grams, the weight of a modest mountain) should have evaporated completely away by now, and black holes a few times heavier than this should still be evaporating strongly. Such black holes have horizons about the size of an atomic nucleus.
A large portion of the energy emitted in the evaporation of such holes should now be in the form of gamma rays (high-energy photons) traveling randomly through the Universe. Such gamma rays do exist, but in amounts and with properties that are readily explained in other ways. The absence of excess gamma rays tells us (according to calculations by Hawking and Page) that there now are no more than about 300 tiny, strongly evaporating black holes in each cubic light-year of space; and this, in turn, tells us that matter in the big bang cannot have had an extremely soft equation of state.
1. It might seem counterintuitive that Hawking’s area-increase theorem permits any of the holes’ mass at all to be emitted as gravitational waves. Readers comfortable with algebra may find satisfaction in the example of two nonspinning holes that coalesce to produce a single, larger nonspinning hole. The surface area of a nonspinning hole is proportional to the square of its horizon circumference, which in turn is proportional to the square of the hole’s mass. Thus, Hawking’s theorem insists that the sum of the squares of the initial holes’ masses must be less than the square of the final hole’s mass. A little algebra shows that this constraint on the masses permits the final hole’s mass to be less than the sum of the initial holes’ masses, and thus permits some of the initial masses to be emitted as gravitational waves.
2. A more precise definition of the apparent horizon is given in Box 12.1 below.
3. The laws of quantum mechanics guarantee that the number of ways to distribute the atoms and molecules is always finite, and never infinite. In defining the entropy, physicists often multiply the logarithm of this number of ways by a constant that will be irrelevant to us, loge10 × k, where loge10 is the “natural logarithm” of 10, that is, 2.30258 . . . , and k is “Boltzmann’s constant,” 1.38062 × 10−16 erg per degree Celsius. Throughout this book I shall ignore this constant.
4. This Planck-Wheeler area is given by the formula Għ/c3, where G = 6.670 × 10−8 dyne-cehtimeter2/gram2 is Newton’s gravitation constant, ħ = 1.055 × 1 0-27 erg-second is Planck’s quantum mechanical constant, and c = 2.998 × 1010 centimeter/second is the speed of light. For related issues, see Footnote 2 in Chapter 13, Footnote 6 in Chapter 14, and the associated discussions in the text of those chapters.
5. The logarithm of 101079 is 1079 (Bekenstein’s conjectured entropy). Note that 101079 is a 1 with 1079 zeroes after it, that is, with nearly as many zeroes as there are atoms in the Universe.
6. Vorobyevskoye Shosse has since been renamed Kosygin Street, and its buildings have been renumbered. In the late 1980s Mikhail Gorbachev had a home at Number 10, several doors west of Zel’dovich.
7. See the last section of Chapter 1: “The Nature of Physical Law.”
8. In technical language, the outer parts are in the “radiation zone” while the inner parts are in the “near zone.”
9. Recall that photons and electromagnetic waves are different aspects of the same thing; see the discussion of wave/particle duality in Box 4.1.
10. This lack of interest was all the more remarkable because in the meantime, Charles Misner in America had shown that real waves (as opposed to Zel’dovich’s vacuum fluctuations) can be amplified by a spinning hole in a manner analogous to Figure 12.1, and this amplification—to which Misner gave the name “superradiance”-—-was generating great interest.
11. The “almost” takes care of certain ambiguities in a procedure called “renormalization,” by which one computes the net energy carried by vacuum fluctuations. These ambiguities, which were identified and codified by Robert Wald (a former student of Wheeler’s), do not influence a black hole’s evaporation, and they probably will not be resolved until the full quantum theory of gravity is in hand.
12. Some readers may already be familiar with these concepts in the context of matter and antimatter, for example, an electron (which is a particle of matter) and a positron (its antiparticle). Just as the electromagnetic field is the field aspect of a photon, so also there exists an electron field which is the field aspect of the electron and the positron. At locations where the electron field’s vacuum fluctuations are momentarily large, a virtual electron and a virtual positron are likely to flash into existence, as a pair; when the field fluctuates down, the electron and positron are likely to annihilate each other and disappear. The photon is its own antiparticle, so virtual photons flash in and out of existence in pairs, and similarly for gravitons
13.Recall that, since mass and energy are totally convertible into each other, they are really just different names for the same concept.
14. The peculiar factor 0.10857 ... is actually 1/(4logel0), where loge10 = 2.30258 . . . results from my choice of “normalization” of the entropy; see Footnote 3 on page 423.