From Eternity to Here (50 page)

HAWKING RADIATION

Along with Wheeler’s group at Princeton, the best work in general relativity in the early 1970s was being done in Great Britain. Stephen Hawking and Roger Penrose, in particular, were inventing and applying new mathematical techniques to the study of curved spacetime. Out of these investigations came the celebrated singularity theorems—when gravity becomes sufficiently strong, as in black holes or near the Big Bang, general relativity necessarily predicts the existence of singularities—as well as Hawking’s result that the area of black-hole event horizons would never decrease.

So Hawking paid close attention to Bekenstein’s work, but he wasn’t very happy with it. For one thing, if you’re going to take the analogy between area and entropy seriously, you should take the other parts of the thermodynamics/black-hole-mechanics analogy just as seriously. In particular, the surface gravity of a black hole (which is large for small black holes with negligible spin and charge, smaller for large black holes or ones with substantial spin or charge) should be proportional to its
temperature
. But that would seem to be, on the face of it, absurd. When you heat things up to high temperature, they glow, like molten metal or a burning flame. But black holes don’t glow; they’re black.
So there,
we can imagine Hawking thinking across the Atlantic.

Inveterate traveler that he is, Hawking visited the Soviet Union in 1973 to talk about black holes. Under the leadership of Yakov Zel’dovich, Moscow featured a group of experts in relativity and cosmology that rivaled those in Princeton or Cambridge. Zel’dovich and his colleague Alexander Starobinsky told Hawking about some work they had done to understand the Penrose process—extracting energy from a rotating black hole—in the light of quantum mechanics. According to the Moscow group, quantum mechanics implied that a spinning black hole would
spontaneously
emit radiation and lose energy; there was no need for a super-advanced civilization to throw things at it.

Hawking was intrigued but didn’t buy the specific arguments that Zel’dovich and Starobinsky had offered.
²¹⁶
So he set out to understand the implications of quantum mechanics in the context of black holes by himself. It’s not a simple problem. “Quantum mechanics” is a very general idea: The space of states consists of wave functions rather than positions and momenta, and you can’t observe the wave function exactly without dramatically altering it. Within that framework, we can think of different types of quantum systems, from individual particles to collections of superstrings. The founders of quantum mechanics focused, sensibly enough, on relatively simple systems, consisting of a small number of atoms moving slowly with respect to one another. That’s still what most physics students learn when they first study quantum mechanics.

When particles become very energetic and start moving at speeds close to the speed of light, we can no longer ignore the lessons of relativity. For one thing, the energy of two particles that collide with each other can become so large that they create multiple new particles, through the miracle of
E
=
mc
²
. Through decades of effort on the part of theoretical physicists, the proper formalism to reconcile quantum mechanics with special relativity was assembled, in the form of “quantum field theory.”

The basic idea of quantum field theory is simple: The world is made of fields, and when we observe the wave functions of those fields, we see particles. Unlike a particle, which exists at some certain point, a field exists everywhere in space; the electric field, the magnetic field, and the gravitational field are all familiar examples. At every single point in space, every field that exists has some particular value (although that value might be zero). According to quantum field theory,
everything
is a field—there is an electron field, various kinds of quark fields, and so on. But when we look at a field, we see particles. When we look at the electric and magnetic fields, for example, we see photons, the particles of electromagnetism. A weakly vibrating electromagnetic field shows up as a small number of photons; a wildly vibrating electromagnetic field shows up as a large number of photons.
²¹⁷

Figure 60:
Fields have a value at every point in space. When we observe a quantum field, we don’t see the field itself, but a collection of particles. A gently oscillating field, at the top, corresponds to a small number of particles; a wildly oscillating field, at the bottom, corresponds to a large number of particles.

Quantum field theory reconciles quantum mechanics with special relativity. This is very different from “quantum gravity,” which would reconcile quantum mechanics with
general
relativity, the theory of gravity and spacetime curvature. In quantum field theory, we imagine that spacetime itself is perfectly classical, whether it’s curved or not; the fields are subject to the rules of quantum mechanics, while spacetime simply acts as a fixed background. In full-fledged quantum gravity, by contrast, we imagine that even spacetime has a wave function and is completely quantum mechanical. Hawking’s work was in the context of quantum field theory in a fixed curved spacetime.

Field theory was not something Hawking was an expert in; despite being lumped with general relativity under the umbrella of “impressive-sounding theories of modern physics that seem inscrutable to outsiders,” the two areas are quite different, and an expert in one might not know much about the other. So he set out to learn. Sir Martin Rees, who is one of the world’s leading theoretical astrophysicists and currently Astronomer Royal of Britain, was at the time a young scientist at Cambridge; like Hawking, he had received his Ph.D. a few years before under the supervision of Dennis Sciama. By this time, Hawking was severely crippled by his disease; he would ask for a book on quantum field theory, and Rees would prop it up in front of him. While Hawking stared silently at the book for hours on end, Rees wondered whether the toll of his condition was simply becoming too much for him.
²¹⁸

Far from it. In fact, Hawking was applying the formalism of quantum field theory to the question of radiation from black holes. He was hoping to derive a formula that would reproduce Zel’dovich and Starobinsky’s result for rotating black holes, but instead he kept finding something unbelievable: Quantum field theory seemed to imply that even nonrotating black holes should radiate. Indeed, they should radiate in exactly the same way as a system in thermal equilibrium at some fixed temperature, with the temperature being proportional to the surface gravity, just as it had been in the analogy between black holes and thermodynamics.

Hawking, much to his own surprise, had proven Bekenstein right. Black holes really do behave precisely as ordinary thermodynamic objects. That means, among other things, that the entropy of a black hole actually is proportional to the area of its event horizon; that connection is not just an amusing coincidence. In fact, Hawking’s calculation (unlike Bekenstein’s argument) allowed him to pinpoint the precise constant of proportionality: ¼. That is, if
L
P
is the Planck length, so
L
P
is the Planck area, the entropy of a black hole is 1/4 of the area of its horizon as measured in units of the Planck area:

S
BH
=
A
/(4
L
P
²
).

You are allowed to imagine that the subscript
BH
stands for “Black Hole” or “Bekenstein-Hawking,” as you prefer. This formula is the single most important clue we have about the reconciliation of gravitation with quantum mechanics.
²¹⁹
And if we want to understand why entropy was small near the Big Bang, we have to understand something about entropy and gravity, so this is a logical starting point.

EVAPORATION

To really understand how Hawking derived the startling result that black holes radiate requires a subtle mathematical analysis of the behavior of quantum fields in curved space. Nevertheless, there is a favorite hand-waving explanation that conveys enough of the essential truth that everyone in the world, including Hawking himself, relies upon it. So why not us?

The primary idea is that quantum field theory implies the existence of “virtual particles” as well as good old-fashioned real particles. We encountered this idea briefly in Chapter Three, when we were talking about vacuum energy. For a quantum field, we might think that the state of lowest energy would be when the field was absolutely constant—just sitting there, not changing from place to place or time to time. If it were a classical field, that would be right, but just as we can’t pin down a particle to one particular position in quantum mechanics, we can’t pin down a field to one particular configuration in quantum field theory. There will always be some intrinsic uncertainty and fuzziness in the value of the quantum field. We can think of this inherent jitter in the quantum field as particles popping in and out of existence, one particle and one antiparticle at a time, so rapidly that we can’t observe them. These virtual particles can never be detected directly; if we see a particle, we know it’s a real one, not a virtual one. But virtual particles can interact with real (non-virtual) ones, subtly altering their properties, and those effects have been observed and studied in great detail. They really are there.

What Hawking figured out is that the gravitational field of a black hole can turn virtual particles into real ones. Ordinarily, virtual particles appear in pairs: one particle and one antiparticle.
²²⁰
They appear, persist for the briefest moment, and then annihilate, and no one is the wiser. But a black hole changes things, due to the presence of the event horizon. When a virtual particle/antiparticle pair pops into existence very close to the horizon, one of the partners can fall in, and obviously has no choice but to continue on to the singularity. The other partner, meanwhile, is now able to escape to infinity. The event horizon has served to rip apart the virtual pair, gobbling up one of the particles. The one that escapes is part of the Hawking radiation.

At this point a crucial property of virtual particles comes into play: Their energy can be anything at all. The total energy of a virtual particle/antiparticle pair is exactly zero, since they must be able to pop into and out of the vacuum. For real particles, the energy is equal to the mass times the speed of light squared when the particle is at rest, and grows larger if the particle is moving; consequently, it can never be negative. So if the real particle that escapes the black hole has positive energy, and the total energy of the original virtual pair was zero, that means the partner that fell into the black hole must have a
negative
energy. When it falls in, the total mass of the black hole goes down.

Eventually, unless extra energy is added from other sources, a black hole will evaporate away entirely. Black holes are not, as it turns out, places were time ends once and for all; they are objects that exist for some period of time before they eventually disappear. In a way, Hawking radiation has made black holes a lot more ordinary than they seemed to be in classical general relativity.

Figure 61:
Hawking radiation. In quantum field theory, virtual particles and antiparticles are constantly popping in and out of the vacuum. But in the vicinity of a black hole, one of the particles can fall into the event horizon, while the other escapes to the outside world as Hawking radiation.

An interesting feature of Hawking radiation is that
smaller
black holes are
hotter
. The temperature is proportional to the surface gravity, which is greater for less massive black holes. The kinds of astrophysical black holes we’ve been talking about, with masses equal to or much greater than that of the Sun, have extremely low Hawking temperatures; in the current universe, they are not evaporating at all, as they are taking in a lot more energy from objects around them than they are losing energy from Hawking radiation. That would be true even if the only external source of energy were the cosmic microwave background, at a temperature of about 3 Kelvin. In order for a black hole to be hotter than the microwave background is today, it would have to be less than about 10
¹⁴
kilograms—about the mass of Mt. Everest, and much smaller than any known black hole.
²²¹
Of course, the microwave background continues to cool down as the universe expands; so if we wait long enough, the black holes will be hotter than the surrounding universe, and begin to lose mass. As they do so, they heat up, and lose mass even faster; it’s a runaway process and, once the black hole has been whittled down to a very small size, the end comes quickly in a dramatic explosion.