increase whenever matter or radiation fell into the black hole Figure 7:2.
Figures 7:2 & 7:3
Or if two black holes collided and merged together to form a single black hole, the area of the event horizon of the final
black hole would be greater than or equal to the sum of the areas of the event horizons of the original black holes
Figure 7:3. This nondecreasing property of the event horizon’s area placed an important restriction on the possible
behavior of black holes. I was so excited with my discovery that I did not get much sleep that night. The next day I rang
up Roger Penrose. He agreed with me. I think, in fact, that he had been aware of this property of the area. However,
he had been using a slightly different definition of a black hole. He had not realized that the boundaries of the black
hole according to the two definitions would be the same, and hence so would their areas, provided the black hole had
settled down to a state in which it was not changing with time.
The nondecreasing behavior of a black hole’s area was very reminiscent of the behavior of a physical quantity called
entropy, which measures the degree of disorder of a system. It is a matter of common experience that disorder will
tend to increase if things are left to themselves. (One has only to stop making repairs around the house to see that!)
One can create order out of disorder (for example, one can paint the house), but that requires expenditure of effort or
energy and so decreases the amount of ordered energy available.
A precise statement of this idea is known as the second law of thermodynamics. It states that the entropy of an isolated
system always increases, and that when two systems are joined together, the entropy of the combined system is
greater than the sum of the entropies of the individual systems. For example, consider a system of gas molecules in a
box. The molecules can be thought of as little billiard balls continually colliding with each other and bouncing off the
walls of the box. The higher the temperature of the gas, the faster the molecules move, and so the more frequently and
harder they collide with the walls of the box and the greater the outward pressure they exert on the walls. Suppose that
initially the molecules are all confined to the left-hand side of the box by a partition. If the partition is then removed, the
molecules will tend to spread out and occupy both halves of the box. At some later time they could, by chance, all be in
the right half or back in the left half, but it is overwhelmingly more probable that there will be roughly equal numbers in
the two halves. Such a state is less ordered, or more disordered, than the original state in which all the molecules were
in one half. One therefore says that the entropy of the gas has gone up. Similarly, suppose one starts with two boxes,
one containing oxygen molecules and the other containing nitrogen molecules. If one joins the boxes together and
removes the intervening wall, the oxygen and the nitrogen molecules will start to mix. At a later time the most probable
state would be a fairly uniform mixture of oxygen and nitrogen molecules throughout the two boxes. This state would
be less ordered, and hence have more entropy, than the initial state of two separate boxes.
The second law of thermodynamics has a rather different status than that of other laws of science, such as Newton's
law of gravity, for example, because it does not hold always, just in the vast majority of cases. The probability of all the
gas molecules in our first box
found in one half of the box at a later time is many millions of millions to one, but it can happen. However, if one has a
black hole around there seems to be a rather easier way of violating the second law: just throw some matter with a lot
of entropy such as a box of gas, down the black hole. The total entropy of matter outside the black hole would go
down. One could, of course, still say that the total entropy, including the entropy inside the black hole, has not gone
down - but since there is no way to look inside the black hole, we cannot see how much entropy the matter inside it
has. It would be nice, then, if there was some feature of the black hole by which observers outside the black hole could
tell its entropy, and which would increase whenever matter carrying entropy fell into the black hole. Following the
discovery, described above, that the area of the event horizon increased whenever matter fell into a black hole, a
research student at Princeton named Jacob Bekenstein suggested that the area of the event horizon was a measure of
the entropy of the black hole. As matter carrying entropy fell into a black hole, the area of its event horizon would go
up, so that the sum of the entropy of matter outside black holes and the area of the horizons would never go down.
This suggestion seemed to prevent the second law of thermodynamics from being violated in most situations.
However, there was one fatal flaw. If a black hole has entropy, then it ought to also have a temperature. But a body
with a particular temperature must emit radiation at a certain rate. It is a matter of common experience that if one heats
up a poker in a fire it glows red hot and emits radiation, but bodies at lower temperatures emit radiation too; one just
does not normally notice it because the amount is fairly small. This radiation is required in order to prevent violation of
the second law. So black holes ought to emit radiation. But by their very definition, black holes are objects that are not
supposed to emit anything. It therefore seemed that the area of the event horizon of a black hole could not be regarded
as its entropy. In 1972 I wrote a paper with Brandon Carter and an American colleague, Jim Bardeen, in which we
pointed out that although there were many similarities between entropy and the area of the event horizon, there was
this apparently fatal difficulty. I must admit that in writing this paper I was motivated partly by irritation with Bekenstein,
who, I felt, had misused my discovery of the increase of the area of the event horizon. However, it turned out in the end
that he was basically correct, though in a manner he had certainly not expected.
In September 1973, while I was visiting Moscow, I discussed black holes with two leading Soviet experts, Yakov
Zeldovich and Alexander Starobinsky. They convinced me that, according to the quantum mechanical uncertainty
principle, rotating black holes should create and emit particles. I believed their arguments on physical grounds, but I did
not like the mathematical way in which they calculated the emission. I therefore set about devising a better
mathematical treatment, which I described at an informal seminar in Oxford at the end of November 1973. At that time I
had not done the calculations to find out how much would actually be emitted. I was expecting to discover just the
radiation that Zeldovich and Starobinsky had predicted from rotating black holes. However, when I did the calculation, I
found, to my surprise and annoyance, that even non-rotating black holes should apparently create and emit particles at
a steady rate. At first I thought that this emission indicated that one of the approximations I had used was not valid. I
was afraid that if Bekenstein found out about it, he would use it as a further argument to support his ideas about the
entropy of black holes, which I still did not like. However, the more I thought about it, the more it seemed that the
approximations really ought to hold. But what finally convinced me that the emission was real was that the spectrum of
the emitted particles was exactly that which would be emitted by a hot body, and that the black hole was emitting
particles at exactly the correct rate to prevent violations of the second law. Since then the calculations have been
repeated in a number of different forms by other people. They all confirm that a black hole ought to emit particles and
radiation as if it were a hot body with a temperature that depends only on the black hole’s mass: the higher the mass,
the lower the temperature.
How is it possible that a black hole appears to emit particles when we know that nothing can escape from within its
event horizon? The answer, quantum theory tells us, is that the particles do not come from within the black hole, but
from the “empty” space just outside the black hole’s event horizon! We can understand this in the following way: what
we think of as “empty” space cannot be completely empty because that would mean that all the fields, such as the
gravitational and electromagnetic fields, would have to be exactly zero. However, the value of a field and its rate of
change with time are like the position and velocity of a particle: the uncertainty principle implies that the more
accurately one knows one of these quantities, the less accurately one can know the other. So in empty space the field
cannot be fixed at exactly zero, because then it would have both a precise value (zero) and a precise rate of change
(also zero). There must be a certain minimum amount of uncertainty, or quantum fluctuations, in the value of the field.
One can think of these fluctuations as pairs of particles of light or gravity that appear together at some time, move
apart, and then come together again and annihilate each other. These particles are virtual particles like the particles
that carry the gravitational force of the sun: unlike real particles, they cannot be observed directly with a particle
detector. However, their indirect effects, such as small changes in the energy of electron orbits in atoms, can be
measured and agree with the theoretical predictions to a remarkable degree of accuracy. The uncertainty principle also
predicts that there will be similar virtual pairs of matter particles, such as electrons or quarks. In this case, however,
one member of the pair will be a particle and the other an antiparticle (the antiparticles of light and gravity are the same
as the particles).
Because energy cannot be created out of nothing, one of the partners in a particle/antiparticle pair will have positive
energy, and the other partner negative energy. The one with negative energy is condemned to be a short-lived virtual
particle because real particles always have positive energy in normal situations. It must therefore seek out its partner
and annihilate with it. However, a real particle close to a massive body has less energy than if it were far away,
because it would take energy to lift it far away against the gravitational attraction of the body. Normally, the energy of
the particle is still positive, but the gravitational field inside a black hole is so strong that even a real particle can have
negative energy there. It is therefore possible, if a black hole is present, for the virtual particle with negative energy to
fall into the black hole and become a real particle or antiparticle. In this case it no longer has to annihilate with its
partner. Its forsaken partner may fall into the black hole as well. Or, having positive energy, it might also escape from
the vicinity of the black hole as a real particle or antiparticle Figure 7:4.
Figure 7:4
To an observer at a distance, it will appear to have been emitted from the black hole. The smaller the black hole, the
shorter the distance the particle with negative energy will have to go before it becomes a real particle, and thus the
greater the rate of emission, and the apparent temperature, of the black hole.
The positive energy of the outgoing radiation would be balanced by a flow of negative energy particles into the black
hole. By Einstein’s equation E = mc2 (where E is energy, m is mass, and c is the speed of light), energy is proportional
to mass. A flow of negative energy into the black hole therefore reduces its mass. As the black hole loses mass, the
area of its event horizon gets smaller, but this decrease in the entropy of the black hole is more than compensated for
by the entropy of the emitted radiation, so the second law is never violated.
Moreover, the lower the mass of the black hole, the higher its temperature. So as the black hole loses mass, its
temperature and rate of emission increase, so it loses mass more quickly. What happens when the mass of the black
hole eventually becomes extremely small is not quite clear, but the most reasonable guess is that it would disappear
completely in a tremendous final burst of emission, equivalent to the explosion of millions of H-bombs.
A black hole with a mass a few times that of the sun would have a temperature of only one ten millionth of a degree
above absolute zero. This is much less than the temperature of the microwave radiation that fills the universe (about
2.7o above absolute zero), so such black holes would emit even less than they absorb. If the universe is destined to go
on expanding forever, the temperature of the microwave radiation will eventually decrease to less than that of such a
black hole, which will then begin to lose mass. But, even then, its temperature would be so low that it would take about
a million million million million million million million million million million million years (1 with sixty-six zeros after it) to
evaporate completely. This is much longer than the age of the universe, which is only about ten or twenty thousand
million years (1 or 2 with ten zeros after it). On the other hand, as mentioned in Chapter 6, there might be primordial
black holes with a very much smaller mass that were made by the collapse of irregularities in the very early stages of
the universe. Such black holes would have a much higher temperature and would be emitting radiation at a much
greater rate. A primordial black hole with an initial mass of a thousand million tons would have a lifetime roughly equal
to the age of the universe. Primordial black holes with initial masses less than this figure would already have
completely evaporated, but those with slightly greater masses would still be emitting radiation in the form of X rays and
gamma rays. These X rays and gamma rays are like waves of light, but with a much shorter wavelength. Such holes
hardly deserve the epithet black: they really are white hot and are emitting energy at a rate of about ten thousand
megawatts.
One such black hole could run ten large power stations, if only we could harness its power. This would be rather
