《时间简史―从大爆炸到黑洞》 ――史蒂芬?霍金

A Brief History of Time - Stephen Hawking... Chapter 6
the body that has collapsed must be lost when a black hole is formed, because afterward all we can possibly measure
about the body is its mass and rate of rotation. The significance of this will be seen in the next chapter.
Black holes are one of only a fairly small number of cases in the history of science in which a theory was developed in
great detail as a mathematical model before there was any evidence from observations that it was correct. Indeed, this
used to be the main argument of opponents of black holes: how could one believe in objects for which the only
evidence was calculations based on the dubious theory of general relativity? In 1963, however, Maarten Schmidt, an
astronomer at the Palomar Observatory in California, measured the red shift of a faint starlike object in the direction of
the source of radio waves called 3C273 (that is, source number 273 in the third Cambridge catalogue of radio sources).
He found it was too large to be caused by a gravitational field: if it had been a gravitational red shift, the object would
have to be so massive and so near to us that it would disturb the orbits of planets in the Solar System. This suggested
that the red shift was instead caused by the expansion of the universe, which, in turn, meant that the object was a very
long distance away. And to be visible at such a great distance, the object must be very bright, must, in other words, be
emitting a huge amount of energy. The only mechanism that people could think of that would produce such large
quantities of energy seemed to be the gravitational collapse not just of a star but of a whole central region of a galaxy. A
number of other similar “quasi-stellar objects,” or quasars, have been discovered, all with large red shifts. But they are
all too far away and therefore too difficult to observe to provide conclusive evidence of black holes.
Further encouragement for the existence of black holes came in 1967 with the discovery by a research student at
Cambridge, Jocelyn Bell-Burnell, of objects in the sky that were emitting regular pulses of radio waves. At first Bell and
her supervisor, Antony Hewish, thought they might have made contact with an alien civilization in the galaxy! Indeed, at
the seminar at which they announced their discovery, I remember that they called the first four sources to be found
LGM 1

Figure 6:2
The best explanation for this phenomenon is that matter has been blown off the surface of the visible star. As it falls
toward the unseen companion, it develops a spiral motion (rather like water running out of a bath), and it gets very hot,
emitting X-rays Figure 6:3.
Figure 6:3
For this mechanism to work, the unseen object has to be very small, like a white dwarf, neutron star, or black hole.
From the observed orbit of the visible star, one can determine the lowest possible mass of the unseen object. In the
case of Cygnus X-l, this is about six times the mass of the sun, which, according to Chandrasekhar’r result, is too great
for the unseen object to be a white dwarf. It is also too large a mass to be a neutron star. It seems, therefore, that it
must be a black hole.
There are other models to explain Cygnus X-1 that do not include a black hole, but they are all rather far-fetched. A
black hole seems to be the only really natural explanation of the observations. Despite this, I had a bet with Kip Thorne
of the California Institute of Technology that in fact Cygnus X-1 does not contain a black hole! This was a form f
insurance policy for me. I have done a lot of work on black holes, and it would all be wasted if it turned out that black
holes do not exist. But in that case, I would have the consolation of winning my bet, which would bring me four years of
the magazine Private Eye. In fact, although the situation with Cygnus X-1 has not changed much since we made the bet
in 1975, there is now so much other observational evidence in favor of black holes that I have conceded the bet. I paid
the specified penalty, which was a one-year subscription to Penthouse, to the outrage of Kip’s liberated wife.
We also now have evidence for several other black holes in systems like Cygnus X-1 in our galaxy and in two
neighboring galaxies called the Magellanic Clouds. The number of black holes, however, is almost certainly very much
higher; in the long history of the universe, many stars must have burned all their nuclear fuel and have had to collapse.
The number of black holes may well be greater even than the number of visible stars, which totals about a hundred
thousand million in our galaxy alone. The extra gravitational attraction of such a large number of black holes could
explain why our galaxy rotates at the rate it does: the mass of the visible stars is insufficient to account for this. We also

A Brief History of Time - Stephen Hawking... Chapter 6
have some evidence that there is a much larger black hole, with a mass of about a hundred thousand times that of the
sun, at the center of our galaxy. Stars in the galaxy that come too near this black hole will be torn apart by the
difference in the gravitational forces on their near and far sides. Their remains and gas that is thrown off other stars, will
fall toward the black hole. As in the case of Cygnus X-l, the gas will spiral inward and will heat up, though not as much
as in that case. It will not get hot enough to emit X rays, but it could account for the very compact source of radio waves
and infrared rays that is observed at the galactic center.
It is thought that similar but even larger black holes, with masses of about a hundred million times the mass of the sun,
occur at the centers of quasars. For example, observations with the Hubble telescope of the galaxy known as M87
reveal that it contains a disk of gas 130 light-years across rotating about a central object two thousand million times the
mass of the sun. This can only be a black hole. Matter falling into such a supermassive black hole would provide the
only source of power great enough to explain the enormous amounts of energy that these objects are emitting. As the
matter spirals into the black hole, it would make the black hole rotate in the same direction, causing it to develop a
magnetic field rather like that of the earth. Very high-energy particles would be generated near the black hole by the
in-falling matter. The magnetic field would be so strong that it could focus these particles into jets ejected outward along
the axis of rotation of the black hole, that is, in the directions of its north and south poles. Such jets are indeed observed
in a number of galaxies and quasars. One can also consider the possibility that there might be black holes with masses
much less than that of the sun. Such black holes could not be formed by gravitational collapse, because their masses
are below the Chandrasekhar mass limit: stars of this low mass can support themselves against the force of gravity
even when they have exhausted their nuclear fuel. Low-mass black holes could form only if matter was compressed to
enormous densities by very large external pressures. Such conditions could occur in a very big hydrogen bomb: the
physicist John Wheeler once calculated that if one took all the heavy water in all the oceans of the world, one could
build a hydrogen bomb that would compress matter at the center so much that a black hole would be created. (Of
course, there would be no one left to observe it!) A more practical possibility is that such low-mass black holes might
have been formed in the high temperatures and pressures of the very early universe. Black holes would have been
formed only if the early universe had not been perfectly smooth and uniform, because only a small region that was
denser than average could be compressed in this way to form a black hole. But we know that there must have been
some irregularities, because otherwise the matter in the universe would still be perfectly uniformly distributed at the
present epoch, instead of being clumped together in stars and galaxies.
Whether the irregularities required to account for stars and galaxies would have led to the formation of a significant
number of “primordial” black holes clearly depends on the details of the conditions in the early universe. So if we could
determine how many primordial black holes there are now, we would learn a lot about the very early stages of the
universe. Primordial black holes with masses more than a thousand million tons (the mass of a large mountain) could
be detected only by their gravitational influence on other, visible matter or on the expansion of the universe. However,
as we shall learn in the next chapter, black holes are not really black after all: they glow like a hot body, and the smaller
they are, the more they glow. So, paradoxically, smaller black holes might actually turn out to be easier to detect than
large ones!

A Brief History of Time - Stephen Hawking... Chapter 7
CHAPTER 7
BLACK HOLES AIN’T SO BLACK
Before 1970, my research on general relativity had concentrated mainly on the question of whether or not there had
been a big bang singularity. However, one evening in November that year, shortly after the birth of my daughter, Lucy,
I started to think about black holes as I was getting into bed. My disability makes this rather a slow process, so I had
plenty of time. At that date there was no precise definition of which points in space-time lay inside a black hole and
which lay outside. I had already discussed with Roger Penrose the idea of defining a black hole as the set of events
from which it was not possible to escape to a large distance, which is now the generally accepted definition. It means
that the boundary of the black hole, the event horizon, is formed by the light rays that just fail to escape from the black
hole, hovering forever just on the edge Figure 7:1. It is a bit like running away from the police and just managing to
keep one step ahead but not being able to get clear away!
Figure 7:1
Suddenly I realized that the paths of these light rays could never approach one another. If they did they must
eventually run into one another. It would be like meeting someone else running away from the police in the opposite
direction

A Brief History of Time - Stephen Hawking... Chapter 7
event horizon had always to be moving parallel to, or away from, each other. Another way of seeing this is that the
event horizon, the boundary of the black hole, is like the edge of a shadow

A Brief History of Time - Stephen Hawking... Chapter 7
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
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 regarde
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

A Brief History of Time - Stephen Hawking... Chapter 7
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

A Brief History of Time - Stephen Hawking... Chapter 7
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
difficult, however: the black hole would have the mass of a mountain compressed into less than a million millionth of an
inch, the size of the nucleus of an atom! If you had one of these black holes on the surface of the earth, there would be
no way to stop it from falling through the floor to the center of the earth. It would oscillate through the earth and back,
until eventually it settled down at the center. So the only place to put such a black hole, in which one might use the
energy that it emitted, would be in orbit around the earth

Figure 7:5
With primordial black holes being so scarce, it might seem unlikely that there would be one near enough for us to
observe as an individual source of gamma rays. But since gravity would draw primordial black holes toward any matter,
they should be much more common in and around galaxies. So although the gamma ray background tells us that there
can be no more than 300 primordial black holes per cubic light-year on average, it tells us nothing about how common
they might be in our own galaxy. If they were, say, a million times more common than this, then the nearest black hole
to us would probably be at a distance of about a thousand million kilometers, or about as far away as Pluto, the farthest
known planet. At this distance it would still be very difficult to detect the steady emission of a black hole, even if it was
ten thousand megawatts. In order to observe a primordial black hole one would have to detect several gamma ray
quanta coming from the same direction within a reasonable space of time, such as a week. Otherwise, they might
simply be part of the background. But Planck’s quantum principle tells us that each gamma ray quantum has a very
high energy, because gamma rays have a very high frequency, so it would not take many quanta to radiate even ten
thousand megawatts. And to observe these few coming from the distance of Pluto would require a larger gamma ray
detector than any that have been constructed so far. Moreover, the detector would have to be in space, because
gamma rays cannot penetrate the atmosphere.
Of course, if a black hole as close as Pluto were to reach the end of its life and blow up, it would be easy to detect the
final burst of emission. But if the black hole has been emitting for the last ten or twenty thousand million years, the
chance of it reaching the end of its life within the next few years, rather than several million years in the past or future,
is really rather small! So in order to have a reasonable chance of seeing an explosion before your research grant ran
out, you would have to find a way to detect any explosions within a distance of about one light-year. In fact bursts of