《时间简史―从大爆炸到黑洞》 ――史蒂芬?霍金

A Brief History of Time - Stephen Hawking... Chapter 3
the stars that we can see in our galaxy and other galaxies, the total is less than one hundredth of the amount
required to halt the expansion of the universe, even for the lowest estimate of the rate of expansion. Our galaxy
and other galaxies, however, must contain a large amount of “dark matter” that we cannot see directly, but
which we know must be there because of the influence of its gravitational attraction on the orbits of stars in the
galaxies. Moreover, most galaxies are found in clusters, and we can similarly infer the presence of yet more
dark matter in between the galaxies in these clusters by its effect on the motion of the galaxies. When we add
up all this dark matter, we still get only about one tenth of the amount required to halt the expansion. However,
we cannot exclude the possibility that there might be some other form of matter, distributed almost uniformly
throughout the universe, that we have not yet detected and that might still raise the average density of the
universe up to the critical value needed to halt the expansion. The present evidence therefore suggests that the
universe will probably expand forever, but all we can really be sure of is that even if the universe is going to
recollapse, it won’t do so for at least another ten thousand million years, since it has already been expanding
for at least that long. This should not unduly worry us: by that time, unless we have colonized beyond the Solar
System, mankind will long since have died out, extinguished along with our sun!
All of the Friedmann solutions have the feature that at some time in the past (between ten and twenty thousand
million years ago) the distance between neighboring galaxies must have been zero. At that time, which we call
the big bang, the density of the universe and the curvature of space-time would have been infinite. Because
mathematics cannot really handle infinite numbers, this means that the general theory of relativity (on which
Friedmann’s solutions are based) predicts that there is a point in the universe where the theory itself breaks
down. Such a point is an example of what mathematicians call a singularity. In fact, all our theories of science
are formulated on the assumption that space-time is smooth and nearly fiat, so they break down at the big bang
singularity, where the curvature of space-time is infinite. This means that even if there were events before the
big bang, one could not use them to determine what would happen afterward, because predictability would
break down at the big bang.
Correspondingly, if, as is the case, we know only what has happened since the big bang, we could not
determine what happened beforehand. As far as we are concerned, events before the big bang can have no
consequences, so they should not form part of a scientific model of the universe. We should therefore cut them
out of the model and say that time had a beginning at the big bang.
Many people do not like the idea that time has a beginning, probably because it smacks of divine intervention.
(The Catholic Church, on the other hand, seized on the big bang model and in 1951officially pronounced it to
be in accordance with the Bible.) There were therefore a number of attempts to avoid the conclusion that there
had been a big bang. The proposal that gained widest support was called the steady state theory. It was
suggested in 1948 by two refugees from Nazi-occupied Austria, Hermann Bondi and Thomas Gold, together
with a Briton, Fred Hoyle, who had worked with them on the development of radar during the war. The idea was
that as the galaxies moved away from each other, new galaxies were continually forming in the gaps in
between, from new matter that was being continually created. The universe would therefore look roughly the
same at all times as well as at all points of space. The steady state theory required a modification of general
relativity to allow for the continual creation of matter, but the rate that was involved was so low (about one
particle per cubic kilometer per year) that it was not in conflict with experiment. The theory was a good scientific
theory, in the sense described in Chapter 1: it was simple and it made definite predictions that could be tested
by observation. One of these predictions was that the number of galaxies or similar objects in any given volume
of space should be the same wherever and whenever we look in the universe. In the late 1950s and early
1960s a survey of sources of radio waves from outer space was carried out at Cambridge by a group of
astronomers led by Martin Ryle (who had also worked with Bondi, Gold, and Hoyle on radar during the war).
The Cambridge group showed that most of these radio sources must lie outside our galaxy (indeed many of
them could be identified with other galaxies) and also that there were many more weak sources than strong
ones. They interpreted the weak sources as being the more distant ones, and the stronger ones as being
nearer. Then there appeared to be less common sources per unit volume of space for the nearby sources than
for the distant ones. This could mean that we are at the center of a great region in the universe in which the
sources are fewer than elsewhere. Alternatively, it could mean that the sources were more numerous in the
past, at the time that the radio waves left on their journey to us, than they are now. Either explanation
contradicted the predictions of the steady state theory. Moreover, the discovery of the microwave radiation by
Penzias and Wilson in 1965 also indicated that the universe must have been much denser in the past. The
steady state theory therefore had to be abandoned.

A Brief History of Time - Stephen Hawking... Chapter 3
Another attempt to avoid the conclusion that there must have been a big bang, and therefore a beginning of
time, was made by two Russian scientists, Evgenii Lifshitz and Isaac Khalatnikov, in 1963. They suggested that
the big bang might be a peculiarity of Friedmann’s models alone, which after all were only approximations to
the real universe. Perhaps, of all the models that were roughly like the real universe, only Friedmann’s would
contain a big bang singularity. In Friedmann’s models, the galaxies are all moving directly away from each
other

A Brief History of Time - Stephen Hawking... Chapter 3
became generally accepted and nowadays nearly everyone assumes that the universe started with a big bang
singularity. It is perhaps ironic that, having changed my mind, I am now trying to convince other physicists that
there was in fact no singularity at the beginning of the universe

A Brief History of Time - Stephen Hawking... Chapter 4
CHAPTER 4
THE UNCERTAINTY PRINCIPLE
The success of scientific theories, particularly Newton’s theory of gravity, led the French scientist the Marquis de Laplace
at the beginning of the nineteenth century to argue that the universe was completely deterministic. Laplace suggested
that there should be a set of scientific laws that would allow us to predict everything that would happen in the universe, if
only we knew the complete state of the universe at one time. For example, if we knew the positions and speeds of the
sun and the planets at one time, then we could use Newton’s laws to calculate the state of the Solar System at any other
time. Determinism seems fairly obvious in this case, but Laplace went further to assume that there were similar laws
governing everything else, including human behavior.
The doctrine of scientific determinism was strongly resisted by many people, who felt that it infringed God’s freedom to
intervene in the world, but it remained the standard assumption of science until the early years of this century. One of the
first indications that this belief would have to be abandoned came when calculations by the British scientists Lord
Rayleigh and Sir James Jeans suggested that a hot object, or body, such as a star, must radiate energy at an infinite
rate. According to the laws we believed at the time, a hot body ought to give off electromagnetic waves (such as radio
waves, visible light, or X rays) equally at all frequencies. For example, a hot body should radiate the same amount of
energy in waves with frequencies between one and two million million waves a second as in waves with frequencies
between two and three million million waves a second. Now since the number of waves a second is unlimited, this would
mean that the total energy radiated would be infinite.
In order to avoid this obviously ridiculous result, the German scientist Max Planck suggested in 1900 that light, X rays,
and other waves could not be emitted at an arbitrary rate, but only in certain packets that he called quanta. Moreover,
each quantum had a certain amount of energy that was greater the higher the frequency of the waves, so at a high
enough frequency the emission of a single quantum would require more energy than was available. Thus the radiation at
high frequencies would be reduced, and so the rate at which the body lost energy would be finite.
The quantum hypothesis explained the observed rate of emission of radiation from hot bodies very well, but its
implications for determinism were not realized until 1926, when another German scientist, Werner Heisenberg,
formulated his famous uncertainty principle. In order to predict the future position and velocity of a particle, one has to be
able to measure its present position and velocity accurately. The obvious way to do this is to shine light on the particle.
Some of the waves of light will be scattered by the particle and this will indicate its position. However, one will not be able
to determine the position of the particle more accurately than the distance between the wave crests of light, so one needs
to use light of a short wavelength in order to measure the position of the particle precisely. Now, by Planck’s quantum
hypothesis, one cannot use an arbitrarily small amount of light; one has to use at least one quantum. This quantum will
measures the position, the shorter the wavelength of the light that one needs and hence the higher the energy of a single
quantum. So the velocity of the particle will be disturbed by a larger amount. In other words, the more accurately you try
to measure the position of the particle, the less accurately you can measure its speed, and vice versa. Heisenberg
showed that the uncertainty in the position of the particle times the uncertainty in its velocity times the mass of the
particle can never be smaller than a certain quantity, which is known as Planck’s constant. Moreover, this limit does not
depend on the way in which one tries to measure the position or velocity of the particle, or on the type of particle:
Heisenberg’s uncertainty principle is a fundamental, inescapable property of the world.
The uncertainty principle had profound implications for the way in which we view the world. Even after more than seventy
years they have not been fully appreciated by many philosophers, and are still the subject of much controversy. The
uncertainty principle signaled an end to Laplace’s dream of a theory of science, a model of the universe that would be
completely deterministic: one certainly cannot predict future events exactly if one cannot even measure the present state
of the universe precisely! We could still imagine that there is a set of laws that determine events completely for some
supernatural being, who could observe the present state of the universe without disturbing it. However, such models of
the universe are not of much interest to us ordinary mortals. It seems better to employ the principle of economy known as
Occam’s razor and cut out all the features of the theory that cannot be observed. This approach led Heisenberg, Erwin
Schrodinger, and Paul Dirac in the 1920s to reformulate mechanics into a new theory called quantum mechanics, based
on the uncertainty principle. In this theory particles no longer had separate, well-defined positions and velocities that
could not be observed, Instead, they had a quantum state, which was a combination of position and velocity.
In general, quantum mechanics does not predict a single definite result for an observation. Instead, it predicts a number
of different possible outcomes and tells us how likely each of these is. That is to say, if one made the same measurement
on a large number of similar systems, each of which started off in the same way, one would find that the result of the

Figure 4:1
A familiar example of interference in the case of light is the colors that are often seen in soap bubbles. These are caused
by reflection of light from the two sides of the thin film of water forming the bubble. White light consists of light waves of
all different wavelengths, or colors, For certain wavelengths the crests of the waves reflected from one side of the soap
film coincide with the troughs reflected from the other side. The colors corresponding to these wavelengths are absent
from the reflected light, which therefore appears to be colored. Interference can also occur for particles, because of the
duality introduced by quantum mechanics. A famous example is the so-called two-slit experiment Figure 4:2.

Figure 4:2
Consider a partition with two narrow parallel slits in it. On one side of the partition one places a source of fight of a
particular color (that is, of a particular wavelength). Most of the light will hit the partition, but a small amount will go
through the slits. Now suppose one places a screen on the far side of the partition from the light. Any point on the screen
will receive waves from the two slits. However, in general, the distance the light has to travel from the source to the
screen via the two slits will be different. This will mean that the waves from the slits will not be in phase with each other
when they arrive at the screen: in some places the waves will cancel each other out, and in others they will reinforce
each other. The result is a characteristic pattern of light and dark fringes.
The remarkable thing is that one gets exactly the same kind of fringes if one replaces the source of light by a source of
particles such as electrons with a definite speed (this means that the corresponding waves have a definite length). It
seems the more peculiar because if one only has one slit, one does not get any fringes, just a uniform distribution of
electrons across the screen. One might therefore think that opening another slit would just increase the number of
electrons hitting each point of the screen, but, because of interference, it actually decreases it in some places. If
electrons are sent through the slits one at a time, one would expect each to pass through one slit or the other, and so
behave just as if the slit it passed through were the only one there

A Brief History of Time - Stephen Hawking... Chapter 4
passing through both slits at the same time!
The phenomenon of interference between particles has been crucial to our understanding of the structure of atoms, the
basic units of chemistry and biology and the building blocks out of which we, and everything around us, are made. At the
beginning of this century it was thought that atoms were rather like the planets orbiting the sun, with electrons (particles
of negative electricity) orbiting around a central nucleus, which carried positive electricity. The attraction between the
positive and negative electricity was supposed to keep the electrons in their orbits in the same way that the gravitational
attraction between the sun and the planets keeps the planets in their orbits. The trouble with this was that the laws of
mechanics and electricity, before quantum mechanics, predicted that the electrons would lose energy and so spiral
inward until they collided with the nucleus. This would mean that the atom, and indeed all matter, should rapidly collapse
to a state of very high density. A partial solution to this problem was found by the Danish scientist Niels Bohr in 1913. He
suggested that maybe the electrons were not able to orbit at just any distance from the central nucleus but only at certain
specified distances. If one also supposed that only one or two electrons could orbit at any one of these distances, this
would solve the problem of the collapse of the atom, because the electrons could not spiral in any farther than to fill up
the orbits with e least distances and energies.
This model explained quite well the structure of the simplest atom, hydrogen, which has only one electron orbiting around
the nucleus. But it was not clear how one ought to extend it to more complicated atoms. Moreover, the idea of a limited
set of allowed orbits seemed very arbitrary. The new theory of quantum mechanics resolved this difficulty. It revealed that
an electron orbiting around the nucleus could be thought of as a wave, with a wavelength that depended on its velocity.
For certain orbits, the length of the orbit would correspond to a whole number (as opposed to a fractional number) of
wavelengths of the electron. For these orbits the wave crest would be in the same position each time round, so the
waves would add up: these orbits would correspond to Bohr’s allowed orbits. However, for orbits whose lengths were not
a whole number of wavelengths, each wave crest would eventually be canceled out by a trough as the electrons went
round; these orbits would not be allowed.
A nice way of visualizing the wave/particle duality is the so-called sum over histories introduced by the American scientist
Richard Feynman. In this approach the particle is not supposed to have a single history or path in space-time, as it would
in a classical, nonquantum theory. Instead it is supposed to go from A to B by every possible path. With each path there
are associated a couple of numbers: one represents the size of a wave and the other represents the position in the cycle
(i.e., whether it is at a crest or a trough). The probability of going from A to B is found by adding up the waves for all the
paths. In general, if one compares a set of neighboring paths, the phases or positions in the cycle will differ greatly. This
means that the waves associated with these paths will almost exactly cancel each other out. However, for some sets of
neighboring paths the phase will not vary much between paths. The waves for these paths will not cancel out Such paths
correspond to Bohr’s allowed orbits.
With these ideas, in concrete mathematical form, it was relatively straightforward to calculate the allowed orbits in more
complicated atoms and even in molecules, which are made up of a number of atoms held together by electrons in orbits
that go round more than one nucleus. Since the structure of molecules and their reactions with each other underlie all of
chemistry and biology, quantum mechanics allows us in principle to predict nearly everything we see around us, within
the limits set by the uncertainty principle. (In practice, however, the calculations required for systems containing more
than a few electrons are so complicated that we cannot do them.)
Einstein’s general theory of relativity seems to govern the large-scale structure of the universe. It is what is called a
classical theory; that is, it does not take account of the uncertainty principle of quantum mechanics, as it should for
consistency with other theories. The reason that this does not lead to any discrepancy with observation is that all the
gravitational fields that we normally experience are very weak. How-ever, the singularity theorems discussed earlier
indicate that the gravitational field should get very strong in at least two situations, black holes and the big bang. In such
strong fields the effects of quantum mechanics should be important. Thus, in a sense, classical general relativity, by
predicting points of infinite density, predicts its own downfall, just as classical (that is, nonquantum) mechanics predicted
its downfall by suggesting that atoms should collapse to infinite density. We do not yet have a complete consistent theory
that unifies general relativity and quantum mechanics, but we do know a number of the features it should have. The
consequences that these would have for black holes and the big bang will be described in later chapters. For the
moment, however, we shall turn to the recent attempts to bring together our understanding of the other forces of nature
into a single, unified quantum theory.

A Brief History of Time - Stephen Hawking... Chapter 5
CHAPTER 5
ELEMENTARY PARTICLES AND THE FORCES OF NATURE
Aristotle believed that all the matter in the universe was made up of four basic elements

A Brief History of Time - Stephen Hawking... Chapter 5
is much larger than the size of an atom, we cannot hope to “look” at the parts of an atom in the ordinary way. We need to
use something with a much smaller wave-length. As we saw in the last chapter, quantum mechanics tells us that all
particles are in fact waves, and that the higher the energy of a particle, the smaller the wavelength of the corresponding
wave. So the best answer we can give to our question depends on how high a particle energy we have at our disposal,
because this determines on how small a length scale we can look. These particle energies are usually measured in units
called electron volts. (In Thomson’s experiments with electrons, we saw that he used an electric field to accelerate the
electrons. The energy that an electron gains from an electric field of one volt is what is known as an electron volt.) In the
nineteenth century, when the only particle energies that people knew how to use were the low energies of a few electron
volts generated by chemical reactions such as burning, it was thought that atoms were the smallest unit. In Rutherford’s
experiment, the alpha-particles had energies of millions of electron volts. More recently, we have learned how to use
electromagnetic fields to give particles energies of at first millions and then thousands of millions of electron volts. And so
we know that particles that were thought to be “elementary” thirty years ago are, in fact, made up of smaller particles.
May these, as we go to still higher energies, in turn be found to be made from still smaller particles? This is certainly
possible, but we do have some theoretical reasons for believing that we have, or are very near to, a knowledge of the
ultimate building blocks of nature.
Using the wave/particle duality discussed in the last chapter, every-thing in the universe, including light and gravity, can
be described in terms of particles. These particles have a property called spin. One way of thinking of spin is to imagine
the particles as little tops spinning about an axis. However, this can be misleading, because quantum mechanics tells us
that the particles do not have any well-defined axis. What the spin of a particle really tells us is what the particle looks like
from different directions. A particle of spin 0 is like a dot: it looks the same from every direction Figure 5:1-i. On the other
hand, a particle of spin 1 is like an arrow: it looks different from different directions Figure 5:1-ii. Only if one turns it round
a complete revolution (360 degrees) does the particle look the same. A particle of spin 2 is like a double-headed arrow
Figure 5:1-iii: it looks the same if one turns it round half a revolution (180 degrees). Similarly, higher spin particles look
the same if one turns them through smaller fractions of a complete revolution. All this seems fairly straightforward, but the
remark-able fact is that there are particles that do not look the same if one turns them through just one revolution: you
have to turn them through two complete revolutions! Such particles are said to have spin ?.

Figure 5:1
All the known particles in the universe can be divided into two groups: particles of spin ?, which make up the matter in
the universe, and particles of spin 0, 1, and 2, which, as we shall see, give rise to forces between the matter particles.
The matter particles obey what is called Pauli’s exclusion principle. This was discovered in 1925 by an Austrian physicist,
Wolfgang Pauli