《时间简史―从大爆炸到黑洞》 ――史蒂芬?霍金

Figure 2:5
What happens at such events can neither affect nor be affected by what happens at P. For example, if the sun
were to cease to shine at this very moment, it would not affect things on earth at the present time because they
would be in the elsewhere of the event when the sun went out Figure 2:6.

Figure 2:6
We would know about it only after eight minutes, the time it takes light to reach us from the sun. Only then
would events on earth lie in the future light cone of the event at which the sun went out. Similarly, we do not
know what is happening at the moment farther away in the universe: the light that we see from distant galaxies
left them millions of years ago, and in the case of the most distant object that we have seen, the light left some
eight thousand million years ago. Thus, when we look at the universe, we are seeing it as it was in the past.
If one neglects gravitational effects, as Einstein and Poincare did in 1905, one has what is called the special
theory of relativity. For every event in space-time we may construct a light cone (the set of all possible paths of
light in space-time emitted at that event), and since the speed of light is the same at every event and in every
direction, all the light cones will be identical and will all point in the same direction. The theory also tells us that
nothing can travel faster than light. This means that the path of any object through space and time must be
represented by a line that lies within the light cone at each event on it (Fig. 2.7). The special theory of relativity
was very successful in explaining that the speed of light appears the same to all observers (as shown by the
Michelson-Morley experiment) and in describing what happens when things move at speeds close to the speed
of light. However, it was inconsistent with the Newtonian theory of gravity, which said that objects attracted
each other with a force that depended on the distance between them. This meant that if one moved one of the
objects, the force on the other one would change instantaneously. Or in other gravitational effects should travel
with infinite velocity, instead of at or below the speed of light, as the special theory of relativity required.
Einstein made a number of unsuccessful attempts between 1908 and 1914 to find a theory of gravity that was
consistent with special relativity. Finally, in 1915, he proposed what we now call the general theory of relativity.
Einstein made the revolutionary suggestion that gravity is not a force like other forces, but is a consequence of
the fact that space-time is not flat, as had been previously assumed: it is curved, or “warped,” by the distribution

A Brief History of Time - Stephen Hawking... Chapter 2
of mass and energy in it. Bodies like the earth are not made to move on curved orbits by a force called gravity;
instead, they follow the nearest thing to a straight path in a curved space, which is called a geodesic. A
geodesic is the shortest (or longest) path between two nearby points. For example, the surface of the earth is a
two-dimensional curved space. A geodesic on the earth is called a great circle, and is the shortest route
between two points (Fig. 2.8). As the geodesic is the shortest path between any two airports, this is the route
an airline navigator will tell the pilot to fly along. In general relativity, bodies always follow straight lines in
four-dimensional space-time, but they nevertheless appear to us to move along curved paths in our
three-dimensional space. (This is rather like watching an airplane flying over hilly ground. Although it follows a
straight line in three-dimensional space, its shadow follows a curved path on the two-dimensional ground.)
The mass of the sun curves space-time in such a way that although the earth follows a straight path in
four-dimensional space-time, it appears to us to move along a circular orbit in three-dimensional space.
Fact, the orbits of the planets predicted by general relativity are almost exactly the same as those predicted by
the Newtonian theory of gravity. However, in the case of Mercury, which, being the nearest planet to the sun,
feels the strongest gravitational effects, and has a rather elongated orbit, general relativity predicts that the long
axis of the ellipse should rotate about the sun at a rate of about one degree in ten thousand years. Small
though this effect is, it had been noticed before 1915 and served as one of the first confirmations of Einstein’s
theory. In recent years the even smaller deviations of the orbits of the other planets from the Newtonian
predictions have been measured by radar and found to agree with the predictions of general relativity.
Light rays too must follow geodesics in space-time. Again, the fact that space is curved means that light no
longer appears to travel in straight lines in space. So general relativity predicts that light should be bent by
gravitational fields. For example, the theory predicts that the light cones of points near the sun would be slightly
bent inward, on account of the mass of the sun. This means that light from a distant star that happened to pass
near the sun would be deflected through a small angle, causing the star to appear in a different position to an
observer on the earth (Fig. 2.9). Of course, if the light from the star always passed close to the sun, we would
not be able to tell whether the light was being deflected or if instead the star was really where we see it.
However, as the earth orbits around the sun, different stars appear to pass behind the sun and have their light
deflected. They therefore change their apparent position relative to other stars. It is normally very difficult to see
this effect, because the light from the sun makes it impossible to observe stars that appear near to the sun the
sky. However, it is possible to do so during an eclipse of the sun, when the sun’s light is blocked out by the
moon. Einstein’s prediction of light deflection could not be tested immediately in 1915, because the First World
War was in progress, and it was not until 1919 that a British expedition, observing an eclipse from West Africa,
showed that light was indeed deflected by the sun, just as predicted by the theory. This proof of a German
theory by British scientists was hailed as a great act of reconciliation between the two countries after the war. It
is ionic, therefore, that later examination of the photographs taken on that expedition showed the errors were as
great as the effect they were trying to measure. Their measurement had been sheer luck, or a case of knowing
the result they wanted to get, not an uncommon occurrence in science. The light deflection has, however, been
accurately confirmed by a number of later observations.
Another prediction of general relativity is that time should appear to slower near a massive body like the earth.
This is because there is a relation between the energy of light and its frequency (that is, the number of waves of
light per second): the greater the energy, the higher frequency. As light travels upward in the earth’s
gravitational field, it loses energy, and so its frequency goes down. (This means that the length of time between
one wave crest and the next goes up.) To someone high up, it would appear that everything down below was
making longer to happen. This prediction was tested in 1962, using a pair of very accurate clocks mounted at
the top and bottom of a water tower. The clock at the bottom, which was nearer the earth, was found to run
slower, in exact agreement with general relativity. The difference in the speed of clocks at different heights
above the earth is now of considerable practical importance, with the advent of very accurate navigation
systems based on signals from satellites. If one ignored the predictions of general relativity, the position that
one calculated would be wrong by several miles!
Newton’s laws of motion put an end to the idea of absolute position in space. The theory of relativity gets rid of
absolute time. Consider a pair of twins. Suppose that one twin goes to live on the top of a mountain while the
other stays at sea level. The first twin would age faster than the second. Thus, if they met again, one would be
older than the other. In this case, the difference in ages would be very small, but it would be much larger if one

A Brief History of Time - Stephen Hawking... Chapter 2
of the twins went for a long trip in a spaceship at nearly the speed of light. When he returned, he would be
much younger than the one who stayed on earth. This is known as the twins paradox, but it is a paradox only if
one has the idea of absolute time at the back of one’s mind. In the theory of relativity there is no unique
absolute time, but instead each individual has his own personal measure of time that depends on where he is
and how he is moving.
Before 1915, space and time were thought of as a fixed arena in which events took place, but which was not
affected by what happened in it. This was true even of the special theory of relativity. Bodies moved, forces
attracted and repelled, but time and space simply continued, unaffected. It was natural to think that space and
time went on forever.
The situation, however, is quite different in the general theory of relativity. Space and time are now dynamic
quantities: when a body moves, or a force acts, it affects the curvature of space and time

A Brief History of Time - Stephen Hawking... Chapter 3
CHAPTER 3
THE EXPANDING UNIVERSE
If one looks at the sky on a clear, moonless night, the brightest objects one sees are likely to be the planets
Venus, Mars, Jupiter, and Saturn. There will also be a very large number of stars, which are just like our own
sun but much farther from us. Some of these fixed stars do, in fact, appear to change very slightly their
positions relative to each other as earth orbits around the sun: they are not really fixed at all! This is because
they are comparatively near to us. As the earth goes round the sun, we see them from different positions
against the background of more distant stars. This is fortunate, because it enables us to measure directly the
distance of these stars from us: the nearer they are, the more they appear to move. The nearest star, called
Proxima Centauri, is found to be about four light-years away (the light from it takes about four years to reach
earth), or about twenty-three million million miles. Most of the other stars that are visible to the naked eye lie
within a few hundred light-years of us. Our sun, for comparison, is a mere light-minutes away! The visible stars
appear spread all over the night sky, but are particularly concentrated in one band, which we call the Milky
Way. As long ago as 1750, some astronomers were suggesting that the appearance of the Milky Way could be
explained if most of the visible stars lie in a single disklike configuration, one example of what we now call a
spiral galaxy. Only a few decades later, the astronomer Sir William Herschel confirmed this idea by
painstakingly cataloging the positions and distances of vast numbers of stars. Even so, the idea gained
complete acceptance only early this century.
Our modern picture of the universe dates back to only 1924, when the American astronomer Edwin Hubble
demonstrated that ours was not the only galaxy. There were in fact many others, with vast tracts of empty
space between them. In order to prove this, he needed to determine the distances to these other galaxies,
which are so far away that, unlike nearby stars, they really do appear fixed. Hubble was forced, therefore, to
use indirect methods to measure the distances. Now, the apparent brightness of a star depends on two factors:
how much light it radiates (its luminosity), and how far it is from us. For nearby stars, we can measure their
apparent brightness and their distance, and so we can work out their luminosity. Conversely, if we knew the
luminosity of stars in other galaxies, we could work out their distance by measuring their apparent brightness.
Hubble noted that certain types of stars always have the same luminosity when they are near enough for us to
measure; therefore, he argued, if we found such stars in another galaxy, we could assume that they had the
same luminosity

Figure 3:1
We live in a galaxy that is about one hundred thousand light-years across and is slowly rotating; the stars in its
spiral arms orbit around its center about once every several hundred million years. Our sun is just an ordinary,
average-sized, yellow star, near the inner edge of one of the spiral arms. We have certainly come a long way
since Aristotle and Ptolemy, when thought that the earth was the center of the universe!
Stars are so far away that they appear to us to be just pinpoints of light. We cannot see their size or shape. So
how can we tell different types of stars apart? For the vast majority of stars, there is only one characteristic
feature that we can observe

A Brief History of Time - Stephen Hawking... Chapter 3
the waves we receive will be the same as the wavelength at which they are emitted (the gravitational field of the
galaxy will not be large enough to have a significant effect). Suppose now that the source starts moving toward
us. When the source emits the next wave crest it will be nearer to us, so the distance between wave crests will
be smaller than when the star was stationary. This means that the wavelength of the waves we receive is
shorter than when the star was stationary. Correspondingly, if the source is moving away from us, the
wavelength of the waves we receive will be longer. In the case of light, therefore, means that stars moving
away from us will have their spectra shifted toward the red end of the spectrum (red-shifted) and those moving
toward us will have their spectra blue-shifted. This relationship between wavelength and speed, which is called
the Doppler effect, is an everyday experience. Listen to a car passing on the road: as the car is approaching, its
engine sounds at a higher pitch (corresponding to a shorter wavelength and higher frequency of sound waves),
and when it passes and goes away, it sounds at a lower pitch. The behavior of light or radio waves is similar.
Indeed, the police make use of the Doppler effect to measure the speed of cars by measuring the wavelength
of pulses of radio waves reflected off them.
ln the years following his proof of the existence of other galaxies, Rubble spent his time cataloging their
distances and observing their spectra. At that time most people expected the galaxies to be moving around
quite randomly, and so expected to find as many blue-shifted spectra as red-shifted ones. It was quite a
surprise, therefore, to find that most galaxies appeared red-shifted: nearly all were moving away from us! More
surprising still was the finding that Hubble published in 1929: even the size of a galaxy’s red shift is not random,
but is directly proportional to the galaxy’s distance from us. Or, in other words, the farther a galaxy is, the faster
it is moving away! And that meant that the universe could not be static, as everyone previously had thought, is
in fact expanding; the distance between the different galaxies is changing all the time.
The discovery that the universe is expanding was one of the great intellectual revolutions of the twentieth
century. With hindsight, it is easy wonder why no one had thought of it before. Newton, and others should have
realized that a static universe would soon start to contract under the influence of gravity. But suppose instead
that the universe is expanding. If it was expanding fairly slowly, the force of gravity would cause it eventually to
stop expanding and then to start contracting. However, if it was expanding at more than a certain critical rate,
gravity would never be strong enough to stop it, and the universe would continue to expand forever. This is a bit
like what happens when one fires a rocket upward from the surface of the earth. If it has a fairly low speed,
gravity will eventually stop the rocket and it will start falling back. On the other hand, if the rocket has more than
a certain critical speed (about seven miles per second), gravity will not be strong enough to pull it back, so it will
keep going away from the earth forever. This behavior of the universe could have been predicted from
Newton’s theory of gravity at any time in the nineteenth, the eighteenth, or even the late seventeenth century.
Yet so strong was the belief in a static universe that it persisted into the early twentieth century. Even Einstein,
when he formulated the general theory of relativity in 1915, was so sure that the universe had to be static that
he modified his theory to make this possible, introducing a so-called cosmological constant into his equations.
Einstein introduced a new “antigravity” force, which, unlike other forces, did not come from any particular
source but was built into the very fabric of space-time. He claimed that space-time had an inbuilt tendency to
expand, and this could be made to balance exactly the attraction of all the matter in the universe, so that a
static universe would result. Only one man, it seems, was willing to take general relativity at face value, and
while Einstein and other physicists were looking for ways of avoiding general relativity’s prediction of a
nonstatic universe, the Russian physicist and mathematician Alexander Friedmann instead set about explaining
it.
Friedmann made two very simple assumptions about the universe: that the universe looks identical in
whichever direction we look, and that this would also be true if we were observing the universe from anywhere
else. From these two ideas alone, Friedmann showed that we should not expect the universe to be static. In
fact, in 1922, several years before Edwin Hubble’s discovery, Friedmann predicted exactly what Hubble found!
The assumption that the universe looks the same in every direction is clearly not true in reality. For example, as
we have seen, the other stars in our galaxy form a distinct band of light across the night sky, called the Milky
Way. But if we look at distant galaxies, there seems to be more or less the same number of them. So the
universe does seem to be roughly the same in every direction, provided one views it on a large scale compared
to the distance between galaxies, and ignores the differences on small scales. For a long time, this was
sufficient justification for Friedmann’s assumption

A Brief History of Time - Stephen Hawking... Chapter 3
description of our universe.
In 1965 two American physicists at the Bell Telephone Laboratories in New Jersey, Arno Penzias and Robert
Wilson, were testing a very sensitive microwave detector. (Microwaves are just like light waves, but with a
wavelength of around a centimeter.) Penzias and Wilson were worried when they found that their detector was
picking up more noise than it ought to. The noise did not appear to be coming from any particular direction.
First they discovered bird droppings in their detector and checked for other possible malfunctions, but soon
ruled these out. They knew that any noise from within the atmosphere would be stronger when the detector
was not pointing straight up than when it was, because light rays travel through much more atmosphere when
received from near the horizon than when received from directly overhead. The extra noise was the same
whichever direction the detector was pointed, so it must come from outside the atmosphere. It was also the
same day and night and throughout the year, even though the earth was rotating on its axis and orbiting around
the sun. This showed that the radiation must come from beyond the Solar System, and even from beyond the
galaxy, as otherwise it would vary as the movement of earth pointed the detector in different directions.
In fact, we know that the radiation must have traveled to us across most of the observable universe, and since
it appears to be the same in different directions, the universe must also be the same in every direction, if only
on a large scale. We now know that whichever direction we look, this noise never varies by more than a tiny
fraction: so Penzias and Wilson had unwittingly stumbled across a remarkably accurate confirmation of
Friedmann’s first assumption. However, because the universe is not exactly the same in every direction, but
only on average on a large scale, the microwaves cannot be exactly the same in every direction either. There
have to be slight variations between different directions. These were first detected in 1992 by the Cosmic
Background Explorer satellite, or COBE, at a level of about one part in a hundred thousand. Small though these
variations are, they are very important, as will be explained in Chapter 8.
At roughly the same time as Penzias and Wilson were investigating noise in their detector, two American
physicists at nearby Princeton University, Bob Dicke and Jim Peebles, were also taking an interest in
microwaves. They were working on a suggestion, made by George Gamow (once a student of Alexander
Friedmann), that the early universe should have been very hot and dense, glowing white hot. Dicke and
Peebles argued that we should still be able to see the glow of the early universe, because light from very
distant parts of it would only just be reaching us now. However, the expansion of the universe meant that this
light should be so greatly red-shifted that it would appear to us now as microwave radiation. Dicke and Peebles
were preparing to look for this radiation when Penzias and Wilson heard about their work and realized that they
had already found it. For this, Penzias and Wilson were awarded the Nobel Prize in 1978 (which seems a bit
hard on Dicke and Peebles, not to mention Gamow!).
Now at first sight, all this evidence that the universe looks the same whichever direction we look in might seem
to suggest there is something special about our place in the universe. In particular, it might seem that if we
observe all other galaxies to be moving away from us, then we must be at the center of the universe. There is,
however, an alternate explanation: the universe might look the same in every direction as seen from any other
galaxy too. This, as we have seen, was Friedmann’s second assumption. We have no scientific evidence for, or
against, this assumption. We believe it only on grounds of modesty: it would be most remarkable if the universe
looked the same in every direction around us, but not around other points in the universe! In Friedmann’s
model, all the galaxies are moving directly away from each other. The situation is rather like a balloon with a
number of spots painted on it being steadily blown up. As the balloon expands, the distance between any two
spots increases, but there is no spot that can be said to be the center of the expansion. Moreover, the farther
apart the spots are, the faster they will be moving apart. Similarly, in Friedmann’s model the speed at which any
two galaxies are moving apart is proportional to the distance between them. So it predicted that the red shift of
a galaxy should be directly proportional to its distance from us, exactly as Hubble found. Despite the success of
his model and his prediction of Hubble’s observations, Friedmann’s work remained largely unknown in the West
until similar models were discovered in 1935 by the American physicist Howard Robertson and the British
mathematician Arthur Walker, in response to Hubble’s discovery of the uniform expansion of the universe.
Although Friedmann found only one, there are in fact three different kinds of models that obey Friedmann’s two
fundamental assumptions. In the first kind (which Friedmann found) the universe is expanding sufficiently
slowly that the gravitational attraction between the different galaxies causes the expansion to slow down and
eventually to stop. The galaxies then start to move toward each other and the universe contracts.

Figure 3:3 Shows the Separation between neighboring galaxies in this model. It starts at zero and eventually
the galaxies are moving apart at a steady speed. Finally, there is a third kind of solution, in which the universe
is expanding only just fast enough to avoid recollapse.

Figure 3:4
In this case the separation, shown in Figure 3:4, also starts at zero and increases forever. However, the speed
at which the galaxies are moving apart gets smaller and smaller, although it never quite reaches zero.
A remarkable feature of the first kind of Friedmann model is that in it the universe is not infinite in space, but
neither does space have any boundary. Gravity is so strong that space is bent round onto itself, making it rather
like the surface of the earth. If one keeps traveling in a certain direction on the surface of the earth, one never
comes up against an impassable barrier or falls over the edge, but eventually comes back to where one
started.
In the first kind of Friedmann model, space is just like this, but with three dimensions instead of two for the
earth’s surface. The fourth dimension, time, is also finite in extent, but it is like a line with two ends or
boundaries, a beginning and an end. We shall see later that when one combines general relativity with the
uncertainty principle of quantum mechanics, it is possible for both space and time to be finite without any edges
or boundaries.
The idea that one could go right round the universe and end up where one started makes good science fiction,
but it doesn’t have much practical significance, because it can be shown that the universe would recollapse to
zero size before one could get round. You would need to travel faster than light in order to end up where you
started before the universe came to an end