《时间简史―从大爆炸到黑洞》 ――史蒂芬?霍金

A Brief History of Time - Stephen Hawking... Chapter 7
gamma rays from space have been detected by satellites originally constructed to look for violations of the Test Ban
Treaty. These seem to occur about sixteen times a month and to be roughly uniformly distributed in direction across
the sky. This indicates that they come from outside the Solar System since otherwise we would expect them to be
concentrated toward the plane of the orbits of the planets. The uniform distribution also indicates that the sources are
either fairly near to us in our galaxy or right outside it at cosmological distances because otherwise, again, they would
be concentrated toward the plane of the galaxy. In the latter case, the energy required to account for the bursts would
be far too high to have been produced by tiny black holes, but if the sources were close in galactic terms, it might be
possible that they were exploding black holes. I would very much like this to be the case but I have to recognize that
there are other possible explanations for the gamma ray bursts, such as colliding neutron stars. New observations in
the next few years, particularly by gravitational wave detectors like LIGO, should enable us to discover the origin of the
gamma ray bursts.
Even if the search for primordial black holes proves negative, as it seems it may, it will still give us important
information about the very early stages of the universe. If the early universe had been chaotic or irregular, or if the
pressure of matter had been low, one would have expected it to produce many more primordial black holes than the
limit already set by our observations of the gamma ray background. Only if the early universe was very smooth and
uniform, with a high pressure, can one explain the absence of observable numbers of primordial black holes.
The idea of radiation from black holes was the first example of a prediction that depended in an essential way on both
the great theories of this century, general relativity and quantum mechanics. It aroused a lot of opposition initially
because it upset the existing viewpoint: “How can a black hole emit anything?” When I first announced the results of
my calculations at a conference at the Rutherford-Appleton Laboratory near Oxford, I was greeted with general
incredulity. At the end of my talk the chairman of the session, John G. Taylor from Kings College, London, claimed it
was all nonsense. He even wrote a paper to that effect. However, in the end most people, including John Taylor, have
come to the conclusion that black holes must radiate like hot bodies if our other ideas about general relativity and
quantum mechanics are correct. Thus, even though we have not yet managed to find a primordial black hole, there is
fairly general agreement that if we did, it would have to be emitting a lot of gamma rays and X rays.
The existence of radiation from black holes seems to imply that gravitational collapse is not as final and irreversible as
we once thought. If an astronaut falls into a black hole, its mass will increase, but eventually the energy equivalent of
that extra mass will be returned to the universe in the form of radiation. Thus, in a sense, the astronaut will be
“recycled.” It would be a poor sort of immortality, however, because any personal concept of time for the astronaut
would almost certainly come to an end as he was torn apart inside the black hole! Even the types of particles that were
eventually emitted by the black hole would in general be different from those that made up the astronaut: the only
feature of the astronaut that would survive would be his mass or energy.
The approximations I used to derive the emission from black holes should work well when the black hole has a mass
greater than a fraction of a gram. However, they will break down at the end of the black hole’s life when its mass gets
very small. The most likely outcome seems to be that the black hole will just disappear, at least from our region of the
universe, taking with it the astronaut and any singularity there might be inside it, if indeed there is one. This was the
first indication that quantum mechanics might remove the singularities that were predicted by general relativity.
However, the methods that I and other people were using in 1974 were not able to answer questions such as whether
singularities would occur in quantum gravity. From 1975 onward I therefore started to develop a more powerful
approach to quantum gravity based on Richard Feynrnan’s idea of a sum over histories. The answers that this
approach suggests for the origin and fate of the universe and its contents, such as astronauts, will be de-scribed in the
next two chapters. We shall see that although the uncertainty principle places limitations on the accuracy of all our
predictions, it may at the same time remove the fundamental unpredictability that occurs at a space-time singularity.

A Brief History of Time - Stephen Hawking... Chapter 8
CHAPTER 8
THE ORIGIN AND FATE OF THE UNIVERSE
Einstein’s general theory of relativity, on its own, predicted that space-time began at the big bang singularity and
would come to an end either at the big crunch singularity (if the whole universe recollapsed), or at a singularity inside
a black hole (if a local region, such as a star, were to collapse). Any matter that fell into the hole would be destroyed
at the singularity, and only the gravitational effect of its mass would continue to be felt outside. On the other hand,
when quantum effects were taken into account, it seemed that the mass or energy of the matter would eventually be
returned to the rest of the universe, and that the black hole, along with any singularity inside it, would evaporate
away and finally disappear. Could quantum mechanics have an equally dramatic effect on the big bang and big
crunch singularities? What really happens during the very early or late stages of the universe, when gravitational
fields are so strong that quantum effects cannot be ignored? Does the universe in fact have a beginning or an end?
And if so, what are they like?
Throughout the 1970s I had been mainly studying black holes, but in 1981 my interest in questions about the origin
and fate of the universe was reawakened when I attended a conference on cosmology organized by the Jesuits in
the Vatican. The Catholic Church had made a bad mistake with Galileo when it tried to lay down the law on a
question of science, declaring that the sun went round the earth. Now, centuries later, it had decided to invite a
number of experts to advise it on cosmology. At the end of the conference the participants were granted an audience
with the Pope. He told us that it was all right to study the evolution of the universe after the big bang, but we should
not inquire into the big bang itself because that was the moment of Creation and therefore the work of God. I was
glad then that he did not know the subject of the talk I had just given at the conference

A Brief History of Time - Stephen Hawking... Chapter 8
About one hundred seconds after the big bang, the temperature would have fallen to one thousand million degrees,
the temperature inside the hottest stars. At this temperature protons and neutrons would no longer have sufficient
energy to escape the attraction of the strong nuclear force, and would have started to combine together to produce
the nuclei of atoms of deuterium (heavy hydrogen), which contain one proton and one neutron. The deuterium nuclei
would then have combined with more protons and neutrons to make helium nuclei, which contain two protons and
two neutrons, and also small amounts of a couple of heavier elements, lithium and beryllium. One can calculate that
in the hot big bang model about a quarter of the protons and neutrons would have been converted into helium nuclei,
along with a small amount of heavy hydrogen and other elements. The remaining neutrons would have decayed into
protons, which are the nuclei of ordinary hydrogen atoms.
This picture of a hot early stage of the universe was first put forward by the scientist George Gamow in a famous
paper written in 1948 with a student of his, Ralph Alpher. Gamow had quite a sense of humor

A Brief History of Time - Stephen Hawking... Chapter 8
atmosphere from the emission of gases from the rocks. This early atmosphere was not one in which we could have
survived. It contained no oxygen, but a lot of other gases that are poisonous to us, such as hydrogen sulfide (the gas
that gives rotten eggs their smell). There are, however, other primitive forms of life that can flourish under such
conditions. It is thought that they developed in the oceans, possibly as a result of chance combinations of atoms into
large structures, called macromolecules, which were capable of assembling other atoms in the ocean into similar
structures. They would thus have reproduced themselves and multiplied. In some cases there would be errors in the
reproduction. Mostly these errors would have been such that the new macromolecule could not reproduce itself and
eventually would have been destroyed. However, a few of the errors would have produced new macromolecules that
were even better at reproducing themselves. They would have therefore had an advantage and would have tended
to replace the original macromolecules. In this way a process of evolution was started that led to the development of
more and more complicated, self-reproducing organisms. The first primitive forms of life consumed various materials,
including hydrogen sulfide, and released oxygen. This gradually changed the atmosphere to the composition that it
has today, and allowed the development of higher forms of life such as fish, reptiles, mammals, and ultimately the
human race.
This picture of a universe that started off very hot and cooled as it expanded is in agreement with all the
observational evidence that we have today. Nevertheless, it leaves a number of important questions unanswered:
1. Why was the early universe so hot?
2. Why is the universe so uniform on a large scale? Why does it look the same at all points of space and in all
directions? In particular, why is the temperature of the microwave back-ground radiation so nearly the same when we
look in different directions? It is a bit like asking a number of students an exam question. If they all give exactly the
same answer, you can be pretty sure they have communicated with each other. Yet, in the model described above,
there would not have been time since the big bang for light to get from one distant region to another, even though the
regions were close together in the early universe. According to the theory of relativity, if light cannot get from one
region to another, no other information can. So there would be no way in which different regions in the early universe
could have come to have the same temperature as each other, unless for some unexplained reason they happened
to start out with the same temperature.
3. Why did the universe start out with so nearly the critical rate of expansion that separates models that recollapse
from those that go on expanding forever, that even now, ten thousand million years later, it is still expanding at nearly
the critical rate? If the rate of expansion one second after the big bang had been smaller by even one part in a
hundred thousand million million, the universe would have recollapsed before it ever reached its present size.
4. Despite the fact that the universe is so uniform and homogeneous on a large scale, it contains local irregularities,
such as stars and galaxies. These are thought to have developed from small differences in the density of the early
universe from one region to another. What was the origin of these density fluctuations?
The general theory of relativity, on its own, cannot explain these features or answer these questions because of its
prediction that the universe started off with infinite density at the big bang singularity. At the singularity, general
relativity and all other physical laws would break down: one couldn’t predict what would come out of the singularity.
As explained before, this means that one might as well cut the big bang, and any events before it, out of the theory,
because they can have no effect on what we observe. Space-time would have a boundary

A Brief History of Time - Stephen Hawking... Chapter 8
one model, to represent our universe.
One such possibility is what are called chaotic boundary conditions. These implicitly assume either that the universe
is spatially infinite or that there are infinitely many universes. Under chaotic boundary conditions, the probability of
finding any particular region of space in any given configuration just after the big bang is the same, in some sense,
as the probability of finding it in any other configuration: the initial state of the universe is chosen purely randomly.
This would mean that the early universe would have probably been very chaotic and irregular because there are
many more chaotic and disordered configurations for the universe than there are smooth and ordered ones. (If each
configuration is equally probable, it is likely that the universe started out in a chaotic and disordered state, simply
because there are so many more of them.) It is difficult to see how such chaotic initial conditions could have given
rise to a universe that is so smooth and regular on a large scale as ours is today. One would also have expected the
density fluctuations in such a model to have led to the formation of many more primordial black holes than the upper
limit that has been set by observations of the gamma ray background.
If the universe is indeed spatially infinite, or if there are infinitely many universes, there would probably be some large
regions somewhere that started out in a smooth and uniform manner. It is a bit like the well-known horde of monkeys
hammering away on typewriters

A Brief History of Time - Stephen Hawking... Chapter 8
laws of science or as support for the strong anthropic principle.
There are a number of objections that one can raise to the strong anthropic principle as an explanation of the
observed state of the universe. First, in what sense can all these different universes be said to exist? If they are
really separate from each other, what happens in another universe can have no observable consequences in our
own universe. We should therefore use the principle of economy and cut them out of the theory. If, on the other
hand, they are just different regions of a single universe, the laws of science would have to be the same in each
region, because otherwise one could not move continuously from one region to another. In this case the only
difference between the regions would be their initial configurations and so the strong anthropic principle would
reduce to the weak one.
A second objection to the strong anthropic principle is that it runs against the tide of the whole history of science. We
have developed from the geocentric cosmologies of Ptolemy and his forebears, through the heliocentric cosmology
of Copernicus and Galileo, to the modern picture in which the earth is a medium-sized planet orbiting around an
average star in the outer suburbs of an ordinary spiral galaxy, which is itself only one of about a million million
galaxies in the observable universe. Yet the strong anthropic principle would claim that this whole vast construction
exists simply for our sake. This is very hard to believe. Our Solar System is certainly a prerequisite for our existence,
hand one might extend this to the whole of our galaxy to allow for an earlier generation of stars that created the
heavier elements. But there does not seem to be any need for all those other galaxies, nor for the universe to be so
uniform and similar in every direction on the large scale.
One would feel happier about the anthropic principle, at least in its weak version, if one could show that quite a
number of different initial configurations for the universe would have evolved to produce a universe like the one we
observe. If this is the case, a universe that developed from some sort of random initial conditions should contain a
number of regions that are smooth and uniform and are suitable for the evolution of intelligent life. On the other hand,
if the initial state of the universe had to be chosen extremely carefully to lead to something like what we see around
us, the universe would be unlikely to contain any region in which life would appear. In the hot big bang model
described above, there was not enough time in the early universe for heat to have flowed from one region to another.
This means that the initial state of the universe would have to have had exactly the same temperature everywhere in
order to account for the fact that the microwave back-ground has the same temperature in every direction we look.
The initial rate of expansion also would have had to be chosen very precisely for the rate of expansion still to be so
close to the critical rate needed to avoid recollapse. This means that the initial state of the universe must have been
very carefully chosen indeed if the hot big bang model was correct right back to the beginning of time. It would be
very difficult to explain why the universe should have begun in just this way, except as the act of a God who intended
to create beings like us.
In an attempt to find a model of the universe in which many different initial configurations could have evolved to
something like the present universe, a scientist at the Massachusetts Institute of Technology, Alan Guth, sugges
that the early universe might have gone through a period of very rapid expansion. This expansion is said to be
“inflationary,” meaning that the universe at one time expanded at an increasing rate rather than the decreasing rate
that it does today. According to Guth, the radius of the universe increased by a million million million million million (1
with thirty zeros after it) times in only a tiny fraction of a second.
Guth suggested that the universe started out from the big bang in a very hot, but rather chaotic, state. These high
temperatures would have meant that the particles in the universe would be moving very fast and would have high
energies. As we discussed earlier, one would expect that at such high temperatures the strong and weak nuclear
forces and the electromagnetic force would all be unified into a single force. As the universe expanded, it would cool,
and particle energies would go down. Eventually there would be what is called a phase transition and the symmetry
between the forces would be broken: the strong force would become different from the weak and electromagnetic
forces. One common example of a phase transition is the freezing of water when you cool it down. Liquid water is
symmetrical, the same at every point and in every direction. However, when ice crystals form, they will have definite
positions and will be lined up in some direction. This breaks water’s symmetry.
In the case of water, if one is careful, one can “supercool” it: that is, one can reduce the temperature below the
freezing point (OoC) without ice forming. Guth suggested that the universe might behave in a similar way: the
temperature might drop below the critical value without the symmetry between the forces being broken. If this
happened, the universe would be in an unstable state, with more energy than if the symmetry had been broken. This
special extra energy can be shown to have an antigravitational effect: it would have acted just like the cosmological
constant that Einstein introduced into general relativity when he was trying to construct a static model of the
universe. Since the universe would already be expanding just as in the hot big bang model, the repulsive effect of

A Brief History of Time - Stephen Hawking... Chapter 8
this cosmological constant would therefore have made the universe expand at an ever-increasing rate. Even in
regions where there were more matter particles than average, the gravitational attraction of the matter would have
been outweighed by the repulsion of the effective cosmological constant. Thus these regions would also expand in
an accelerating inflationary manner. As they expanded and the matter particles got farther apart, one would be left
with an expanding universe that contained hardly any particles and was still in the supercooled state. Any
irregularities in the universe would simply have been smoothed out by the expansion, as the wrinkles in a balloon are
smoothed away when you blow it up. Thus the present smooth and uniform state of the universe could have evolved
from many different non-uniform initial states.
In such a universe, in which the expansion was accelerated by a cosmological constant rather than slowed down by
the gravitational attraction of matter, there would be enough time for light to travel from one region to another in the
early universe. This could provide a solution to the problem, raised earlier, of why different regions in the early
universe have the same properties. Moreover, the rate of expansion of the universe would automatically become
very close to the critical rate determined by the energy density of the universe. This could then explain why the rate
of expansion is still so close to the critical rate, without having to assume that the initial rate of expansion of the
universe was very carefully chosen.
The idea of inflation could also explain why there is so much matter in the universe. There are something like ten
million million million million million million million million million million million million million million (1 with eighty
zeros after it) particles in the region of the universe that we can observe. Where did they all come from? The answer
is that, in quantum theory, particles can be created out of energy in the form of particle/antiparticle pairs. But that just
raises the question of where the energy came from. The answer is that the total energy of the universe is exactly
zero. The matter in the universe is made out of positive energy. However, the matter is all attracting itself by gravity.
Two pieces of matter that are close to each other have less energy than the same two pieces a long way apart,
because you have to expend energy to separate them against the gravitational force that is pulling them together.
Thus, in a sense, the gravitational field has negative energy. In the case of a universe that is approximately uniform
in space, one can show that this negative gravitational energy exactly cancels the positive energy represented by the
matter. So the total energy of the universe is zero.
Now twice zero is also zero. Thus the universe can double the amount of positive matter energy and also double the
negative gravitational energy without violation of the conservation of energy. This does not happen in the normal
expansion of the universe in which the matter energy density goes down as the universe gets bigger. It does happen,
however, in the inflationary expansion because the energy density of the supercooled state remains constant while
the universe expands: when the universe doubles in size, the positive matter energy and the negative gravitational
energy both double, so the total energy remains zero. During the inflationary phase, the universe increases its size
by a very large amount. Thus the total amount of energy available to make particles becomes very large. As Guth
has remarked, “It is said that there’s no such thing as a free lunch. But the universe is the ultimate free lunch.”
The universe is not expanding in an inflationary way today. Thus there has to be some mechanism that woul
eliminate the very large effective cosmological constant and so change the rate of expansion from an accelerhat
eventually the symmetry between the forces would be broken, just as super-cooled water always freezes in the end.
The extra energy of the unbroken symmetry state would then be released and would reheat the universe to a
temperature just below the critical temperature for symmetry between the forces. The universe would then go on to
expand and cool just like the hot big bang model, but there would now be an explanation of why the universe was
expanding at exactly the critical rate and why different regions had the same temperature.
In Guth’s original proposal the phase transition was supposed to occur suddenly, rather like the appearance of ice
crystals in very cold water. The idea was that “bubbles” of the new phase of broken symmetry would have formed in
the old phase, like bubbles of steam surrounded by boiling water. The bubbles were supposed to expand and meet
up with each other until the whole universe was in the new phase. The trouble was, as I and several other people
pointed out, that the universe was expanding so fast that even if the bubbles grew at the speed of light, they would
be moving away from each other and so could not join up. The universe would be left in a very non-uniform state,
with some regions still having symmetry between the different forces. Such a model of the universe would not

correspond to what we see.
In October 1981, I went to Moscow for a conference on quantum gravity. After the conference I gave a seminar on
the inflationary model and its problems at the Sternberg Astronomical Institute. Before this, I had got someone else
to give my lectures for me, because most people could not understand my voice. But there was not time to prepare
this seminar, so I gave it myself, with one of my graduate students repeating my words. It worked well, and gave me

A Brief History of Time - Stephen Hawking... Chapter 8
much more contact with my audience. In the audience was a young Russian, Andrei Linde, from the Lebedev
Institute in Moscow. He said that the difficulty with the bubbles not joining up could be avoided if the bubbles were so
big that our region of the universe is all contained inside a single bubble. In order for this to work, the change from
symmetry to broken symmetry must have taken place very slowly inside the bubble, but this is quite possible
according to grand unified theories. Linde’s idea of a slow breaking of symmetry was very good, but I later realized
that his bubbles would have to have been bigger than the size of the universe at the time! I showed that instead the
symmetry would have broken everywhere at the same time, rather than just inside bubbles. This would lead to a
uniform universe, as we observe. I was very excited by this idea and discussed it with one of my students, Ian Moss.
As a friend of Linde’s, I was rather embarrassed, however, when I was later sent his paper by a scientific journal and
asked whether it was suitable for publication. I replied that there was this flaw about the bubbles being bigger than
the universe, but that the basic idea of a slow breaking of symmetry was very good. I recommended that the paper ?
published as it was because it would take Linde several months to correct it, since anything he sent to the West
would have to be passed by Soviet censorship, which was neither very skillful nor very quick with scientific papers.
Instead, I wrote a short paper with Ian Moss in the same journal in which we pointed out this problem with the bubble
and showed how it could be resolved.
The day after I got back from Moscow I set out for Philadelphia, where I was due to receive a medal from the
Franklin Institute. My secretary, Judy Fella, had used her not inconsiderable charm to persuade British Airways to
give herself and me free seats on a Concorde as a publicity venture. However, I .was held up on my way to the
airport by heavy rain and I missed the plane. Nevertheless, I got to Philadelphia in the end and received my medal. I
was then asked to give a seminar on the inflationary universe at Drexel University in Philadelphia. I gave the same
seminar about the problems of the inflationary universe, just as in Moscow.
A very similar idea to Linde’s was put forth independently a few months later by Paul Steinhardt and Andreas
Albrecht of the University of Pennsylvania. They are now given joint credit with Linde for what is called “the new
inflationary model,” based on the idea of a slow breaking of symmetry. (The old inflationary model was Guth’s
original suggestion of fast symmetry breaking with the formation of bubbles.)
The new inflationary model was a good attempt to explain why the universe is the way it is. However, I and several
other people showed that, at least in its original form, it predicted much greater variations in the temperature of the
microwave background radiation than are observed. Later work has also cast doubt on whether there could be a
phase transition in the very early universe of the kind required. In my personal opinion, the new inflationary model is
now dead as a scientific theory, although a lot of people do not seem to have heard of its demise and are still writing
papers as if it were viable. A better model, called the chaotic inflationary model, was put forward by Linde in 1983. In
this there is no phase transition or supercooling. Instead, there is a spin 0 field, which, because of quantum
fluctuations, would have large values in some regions of the early universe. The energy of the field in those regions
would behave like a cosmological constant. It would have a repulsive gravitational effect, and thus make those
regions expand in an inflationary manner. As they expanded, the energy of the field in them would slowly decrease
until the inflationary expansion changed to an expansion like that in the hot big bang model. One of these regions
would become what we now see as the observable universe. This model has all the advantages of the earlier
inflationary models, but it does not depend on a dubious phase transition, and it can moreover give a reasonable size
for the fluctuations in the temperature of the microwave background that agrees with observation.
This work on inflationary models showed that the present state of the universe could have arisen from quite a large
number of different initial configurations. This is important, because it shows that the initial state of the part of the
universe that we inhabit did not have to be chosen with great care. So we may, if we wish, use the weak anthropic
principle to explain why the universe looks the way it does now. It cannot be the case, however, that every initial
configuration would have led to a universe like the one we observe. One can show this by considering a very
different state for the universe at the present time, say, a very lumpy and irregular one. One could use the laws of
science to evolve the universe back in time to determine its configuration at earlier times. According to the singularity
theorems of classical general relativity, there would still have been a big bang singularity. If you evolve such a
universe forward in time according to the laws of science, you will end up with the lumpy and irregular state you
started with. Thus there must have been initial configurations that would not have given rise to a universe like the
one we see today. So even the inflationary model does not tell us why the initial configuration was not such as to
produce something very different from what we observe. Must we turn to the anthropic principle for an explanation?
Was it all just a lucky chance? That would seem a counsel of despair, a negation of all our hopes of understanding
the underlying order of the universe.
In order to predict how the universe should have started off, one needs laws that hold at the beginning of time. If the
classical theory of general relativity was correct, the singularity theorems that Roger Penrose and I proved show that

A Brief History of Time - Stephen Hawking... Chapter 8
the beginning of time would have been a point of infinite density and infinite curvature of space-time. All the known
laws of science would break down at such a point. One might suppose that there were new laws that held at
singularities, but it would be very difficult even to formulate such laws at such badly behaved points, and we would
have no guide from observations as to what those laws might be. However, what the singularity theorems really
indicate is that the gravitational field becomes so strong that quantum gravitational effects become important:
classical theory is no longer a good description of the universe. So one has to use a quantum theory of gravity to
discuss the very early stages of the universe. As we shall see, it is possible in the quantum theory for the ordinary
laws of science to hold everywhere, including at the beginning of time: it is not necessary to postulate new laws for
singularities, because there need not be any singularities in the quantum theory.
We don’t yet have a complete and consistent theory that combines quantum mechanics and gravity. However, we
are fairly certain of some features that such a unified theory should have. One is that it should incorporate
Feynman’s proposal to formulate quantum theory in terms of a sum over histories. In this approach, a particle does
not have just a single history, as it would in a classical theory. Instead, it is supposed to follow every possible path in
space-time, and with each of these histories there are associated a couple of numbers, one represent-ing the size of
a wave and the other representing its position in the cycle (its phase). The probability that the particle, say, passes
through some particular point is found by adding up the waves associated with every possible history that passes
through that point. When one actually tries to perform these sums, however, one runs into severe technical
problems. The only way around these is the following peculiar prescription: one must add up the waves for particle
histories that are not in the “real” time that you and I experience but take place in what is called imaginary time.
Imaginary time may sound like science fiction but it is in fact a well-defined mathematical concept. If we take any
ordinary (or “real”) number and multiply it by itself, the result is a positive number. (For example, 2 times 2 is 4, but
so is

A Brief History of Time - Stephen Hawking... Chapter 8
In the classical theory of gravity, which is based on real space-time, there are only two possible ways the universe
can behave: either it has existed for an infinite time, or else it had a beginning at a singularity at some finite time in
the past. In the quantum theory of gravity, on the other hand, a third possibility arises. Because one is using
Euclidean space-times, in which the time direction is on the same footing as directions in space, it is possible for
space-time to be finite in extent and yet to have no singularities that formed a boundary or edge. Space-time would
be like the surface of the earth, only with two more dimensions. The surface of the earth is finite in extent but it
doesn’t have a boundary or edge: if you sail off into the sunset, you don’t fall off the edge or run into a singularity. (I
know, because I have been round the world!)
If Euclidean space-time stretches back to infinite imaginary time, or else starts at a singularity in imaginary time, we
have the same problem as in the classical theory of specifying the initial state of the universe: God may know how
the universe began, but we cannot give any particular reason for thinking it began one way rather than another. On
the other hand, the quantum theory of gravity has opened up a new possibility, in which there would be no boundary
to space-time and so there would be no need to specify the behavior at the boundary. There would be no
singularities at which the laws of science broke down, and no edge of space-time at which one would have to appeal
to God or some new law to set the boundary conditions for space-time. One could say: “The boundary condition of
the universe is that it has no boundary.” The universe would be completely self-contained and not affected by
anything outside itself. It would neither be created nor destroyed, It would just BE.
It was at the conference in the Vatican mentioned earlier that I first put forward the suggestion that maybe time and
space together formed a surface that was finite in size but did not have any boundary or edge. My paper was rather
mathematical, however, so its implications for the role of God in the creation of the universe were not generally
recognized at the time (just as well for me). At the time of the Vatican conference, I did not know how to use the “no
boundary” idea to make predictions about the universe. However, I spent the following sum-mer at the University of
California, Santa Barbara. There a friend and colleague of mine, Jim Hartle, worked out with me what conditions the
universe must satisfy if space-time had no boundary. When I returned to Cambridge, I continued this work with two of
my research students, Julian Luttrel and Jonathan Halliwell.
I’d like to emphasize that this idea that time and space should be finite “without boundary” is just a proposal: it cannot
be deduced from some other principle. Like any other scientific theory, it may initially be put forward for aesthetic or
metaphysical reasons, but the real test is whether it makes predictions that agree with observation. This, how-ever, is
difficult to determine in the case of quantum gravity, for two reasons. First, as will be explained in Chapter 11, we are
not yet sure exactly which theory successfully combines general relativity and quantum mechanics, though we know
quite a lot about the form such a theory must have. Second, any model that described the whole universe in detail
would be much too complicated mathematically for us to be able to calculate exact predictions. One therefore has to
make simplifying assumptions and approximations