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 – and even then, the problem of extracting predictions remains a
formidable one.
Each history in the sum over histories will describe not only the space-time but everything in it as well, including any
complicated organisms like human beings who can observe the history of the universe. This may provide another
justification for the anthropic principle, for if all the histories are possible, then so long as we exist in one of the
histories, we may use the anthropic principle to explain why the universe is found to be the way it is. Exactly what
meaning can be attached to the other histories, in which we do not exist, is not clear. This view of a quantum theory
of gravity would be much more satisfactory, however, if one could show that, using the sum over histories, our
universe is not just one of the possible histories but one of the most probable ones. To do this, we must perform the
sum over histories for all possible Euclidean space-times that have no boundary.
Under the “no boundary” proposal one learns that the chance of the universe being found to be following most of the
possible histories is negligible, but there is a particular family of histories that are much more probable than the
others. These histories may be pictured as being like the surface of the earth, with the distance from the North Pole
representing imaginary time and the size of a circle of constant distance from the North Pole representing the spatial
size of the universe. The universe starts at the North Pole as a single point. As one moves south, the circles of
latitude at constant distance from the North Pole get bigger, corresponding to the universe expanding with imaginary
time Figure 8:1. The universe would reach a maximum size at the equator and would contract with increasing
imaginary time to a single point at the South Pole. Ever though the universe would have zero size at the North and
South Poles, these points would not be singularities, any more than the North aid South Poles on the earth are
singular. The laws of science will hold at them, just as they do at the North and South Poles on the earth.
Figure 8:1
The history of the universe in real time, however, would look very different. At about ten or twenty thousand million
years ago, it would have a minimum size, which was equal to the maximum radius of the history in imaginary time. At
later real times, the universe would expand like the chaotic inflationary model proposed by Linde (but one would not
now have to assume that the universe was created somehow in the right sort of state). The universe would expand to
a very large size Figure 8:1 and eventually it would collapse again into what looks like a singularity in real time. Thus,
in a sense, we are still all doomed, even if we keep away from black holes. Only if we could picture the universe in
terms of imaginary time would there be no singularities.
If the universe really is in such a quantum state, there would be no singularities in the history of the universe in
imaginary time. It might seem therefore that my more recent work had completely undone the results of my earlier
work on singularities. But, as indicated above, the real importance of the singularity theorems was that they showed
that the gravitational field must become so strong that quantum gravitational effects could not be ignored. This in turn
led to the idea that the universe could be finite in imaginary time but without boundaries or singularities. When one
goes back to the real time in which we live, however, there will still appear to be singularities. The poor astronaut
who falls into a black hole will still come to a sticky end; only if he lived in imaginary time would he encounter no
singularities.
This might suggest that the so-called imaginary time is really the real time, and that what we call real time is just a
figment of our imaginations. In real time, the universe has a beginning and an end at singularities that form a
boundary to space-time and at which the laws of science break down. But in imaginary time, there are no
singularities or boundaries. So maybe what we call imaginary time is really more basic, and what we call real is just
an idea that we invent to help us describe what we think the universe is like. But according to the approach I
described in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: it
exists only in our minds. So it is meaningless to ask: which is real, “real” or “imaginary” time? It is simply a matter of
which is the more useful description.
One can also use the sum over histories, along with the no boundary proposal, to find which properties of the
universe are likely to occur together. For example, one can calculate the probability that the universe is expanding at
nearly the same rate in all different directions at a time when the density of the universe has its present value. In the
simplified models that have been examined so far, this probability turns out to be high; that is, the proposed no
boundary condition leads to the prediction that it is extremely probable that the present rate of expansion of the
universe is almost the same in each direction. This is consistent with the observations of the microwave background
radiation, which show that it has almost exactly the same intensity in any direction. If the universe were expanding
faster in some directions than in others, the intensity of the radiation in those directions would be reduced by an
additional red shift.
Further predictions of the no boundary condition are currently being worked out. A particularly interesting problem is
the size of the small departures from uniform density in the early universe that caused the formation first of the
galaxies, then of stars, and finally of us. The uncertainty principle implies that the early universe cannot have been
completely uniform because there must have been some uncertainties or fluctuations in the positions and velocities
of the particles. Using the no boundary condition, we find that the universe must in fact have started off with just the
minimum possible non-uniformity allowed by the uncertainty principle. The universe would have then undergone a
period of rapid expansion, as in the inflationary models. During this period, the initial non-uniformities would have
been amplified until they were big enough to explain the origin of the structures we observe around us. In 1992 the
Cosmic Background Explorer satellite (COBE) first detected very slight variations in the intensity of the microwave
background with direction. The way these non-uniformities depend on direction seems to agree with the predictions
of the inflationary model and the no boundary proposal. Thus the no boundary proposal is a good scientific theory in
the sense of Karl Popper: it could have been falsified by observations but instead its predictions have been
confirmed. In an expanding universe in which the density of matter varied slightly from place to place, gravity would
have caused the denser regions to slow down their expansion and start contracting. This would lead to the formation
of galaxies, stars, and eventually even insignificant creatures like ourselves. Thus all the complicated structures that
we see in the universe might be explained by the no boundary condition for the universe together with the uncertainty
principle of quantum mechanics.
The idea that space and time may form a closed surface without boundary also has profound implications for the role
of God in the affairs of the universe. With the success of scientific theories in describing events, most people have
come to believe that God allows the universe to evolve according to a set of laws and does not intervene in the
universe to break these laws. However, the laws do not tell us what the universe should have looked like when it
started – it would still be up to God to wind up the clockwork and choose how to start it off. So long as the universe
had a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having no
boundary or edge, it would have neither beginning nor end: it would simply be. What place, then, for a creator?
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PREVIOUS NEXT
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CHAPTER 9
THE ARROW OF TIME
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In previous chapters we have seen how our views of the nature of time have changed over the years. Up to the
beginning of this century people believed in an absolute time. That is, each event could be labeled by a number
called “time” in a unique way, and all good clocks would agree on the time interval between two events.
However, the discovery that the speed of light appeared the same to every observer, no matter how he was
moving, led to the theory of relativity – and in that one had to abandon the idea that there was a unique
absolute time. Instead, each observer would have his own measure of time as recorded by a clock that he
carried: clocks carried by different observers would not necessarily agree. Thus time became a more personal
concept, relative to the observer who measured it.
When one tried to unify gravity with quantum mechanics, one had to introduce the idea of “imaginary” time.
Imaginary time is indistinguishable from directions in space. If one can go north, one can turn around and head
south; equally, if one can go forward in imaginary time, one ought to be able to turn round and go backward.
This means that there can be no important difference between the forward and backward directions of
imaginary time. On the other hand, when one looks at “real” time, there’s a very big difference between the
forward and backward directions, as we all know. Where does this difference between the past and the future
come from? Why do we remember the past but not the future?
The laws of science do not distinguish between the past and the future. More precisely, as explained earlier,
the laws of science are unchanged under the combination of operations (or symmetries) known as C, P, and T.
(C means changing particles for antiparticles. P means taking the mirror image, so left and right are
interchanged. And T means reversing the direction of motion of all particles: in effect, running the motion
backward.) The laws of science that govern the behavior of matter under all normal situations are unchanged
under the combination of the two operations C and P on their own. In other words, life would be just the same
for the inhabitants of another planet who were both mirror images of us and who were made of antimatter,
rather than matter.
If the laws of science are unchanged by the combination of operations C and P, and also by the combination C,
P, and T, they must also be unchanged under the operation T alone. Yet there is a big difference between the
forward and backward directions of real time in ordinary life. Imagine a cup of water falling off a table and
breaking into pieces on the floor. If you take a film of this, you can easily tell whether it is being run forward or
backward. If you run it backward you will see the pieces suddenly gather themselves together off the floor and
jump back to form a whole cup on the table. You can tell that the film is being run backward because this kind
of behavior is never observed in ordinary life. If it were, crockery manufacturers would go out of business.
The explanation that is usually given as to why we don’t see broken cups gathering themselves together off the
floor and jumping back onto the table is that it is forbidden by the second law of thermodynamics. This says that
in any closed system disorder, or entropy, always increases with time. In other words, it is a form of Murphy’s
law: things always tend to go wrong! An intact cup on the table is a state of high order, but a broken cup on the
floor is a disordered state. One can go readily from the cup on the table in the past to the broken cup on the
floor in the future, but not the other way round.
The increase of disorder or entropy with time is one example of what is called an arrow of time, something that
distinguishes the past from the future, giving a direction to time. There are at least three different arrows of
time. First, there is the thermodynamic arrow of time, the direction of time in which disorder or entropy
increases. Then, there is the psychological arrow of time. This is the direction in which we feel time passes, the
direction in which we remember the past but not the future. Finally, there is the cosmological arrow of time. This
is the direction of time in which the universe is expanding rather than contracting.
In this chapter I shall argue that the no boundary condition for the universe, together with the weak anthropic
principle, can explain why all three arrows point in the same direction – and moreover, why a well-defined arrow
