Page 69 - A Brief History of Time - Stephen Hawking

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 – 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.

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