Page 60 - Nature Of Space And Time

physics: the universe would not return to a smooth state in the collapse. This would mean
that the arrow of time would not reverse. It would continue pointing in the same direction
as in the expansion.
How can the two ends of time be di erent. Why should perturbations be small at
one end but not the other. The reason is there are two possible complex solutions of

the eld equations that match on to a small three sphere boundary. One is as I have
described earlier: it is approximately half the Euclidean four sphere joined to a small
part of the Lorentzian-de Sitter solution. The other possible solution has the same half

Euclidean four sphere joined to a Lorentzian solution that expands to a very large radius
and then contracts again to the small radius of the given boundary. Obviously, one solution
corresponds to one end of time and the other to the other. The di erence between the two
ends comes from the fact that perturbations in the three metric h ij are heavily damped in
the case of the rst solution with only a short Lorentzian period. However the perturbations

can be very large without being signi cantly damped in the case of the solution that
expands and contracts again. This gives rise to the di erence between the two ends of
time that Roger has pointed out. At one end the universe was very smooth and the Weyl
tensor was very small. It could not, however, be exactly zero for that would have been a

violation of the Uncertainty Principle. Instead there would have been small
uctuations
which later grew into galaxies and bodies like us. By contrast, the universe would have
been very irregular and chaotic at the other end of time with a Weyl tensor that was
typically large. This would explain the observed arrow of time and why cups fall o tables

and break rather than mend themselves and jump back on.
As the arrow of time is not going to reverse, and as I have gone over time, I better draw
my lecture to a close. I have emphasized what I consider the two most remarkable features
that I have learnt in my research on space and time: rst, that gravity curls up spacetime

so that it has a begining and an end. Second, that there is a deep connection between
gravity and thermodynamics that arises because gravity itself determines the topology of
the manifold on which it acts.
The positive curvature of spacetime produced singularities at which classical general

relativity broke down. Cosmic Censorship may shield us from black hole singularities but
we see the Big Bang in full frontal nakedness. Classical general relativity cannot predict
how the universe will begin. However quantum general relativity, together with the no
boundary proposal, predicts a universe like we observe and even seems to predict the

observed spectrum of
uctuations in the microwave background. However, although the
quantum theory restores the predictability that the classical theory lost, it does not do so
completely. Because we can not see the whole of spacetime on account of black hole and

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