convinced me that I had made my greatest mistake, or at least my greatest mistake in
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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 different. Why should perturbations be small at
one end but not the other. The reason is there are two possible complex solutions of
the field 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 difference between the two
ends comes from the fact that perturbations in the three metric hij are heavily damped in the case of the first solution with only a short Lorentzian period. However the perturbations can be very large without being significantly damped in the case of the solution that
expands and contracts again. This gives rise to the difference 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 fluctuations
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 off 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: first, 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 fluctuations 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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cosmological event horizons, our observations are described by an ensemble of quantum states rather than by a single state. This introduces an extra level of unpredictability but
it may also be why the universe appears classical. This would rescue Schrödinger’s cat
from being half alive and half dead.
To have removed predictability from physics and then to have put it back again, but
in a reduced sense, is quite a success story. I rest my case.