would mend themselves and jump back on the table. If only real life were like that.
The local laws that physical fields obey are time symmetric, or more precisely, CPT
invariant. Thus the observed difference between the past and the future must come from
the boundary conditions of the universe. Let us take it that the universe is spatially closed and that it expands to a maximum size and collapses again. As Roger has emphasized, the
universe will be very different at the two ends of this history. At what we call the begining of the universe, it seems to have been very smooth and regular. However, when it collapses
again, we expect it to be very disordered and irregular. Because there are so many more
disordered configurations than ordered ones, this means that the initial conditions would
have had to be chosen incredibly precisely.
It seems, therefore, that there must be different boundary conditions at the two ends
57
final
observers
singularity
event horizon
world line
of observer
maximum area
of event horizon
centre of
centre of
symmetry
symmetry
Euclidean
region
of time. Roger’s proposal is that the Weyl tensor should vanish at one end of time but not
the other. The Weyl tensor is that part of the curvature of spacetime that is not locally
determined by the matter through the Einstein equations. It would have been small in
the smooth ordered early stages. But large in the collapsing universe. Thus this proposal
would distinguish the two ends of time and so might explain the arrow of time.
universe irregular,
Weyl tensor large
universe smooth,
Weyl tensor small
58
I think Roger’s proposal is Weyl in more than one sense of the word. First, it is not
CPT invariant. Roger sees this as a virtue but I feel one should hang on to symmetries
unless there are compelling reasons to give them up. As I shall argue, it is not necessary
to give up CPT. Second, if the Weyl tensor had been exactly zero in the early universe it
would have been exactly homogeneous and isotropic and would have remained so for all
time. Roger’s Weyl hypothesis could not explain the fluctuations in the background nor
the perturbations that gave rise to galaxies and bodies like ourselves.
Objections to Weyl tensor hypothesis
1. Not CPT invariant.
2. Weyl tensor cannot have been exactly zero. Doesn’t explain small fluctu-
ations.
Despite all this, I think Roger has put his finger on an important difference between
the two ends of time. But the fact that the Weyl tensor was small at one end should
not be imposed as an ad hoc boundary condition, but should be deduced from a more
fundamental principle, the no boundary proposal. As we have seen, this implies that
perturbations about half the Euclidean four sphere joined to half the Lorentzian-de Sitter
solution are in their ground state. That is, they are as small as they can be, consistent
with the Uncertainty Principle. This then would imply Roger’s Weyl tensor condition: the
Weyl tensor wouldn’t be exactly zero but it would be as near to zero as it could be.
At first I thought that these arguments about perturbations being in their ground state
would apply at both ends of the expansion contraction cycle. The universe would start
smooth and ordered and would get more disordered and irregular as it expanded. However,
I thought it would have to return to a smooth and ordered state as it got smaller. This
would have implied that the thermodynamic arrow of time would have to reverse in the
contracting phase. Cups would mend themselves and jump back on the table. People
would get younger, not older, as the universe got smaller again. It is not much good
waiting for the universe to collapse again to return to our youth because it will take too
long. But if the arrow of time reverses when the universe contracts, it might also reverse
inside black holes. However, I wouldn’t recommend jumping into a black hole as a way of
prolonging one’s life.
I wrote a paper claiming that the arrow of time would reverse when the universe
contracted again. But after that, discussions with Don Page and Raymond Laflamme