in which singularities are predicted is in the past at the begining of the present expansion of the universe. This led to the abandonment of attempts (mainly by the Russians) to argue
that there was a previous contracting phase and a non singular bounce into expansion.
Instead almost everyone now believes that the universe, and time itself, had a begining at
the Big Bang. This is a discovery far more important than a few miscellaneous unstable
particles but not one that has been so well recognized by Nobel prizes.
The prediction of singularities means that classical general relativity is not a complete
theory. Because the singular points have to be cut out of the spacetime manifold one can
not define the field equations there and can not predict what will come out of a singularity.
With the singularity in the past the only way to deal with this problem seems to be to
appeal to quantum gravity. I shall return to this in my third lecture. But the singularities
that are predicted in the future seem to have a property that Penrose has called, Cosmic
Censorship. That is they conveniently occur in places like black holes that are hidden
from external observers. So any break down of predictability that may occur at these
singularities won’t affect what happens in the outside world, at least not according to
classical theory.
Cosmic Censorship
Nature abhors a naked singularity
However, as I shall show in the next lecture, there is unpredictability in the quantum
theory. This is related to the fact that gravitational fields can have intrinsic entropy which is not just the result of coarse graining. Gravitational entropy, and the fact that time has
a begining and may have an end, are the two themes of my lectures because they are the
ways in which gravity is distinctly different from other physical fields.
The fact that gravity has a quantity that behaves like entropy was first noticed in the
purely classical theory. It depends on Penrose’s Cosmic Censorship Conjecture. This is
unproved but is believed to be true for suitably general initial data and equations of state.
I shall use a weak form of Cosmic Censorship.
One makes the approximation of treating the region around a collapsing star as asymptoti-
cally flat. Then, as Penrose showed, one can conformally embed the spacetime manifold M
in a manifold with boundary ¯
M . The boundary ∂M will be a null surface and will consist
of two components, future and past null infinity, called I+ and I−. I shall say that weak Cosmic Censorship holds if two conditions are satisfied. First, it is assumed that the null
15
no future end points for
generators of event horizon
black hole
singularity
event horizon
+
+
past end point of
generators of event horizon
_
I ( )
+
_
_
geodesic generators of I+ are complete in a certain conformal metric. This implies that observers far from the collapse live to an old age and are not wiped out by a thunderbolt
singularity sent out from the collapsing star. Second, it is assumed that the past of I+
is globally hyperbolic. This means there are no naked singularities that can be seen from
large distances. Penrose has a stronger form of Cosmic Censorship which assumes that the
whole spacetime is globally hyperbolic. But the weak form will suffice for my purposes.
Weak Cosmic Censorship
1. I+ and I− are complete.
2. I−( I+) is globally hyperbolic.
If weak Cosmic Censorship holds the singularities that are predicted to occur in grav-
itational collapse can’t be visible from I+. This means that there must be a region of spacetime that is not in the past of I+. This region is said to be a black hole because no light or anything else can escape from it to infinity. The boundary of the black hole region
is called the event horizon. Because it is also the boundary of the past of I+ the event horizon will be generated by null geodesic segments that may have past end points but
don’t have any future end points. It then follows that if the weak energy condition holds
16
the generators of the horizon can’t be converging. For if they were they would intersect