Why does the cosmic censorship conjecture hold so well? Why does the cosmic censorship conjecture hold so well?
Penrose proposed spacelike singularities and closed timelike curves are always hidden behind event horizons in general relativity. His conjecture appears to hold pretty well. But he only assumed some energy condition (null, weak, dominant) for it to hold, and some topological assumptions on the initial conditions? That makes it a mathematical conjecture? If so, why does it hold so well?
 A: There is a misconception in your answer--- Penrose's censorship conjecture doesn't "hold pretty well", it just holds (or else it doesn't hold). It's a precise conjecture regarding classical solutions to GR obeying energy conditions.
The formation of CTC's is not related to Penrose's conjecture in on obvious way. You could just forbid CTC's from ever forming, they don't emerge in a canonical formulation, where you produce the manifold step by step in time. Penrose didn't just talk about spacelike singularities, timelike singularities are supposed to be cloaked too.
The evidence for cosmic censorship is that to produce a naked singularity, you have to pass through an extremal limit first. Extremal limits are at zero temperature, so this violates the third law of thermodynamics.
For example, if you try to push charge into a black hole to make Q>M, you get a naked singularity. But Q=M is an extremal limit, and it is at zero temperature, so you just end up getting closer and closer to Q=M, and you have to add charge adiabatically, but in adding charge to an extremal black hole, you always end up adding more mass than charge.
You can see that this is true by considering two extremal black holes with charge Q and mass Q. These black holes slide past each other, and come together adiabatically with no force between them. The process of combining them produces a charge 2Q black hole with mass 2Q (in natural units), so that there is no gain in entropy.
For any other process, like slamming the charge Q into the second black hole, you have more energy than the adiabatic sliding of the black holes together, so there is more energy in the system than in the two forceless extremal black hole case.
This is the reason to believe censorship--- the extremal limits are zero temperature, and so maximally unstable to adding energy, and pushing them away from extremality. Naked singularity solutions are generally separated from usual solutions by an extremal boundary which is uncrossable. This is all that Penrose's conjecture is saying.
