Relationship between Weak Cosmic Censorship and Topological Censorship

Question

The weak cosmic censorship states that any singularity cannot be in the causual past of null infinity (reference).

The topological censorship hypothesis states that in a globally hyperbolic, asymptotically flat spacetime, every causal curve from $J^-$ to $J^+$ is homotopic to a topological trivial curve between the two points. (reference)

I am curious as to the relationship between these two; will a found violation of weak cosmic censorship necessarily mean a violation of topological censorship? [edit: or the converse]

Answer 1

No.

Firstly, weak cosmic censorship can only hold in the generic sense, as there are known examples of nakedly singular space-times. (See, e.g. Christodoulou 1993, and Christodoulou 1999.)

Observe in particular that the nakedly singular space-time constructed in the 1993 paper is spherically symmetric with a central axis, and the initial data is prescribed on a set homeomorphic to $\mathbb{R}^3$. So the maximal globally hyperbolic development of this data, which leads to a naked singularity (hence violating cosmic censorship), is simply connected (homeomorphic to $\mathbb{R}^4$ actually). And hence must satisfy topological censorship.

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BTW, if the implication you want were actually true, then given that nakedly singular solutions are known in the literature, it would be rather difficult to have topological censorship be a proven theorem in the generality that it is usually stated.

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However, it is interesting to note that the converse (or something quite close to it) of the statement you are interested in actually holds. By a result of Galloway and Woolgar you have that, roughly speaking: weak cosmic censorship + null energy condition implies topological censorship. The contrapositive of which would say that failure of topological censorship + null energy condition holding will imply weak cosmic censorship is false.