After decades of inconclusive work, it seems that there may have been some dramatic progress within the last few years on the cosmic censorship conjecture (CCC). Joshi and Malafarina claim in a 2014 paper that collapse of a spherically symmetric cloud of dust (i.e., a pressureless perfect fluid) results in a locally naked singularity except if the initial conditions are fine-tuned. (There is also an earlier long review article by the same authors.)

This does not necessarily mean on the face of it that CCC is dead, since spacetimes with spherical symmetry are themselves finely tuned in some sense, but it is rather a dramatic development, since people had imagined for 75 years, based on the Oppenheimer-Snyder calculations for homogeneous dust, that a black hole was the generic result of runaway gravitational collapse. CCC is in a sense impossible to disprove, since part of the research program is to find the most appropriate definition of the conjecture, but these results suggest that if it is to be true, then it has to be weakened so much as to be of little interest.

Reading the 2014 Joshi-Malafarina review article, I find myself frustrated because there are very few equations, diagrams, or rigorous definitions, except in the section where they discusses some specific spherically symmetric paper-and-pencil models in coordinate-dependent form. This makes it hard to be sure what they mean in some of their vague, wordy discussion of topics like classification of singularities and how they evolve.

Can anyone point me to any references that would help me to resolve the following questions?

Are there rigorous, widely known definitions standing behind some of the terminology the authors use such as "simultaneous singularity" or "event-like singularity?" (This type of definition is nontrivial, since we don't have a metric at a singularity.)

They make various statements that I'm having a hard time evaluating as to whether they are coordinate-dependent. E.g., "Generically [...], the mass function has a vanishing or positive value in the limit of approach to the singularity which is visible."

Does numerical work shed any light on whether these violations of CCC are stable with respect to breaking of spherical symmetry?

Are these calculations being continued past a Cauchy horizon, and if so, are reliable criteria being used to decide how to do so? They seem to dismiss such issues cavalierly with comments like this: "Questions such as what will come out of a naked singularity [that forms at the end of a gravitational collapse] are then not really meaningful; 'things' do not have to come out of it."

Is there a more rigorous literature on topics such as the classification of such singularities in GR and what it would mean for them to evolve? My impression is that this would have to be approached through boundary constructions, which seem to be a vexed and controversial field.

Violation of cosmic censorship in the gravitational collapse of a dust cloud. Commun. Math. Phys. 93 (1984), 171–195. (This is not the scalar field model mentioned above.) A good summary of existing work up to 2011 can be found in Kommemi's paper which incidentally answers your question about "boundaries" for all matter models that he calls "strongly/weakly tame". (Note that dust model is explicitly not tame if you treat shell crossings as singularities.) $\endgroup$ – Willie Wong Oct 23 '17 at 19:36