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I want to get a grip about the technical problems related to proving Penrose conjecture

Intuitively, it seems it should not be too difficult; just start by classifying it in 0-dimensional (i.e: Schwarschild singularity) and 1-dimensional singularities (i.e: Kerr singularities).

Assume we have a naked 0-dimensional singularity, take a neighbourhood small enough around it, how does the field look as the neighbourhood shrinks? can't i say that the asymptotic symmetry around the singularity is spherical? why can't i use this symmetry effectively to prove that there must be a schwarschild-like event horizon?

Assume now we have a naked 1-dimensional singularity, a neighbourhood small enough should have cylindrical symmetry. Find appropiate solutions given from symmetry, repeat and rinse

Where is currently stuck the programme for proving this conjecture? is there some specific step that is problematic, or this whole approach doesn't work for some deeper reasons?

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A couple of things:

1) In the way you phrase it, it will almost certainly not be possible to disprove cosmic censorship, since the singularity you could start with could be, for example, a Kerr solution with $a>M$, in which case, the horizon that would typically appear at $r=M\pm \sqrt{M^{2}-a^{2}}$ would disappear, but there would be no apparent contradiction with Einstein's equation whatsoever, as these solutions are perfectly valid solutions of Einstein's equation. It's the evolution of such a solution at "late" times from "physically reasonable" matter without any black holes or horizons which is the issue.

2) There are fine-tunings of physically realistic initial data where cosmic censorship is false. This was originally discovered by numerical studies by Choptuik regarding spherically symmetric collapse of Klein-Gordon fields. Modern restatings of cosmic censorship attempt to show the much weaker claim that the complement of the set of initial data that evolves into naked singularities forms a dense subset of the set of all initial data. In other words, the claim is that you have to do an exact, and unstable, fine-tuning of initial data in order to produce a naked singularity.

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interesting stuff, i didn't know any of that (i mean the part where the censorship conjecture was false for some "cherrypicked" initial states) –  lurscher Oct 4 '11 at 17:46
    
it would seem extraordinarily hard to prove this weaker version of the theorem, since system evolution intrinsically maps open sets chaotically –  lurscher Oct 4 '11 at 18:26
    
@lurscher: yep. That's why it isn't proven yet. One of those weird results that is almost certainly true, but that doesn't have good proof. –  Jerry Schirmer Oct 4 '11 at 23:04
    
@Jerry: I think that the interpretation given here for Choptuik's results, that they represent a failure of cosmic censorship, is no good. Although these solutions are fine-tuned to be just on the boundary of producing a black hole. The Choptuik results give a critical behavior for the scaler-gravitational system which is interestingly self-similar, but it's implications for censorship are minimal. –  Ron Maimon Oct 5 '11 at 4:22
    
@RonMaimon: It was enough for Stephen Hakwing to cede his cosmic censorship bet to Kip Thorne. The critical solution is a naked singularity. –  Jerry Schirmer Oct 5 '11 at 10:36
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The method you suggest for solving the censorship conjecture starts with a singularity and attempts to show that it must be surrounded by a horizon. This cannot work, for reasons described by Jerry Schermer. There are many good solutions of Einstein's equations with a naked singularity, for example, the Schwartschild solution for negative mass:

$$ ds^2 = -(1+{2m\over r})dt^2 + {1\over 1+{2m\over r}} dr^2 + r^2 d\Omega$$

Which is spherically symmetric, satisfies Einstein's equations, and is manifestly horizon free and nonsingular away from $r=0$, which is naked. The point of the censorship conjecture is to prove that such a solution can never arise from sensible initial conditions. This example is excluded by the positive mass theorem, for example.

The boundary between black hole and nakedly singular solutions is the extremal solutions, which are thermodynamically zero temperature, and are as hard to realize as any other zero-temperature limit, by the third law of thermodynamics. To get a black hole to extremality, you need to cool it down carefully. One reason Penrose conjectured that naked singularities do not form is the difficulty of passing to the extremal limit, let alone through it, as you would need to do for a naked singularity.

The so-called counterexamples to censorship are cases which are fine-tuned to be on the boundary of forming a black hole. These results are very important in themselves, but have no bearing on the truth of the conjecture. The Choptuik results show that you can form a self-gravitating marginally stable system by fine tuning the initial conditions for an infalling scalar be just on the edge of forming a black-hole. The interesting result is that there is a self-similar powerlaw of scalar field energy density with a singular blow-up near $r=0$, but this naked singularity obviously resolves into a black hole at infinitesimally greater concentrations of energy, and resolves into a nonsingular solution for slightly weaker concentrations. The point is that there is a one-parameter tuning that can produce these things.

Cosmic censorship pretty much implies that the evolution equations can be globally extended on the exterior of closed trapped surfaces from any asymptotically flat initial conditions to give a regular manifold with a time/space slicing. This is as difficult as any other global problem.

But perhaps there is a physical shortcut. In quantum gravity, using AdS/CFT, you can find a non-gravitational dual, which is flat in the large-N limit. Then you must be able to translate any physical classical initial conditions with dust into a quantum field theory state, and try to extract the space-time behavior from the field, and show it is not singular outside of the horizon. The AdS/CFT description is always exterior to the horizon anyway, since the black-holes are evaporating, and it might be possible to prove that the gravitational field is nonsingular. But the problem of mapping classical gravitational solutions to AdS/CFT states is not sufficiently well understood to carry out this program.

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