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As I understand it, the basic motivation behind ruling out a naked singularity is that we don't know what is happening at a singularity and thus, we won't be able to predict anything in the universe if there is no horizon around such an unknown region. But the reason we don't understand what is happening at the singularity is that we don't have a theory of quantum gravity. But when we have a theory of quantum gravity, this limitation should go away. And thus, causality should be preserved even with naked singularities.

It is a very cultural fact that we don't know how to deal with singularities without horizons at this stage. Thus, it seems quite naive to assume that causality would actually be violated if horizons don't cover the singularity. Though, I believe under some restricted energy conditions the censorship conjecture has been proven and thus, the censorship might be correct due to some other than causality reasons but causality doesn't seem to force the censorship.

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  • $\begingroup$ Kerr solutions have closed timelike curves. $\endgroup$ – tfb Jun 15 '17 at 19:52
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The reason naked singularities are a problem is not that they imply causality violation in the sense of closed timelike curves existing (although sometimes they do: see below), it is that they imply that GR is not a useful theory, even in the cases where it ought to be useful, because the future can't be predicted from the past in many cases. So, in particular, if GR predicts that uncensored singularities arise when starting from physically-reasonable initial conditions, then GR is not useful at predicting what happens in those cases: you need a better theory which makes useful predictions about what happens when GR predicts a singularity.

If cosmic censorship fails, then GR thus fails to be a usefully predictive theory in many cases. In particular it ceases (or may cease) to be a usefully predictive theory for cosmology. Well, we would like it to be useful for cosmology of course.

So the question that cosmic censorship seeks to answer is 'is GR, which we know is not a completely correct theory, still usable in the regimes where we would like it to be a good approximation, or does it fail even there?'.

Note that a reasonable (indeed common) definition of 'causality violating' is 'usefully predictive', as Ben Crowell says in a comment: in that sense naked singularities always violate causality.

However it is actually worse than that. As mentioned in other answers some solutions (Kerr) can have both naked singularities and CTCs while some (Reissner-Nordström) have only naked singularities.

But these are two different pathologies. So it is not sufficient to have some QG theory which fixes the singularities: that theory would also need to fix the CTCs.

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  • $\begingroup$ Thank you for your answer. Although it's not related to my original question, can you elaborate why CTCs are considered highly pathological? Except for messing with the human intuition of not being able to going into one's own past, does it create any concrete theoretical/mathematical issues that an "intuition-less" theoretical physicist would appreciate? $\endgroup$ – Dvij Mankad Jun 29 '17 at 14:54
  • $\begingroup$ I think that might be worth an independent question: it's interesting enough, and you will get better answers than this as more people will see it. However I think the problem is that, since there are now events which are in their own pasts, it becomes impossible to predict the future in the way you would like: so if I take some suitable spacelike surface (a Cauchy surface) I can no longer predict the future from it. $\endgroup$ – tfb Jun 29 '17 at 16:34
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    $\begingroup$ The reason naked singularities are a problem is not that they imply causality violation (although sometimes they do: see below) A naked singularity always implies causality violation, if you're using the (AFAIK) standard condition that the spacetime should be globally hyperbolic. If you lack global hyperbolicity, then you don't have existence and uniqueness for solutions to Cauchy problems, and that's pretty much the definition of violating causality. $\endgroup$ – Ben Crowell Jun 29 '17 at 16:41
  • $\begingroup$ @BenCrowell: I agree with that. I was using a definition in the sense of 'closed timelike curves existing', but I had not stated that. I've elaborated the answer to be, I hope, more satisfactory (at least it now says what I mean!) $\endgroup$ – tfb Jun 29 '17 at 17:12
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There are closed timelike curves in the interior of the Kerr horizon. The obvious way to see this is if you go through the center of the ring singularity (thus, not intersecting the ring singularity), the Boyer-Lindquist $r$ goes negative, and the Boyer-Lindquist $\phi$ becomes timelike. Since, by construction, the orbits of $\phi$ are closed, this means that they are closed timelike curves.

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    $\begingroup$ Thank you for your answer! But I don't understand how closed timelike curves are related to naked singularities. Can you explain a bit? $\endgroup$ – Dvij Mankad Jun 16 '17 at 5:38
  • $\begingroup$ @Dvij There are closed timelike curves in the interior of the kerr horizon. What's there to explain? If you strip away the horizon, there are regions where the past flows into the future. $\endgroup$ – Jerry Schirmer Jun 16 '17 at 16:12
  • $\begingroup$ @Dvij: I guess what I"m saying is that there is a known class of GR solutions (namely, the $a > M$ kerr models), that has a naked singularity and that has closed timelike curves. Therefore, if it were possible to "spin up" a Kerr hole such that $a > M$, then it would also be possible to create causality violations. The only way we prevent this is through cosmic censorship. $\endgroup$ – Jerry Schirmer Jun 26 '17 at 15:03
  • $\begingroup$ Ok. I understand it that in the case of the Kerr solutions, the only way to prevent causality (i.e. preclude naked CTCs) is to avoid naked singularities. But does this imply that we must avoid naked singularities in all the cases? I mean, for example, in a pure RN solution if we admit the super-extremal case then there are no CTCs but we have the naked singularity. How does a naked singularity (by itself) violate causality (considering that actually there exists a quantum gravity theory that is in principle capable of figuring out what is happening at the center)? $\endgroup$ – Dvij Mankad Jun 26 '17 at 15:18
  • $\begingroup$ Why the downvotes? $\endgroup$ – Jerry Schirmer Jun 30 '17 at 3:36
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To my knowledge a naked singularity doesn't imply closed time like curves or other alteration of the ordering of events. I agree with the OP that a primary example is an overcharged Reisser-Nordstrom.

Still, a naked singularity is a problem, so an actual theory of quantum gravity will need to remove this pathologies. To be more explicit, a naked singularity means that the space it's not globally hyperbolic, that is there isn't a Cauchy surface, that is given a set of valid and complete initial conditions I cannot predict the future, since singularities act as disturbance points in your equations. See Wald for more info.

I personally found solutions to supergravity (related to some string theory configurations of branes) with the same asymptotic charges of a naked singularity, but without actual singularities (https://arxiv.org/pdf/1701.05520.pdf, but it's technical, you have been warned!).

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  • $\begingroup$ Thank you for your answer and the reference therein. Can you elaborate how "a naked singularity is a problem" on its own? As you agree, a naked singularity doesn't necessarily imply CTCs. And if we have a proper theory of QG (which the nature itself presumably has) then what is going to come out of the naked singularity is not really indeterminate. It would be dictated by the laws of QG. And thus, I believe naked singularities shouldn't cause a problem of broken down predictability. Can you elaborate what kind of problems you have in mind that a naked singularity can cause? $\endgroup$ – Dvij Mankad Jun 29 '17 at 13:45
  • $\begingroup$ In reference to some recent literature (arxiv.org/pdf/1702.05490.pdf), existence of naked singularities might mean some problems for the weak gravity conjecture. The weak gravity conjecture, I feel, is most probably correct based on many impressive restricted proofs that we have obtained for it so far. $\endgroup$ – Dvij Mankad Jun 29 '17 at 13:52
  • $\begingroup$ A naked singularity does imply causality violation. When you have a naked singularity, the spacetime is not globally hyperbolic. Global hyperbolicity is the condition that's needed if you want solutions to Cauchy problems to exist and be unique. $\endgroup$ – Ben Crowell Jun 29 '17 at 16:38
  • $\begingroup$ It is a matter of how you define causality. Here the OP was not considering the rigorous definition of causality, but the more common meaning of "well ordered causal flow". I agree that it can be misleading. $\endgroup$ – Rexcirus Jun 29 '17 at 17:39

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