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Many consider naked singularities as a fundamental problem and that it should be always covered by a horizon (Cosmic censorship hypothesis). But why naked singularities are really a problem?

If we consider electrodynamics and the Coulomb potential, we have a singularity at $r=0$ but quantum electrodynamics solves the problem. General relativity being a classical theory we have also a singularity and with the hope that quantum gravity will remove it. But we don't need necessarily a horizon.

At classical level, naked singularities should not scare us. Why are they always disregarded?

Edit after the answer by John Rennie:

Thanks John Rennie for your answer. But we actually express exactly the same thing. I'm not saying that a singularity is not a problem, of course it is. But that singularity is not a problem within a classical theory because we expect or hope that the problem will be solved in the quantum regime.

Going back to my previous example, classical electrodynamics, no-one tries to hide a singularity behind a horizon but in general relativity we try to solve the singularity problem within the classical theory. The "Cosmic censorship hypothesis" tries to "solve" (most exactly hide) the problem within the classical regime.

My question then is, why not thinking that a naked singularity is a fair solution in the classical theory but the singularity would disappear in the quantum regime without necessarily imposing a horizon in the classical theory.

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    $\begingroup$ "At classical level, naked singularities should not scare us." Why not? $\endgroup$
    – probably_someone
    Commented Dec 7, 2018 at 17:22
  • $\begingroup$ Because as for the Coulomb potential, the singularity disappears once we add quantum corrections. So we know that the singularity exists only because we don't have a full description of the reality. In the same way, quantum gravity should eliminate the singularity so we should care about it at classical level. $\endgroup$
    – ziususdra
    Commented Dec 7, 2018 at 17:40
  • $\begingroup$ "quantum gravity" - sure, but we have no consistent description of such a thing. Such a thing may not exist at all. $\endgroup$
    – Maury Markowitz
    Commented Dec 7, 2018 at 17:54
  • $\begingroup$ @ziususdra What makes you so sure that quantum corrections will make this singularity disappear? Without the full theory it's impossible to know that. $\endgroup$
    – probably_someone
    Commented Dec 7, 2018 at 18:05
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    $\begingroup$ @ziususdra Just because one possible theory doesn't have these singularities doesn't mean that the theory that actually describes reality doesn't have them. As of yet, there is no evidence that whatever flavor of string theory you're talking about makes more accurate predictions than the current Standard Model and/or GR. $\endgroup$
    – probably_someone
    Commented Dec 7, 2018 at 20:12

2 Answers 2

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Your initial assumption is wrong. Classical singularities do scare us, but we have a resolution for that problem because quantum mechanics modifies the classical behaviour at short distances. For general relativity no such safety net currently exists, though most of us believe quantum mechanics will remove the singularities in GR as well.

It is a basic requirement of a theory that if we know the state of a system then our theory can predict its future evolution. Technically this property of a theory is called global hyperbolicity. If a theory is not globally hyperbolic then causality breaks down because we cannot predict what cause will have what effect.

The problem with a singularity is that while we can calculate the trajectory of an infalling particle up to the moment it reaches the singularity we cannot predict what happens at the singularity or for any later time. This happens because the curvature tends to infinity as we approach the singularity and we can't do arithmetic with infinity.

However provided the singularity is hidden behind an event horizon the unpredictability doesn't matter because everything behind the horizon is causally disconnected from us - the unpredictability can never affect anything that we can observe. But if the singularity is not behind a horizon, i.e. it is naked, then what happens there can and will affect us. That means our theory (GR) is no longer globally hyperbolic and therefore cannot predict the future. We're in trouble!

This also happens in classical physics, and you use the example of the Coulomb potential. If we consider a positive and negative charge on a direct collision course then their equations of motion also become singular when the distance between them falls to zero, and there is no way to calculate what happens afterwards. But of course we know that we have to resort to quantum field theory at very short distances, and this removes the singularity. Panic over.

The problem is that quantum mechanics does not (currently) come to our rescue in GR because we have no theory of quantum gravity. As I mentioned at the start, I doubt you'd find a theoretical physicist who really believes singularities exist - we all think some form of quantum effect will remove them. But this is currently only wishful thinking and there is zero evidence to support it.

There is one final point to be made about causality. GR is time symmetric, and that means if we cannot predict what happens for particles hitting a singularity that also means we cannot predict what comes out of the singularity. If we observed a naked singularity we simply could not tell what it would do next.

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  • $\begingroup$ Isn't the Big Bang a naked singularity? The whole universe got out of that singularity, and of course it wasn't "predictible"! $\endgroup$
    – Cham
    Commented Dec 13, 2018 at 12:30
  • $\begingroup$ @Cham: A spacetime is globally hyperbolic if (1) strong causality holds (essentially no CTCs), and (2) $\forall p$, $q$, the future timelike light cone of $p$ intersected with the past timelike light cone of $q$ is compact. By this definition, cosmological spacetimes are globally hyperbolic, despite the presence of the big bang singularity. That means we have predictability, in the sense of existence and uniqueness for Cauchy problems. You can define the big bang as a naked singularity if you like, but what matters is that it's not timelike. $\endgroup$
    – user4552
    Commented Dec 13, 2018 at 14:45
  • $\begingroup$ Yes, I agree that the Big Bang is a spacelike singularity. But it is "removed" from the manifold (like most singularities) so of course the manifold is globally hyperbolic. But the Big Bang is still a naked singularity out of which everything came out. This is the kind of "troubles" we could get from a naked singularity, like the Reisner-Nordstrom solution with $Q > M$ for example (if I remember well, there's a gravitational repulsion, close to the naked singularity in this case). $\endgroup$
    – Cham
    Commented Dec 13, 2018 at 15:20
  • $\begingroup$ @Cham yes, I agree, though I suspect this is going beyond what the OP was asking about. $\endgroup$
    – John Rennie
    Commented Dec 13, 2018 at 16:18
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    $\begingroup$ @Cham : The big bang is not a naked singularity. And when you remove the singularity from the manifold (more accurately it was never a part of the manifold) you don't always get a globally hyperbolic space-time. $\endgroup$
    – MBN
    Commented Dec 14, 2018 at 9:47
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It's also worth noting that black hole solutions with naked singularities tend to have other pathologies, like closed timelike curves, that don't directly involve the singularity, and even if you removed the singularity with a patch, these other pathologies would remain.

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  • $\begingroup$ That's interesting. Do you have some reference linking naked singularity and closed timelike curves. Because at least we can give some counterexamples in both sides. A solution which has CTC's and which is not a naked singularity: the Godel universe. And a solution which is a naked singularity and doesn't have CTC like Schwarzschild with M<0. $\endgroup$
    – ziususdra
    Commented Dec 14, 2018 at 19:08
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    $\begingroup$ @ziususdra: Cosmic censorship doesn't fix the CTC problem in general (to my knowledge, there isn't a general proof that shows when CTC will show up), but it's somethign to consider in about some of these naked singularity solutions. The presence of CTC in the interior of Kerr solutions well known, but see: arxiv.org/pdf/0708.2324.pdf $\endgroup$
    – Zo the Relativist
    Commented Dec 14, 2018 at 20:22
  • $\begingroup$ It's written by unknown people with only 1 citation and their conclusion is trivial and not realistic. Trivial because they consider a motion in opposite direction to BH within the ergosphere, so of course this observer would need a velocity larger than the speed of light which permits CTC. But if we assume that no-one could go faster than speed of light, this observer doesn't exist. At the same time, in realistic solution, there is a fluid collapsing, which eliminates the multiple copies of spacetime in the maximal extension and therefore I'm not sure that it would really produce such effects $\endgroup$
    – ziususdra
    Commented Dec 14, 2018 at 21:08
  • $\begingroup$ @ziususdra I googled for the first result. It is very well known that you get closed timelike curves by traversing through the ring singularity, and therefore, in a naked singularity solution, you expose them to the general spacetime. References are readily available to any level of rigor you want. $\endgroup$
    – Zo the Relativist
    Commented Dec 14, 2018 at 22:07
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    $\begingroup$ Yes but it is not physical, Kerr says it better than me (with my example of matter collapsing) youtube.com/watch?v=LeLkmS3PZ5g&t=26m $\endgroup$
    – ziususdra
    Commented Dec 15, 2018 at 1:12

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