# Naked singularities, timelike singularities

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## Main Question or Discussion Point

Is there a logical connection between the concept of a naked singularity and the concept of a timelike singularity?

On a Penrose diagram, black hole and big bang singularities are always spacelike.

Global hyperbolicity (Hawking and Ellis, p. 206) basically means two conditions: (1) no CTCs, and (2) $\forall p,q\ J^+(p) \cap J^-(q)$ is compact. (J+ and J- mean future and past timelike lightcones, and compactness essentially says the set doesn't contain any singularities or points at infinity.) The point of global hyperbolicity is that it guarantees existence and uniqueness of solutions to Cauchy problems.

If I try to imagine Penrose diagrams where condition 2 fails, it seems like I need to have a timelike singularity. It doesn't seem like it matters whether there's a horizon, and yet, it seems like the whole reason we care about naked singularities is that they violate uniqueness of solutions of Cauchy problems. (I.e., anything can pop out of a naked singularity, even "green slime and lost socks.") So I'm thinking there must be some link between timelike singularities and naked ones.

I got started on trying to understand these definitions because someone told me that the generalization of the Hopf-Rinow theorem to semi-Riemannian spaces requires hyperbolicity. Hopf-Rinow in a Riemannian space basically says that any two points can be connected by a geodesic, and that geodesic's length is extremal. If I try to see why this would fail if hyperbolicity fails in the semi-Riemannian case, it seems like it could clearly fail if I had a timelike singularity, because the singularity could lie between two spacelike-related points p and q, blocking what would have otherwise been a spacelike geodesic connecting them.

On a separate but somewhat related topic, can anyone help me understand the distinction between the weak and strong causality conditions (H&E pp. 190, 192)? I don't understand what their definition of the latter is trying to express, and I don't understand the example they give in a figure where only weak causality holds. Violating it would seem to mean that a nonspacelike curve would have to pass through a neighborhood twice, but I don't see that happening in the figure.

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PAllen
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I got started on trying to understand these definitions because someone told me that the generalization of the Hopf-Rinow theorem to semi-Riemannian spaces requires hyperbolicity. Hopf-Rinow in a Riemannian space basically says that any two points can be connected by a geodesic, and that geodesic's length is extremal. If I try to see why this would fail if hyperbolicity fails in the semi-Riemannian case, it seems like it could clearly fail if I had a timelike singularity, because the singularity could lie between two spacelike-related points p and q, blocking what would have otherwise been a spacelike geodesic connecting them.
I would think there needs to something more to the generalization, at least the part about length being extremal. Maybe: if two events are causally connected, there is at least one timelike geodesic connecting them, and any such geodesic is a local maximum in the variational sense; one of them will be a global maximum. Most any stronger statement I can conceive of is trivially false in perfectly normal spacetimes (including attempting to pose any extremal property for spacelike geodesics is false even in Minkowski space).

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I would think there needs to something more to the generalization, at least the part about length being extremal. Maybe: if two events are causally connected, there is at least one timelike geodesic connecting them, and any such geodesic is a local maximum in the variational sense; one of them will be a global maximum. Most any stronger statement I can conceive of is trivially false in perfectly normal spacetimes (including attempting to pose any extremal property for spacelike geodesics is false even in Minkowski space).
I'm sure I'm getting the details wrong. I don't have a source that gives the exact statement of the generalization. All I'm working from is a general, brief summary by someone else: http://math.stackexchange.com/questions/194353/geodesic-flows-and-curvature/194827#194827 (See user "student's" comment on the answer.) I would actually be grateful if anyone could point to a complete formulation.

PAllen
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I'm sure I'm getting the details wrong. I don't have a source that gives the exact statement of the generalization. All I'm working from is a general, brief summary by someone else: http://math.stackexchange.com/questions/194353/geodesic-flows-and-curvature/194827#194827 (See user "student's" comment on the answer.) I would actually be grateful if anyone could point to a complete formulation.
He says:

" In any case, what is generally understood to be the "equivalent" of Hopf-Rinow in Lorentzian geometry is the theorem that states that in any globally hyperbolic spacetime, two causally related points can always be joined by a maximizing geodesic. Global hyperbolicity here would be the substitute for completeness."

Which is basically identical to what I guessed above, without ever having heard of Hopf-Rinow by that name.

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He says:

" In any case, what is generally understood to be the "equivalent" of Hopf-Rinow in Lorentzian geometry is the theorem that states that in any globally hyperbolic spacetime, two causally related points can always be joined by a maximizing geodesic. Global hyperbolicity here would be the substitute for completeness."

Which is basically identical to what I guessed above, without ever having heard of Hopf-Rinow by that name.
Would "maximizing" mean maximizing the arc length in a -+++ signature, i.e., minimizing proper time? Or should this really be "extremizing?"

Would spacelike and timelike geodesics be treated differently? The definition of a hyperbolic spacetime doesn't treat them symmetrically, and, e.g., we expect to have closed spacelike curves even if we don't have CTCs.

PAllen
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Would "maximizing" mean maximizing the arc length in a -+++ signature, i.e., minimizing proper time? Or should this really be "extremizing?"

Would spacelike and timelike geodesics be treated differently? The definition of a hyperbolic spacetime doesn't treat them symmetrically, and, e.g., we expect to have closed spacelike curves even if we don't have CTCs.
As he(?) states the generalization, it doesn't apply to spacelike geodesics, as I guessed. The key phrase is:

"two causally related points" - his formulation
"if two events are causally connected" - my formulation.

Maximizing would be relative to a +--- signature, for causally connected events.

Is there a logical connection between the concept of a naked singularity and the concept of a timelike singularity?
Roger Penrose says more than that in THE ROAD TO REALITY:
There are various slightly different technical ways of specifying what is meant by the term 'naked singularity' and I do not propose to enter into the distinction here. Sufficient fo0r our purposes would be to say that a naked singularity is 'timelike', in the sense that signals can both enter and leave the singularity....as indicated in Fig 28.17...If we assume cosmic censorship..as yet neither proved nor refuted....then physical spacetime singularities have to be 'spacelike' (or null) but never 'timelike'. There are two kinds od of spacelike (or null) singularities, namely 'initial' or 'final' ones, depending upon whether timelike curves can escape from the singularity into the future or wenter it from the past.
and he goes on to discuss his Weyle curvature hypothesis applicable to the physical universe.

gotta go be back later or tomorrow....

PAllen
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Roger Penrose says more than that in THE ROAD TO REALITY:

and he goes on to discuss his Weyle curvature hypothesis applicable to the physical universe.

gotta go be back later or tomorrow....
Except that Hawking himself, in conceding a bet, admitted that naked singularities are possible without exotic matter for implausibly symmetric initial conditions. His restated challenge is to require that initial conditions (in the ADM initial value formulation) leading to a naked singularity form an open set. This is his mathematical formulation of 'generic initial conditions'.

I checked and could find no further information in the Penrose book, but here is a contrary view to the Penrose quote above, from a paper previously referenced in these forums:

http://arxiv.org/pdf/gr-qc/0109051v2.pdf
Why do naked singularities form in gravitational collapse?
Pankaj S. Joshi∗, Naresh Dadhich† and Roy Maartens‡

When the dust density is homogeneous, the apparent
horizon starts developing earlier than the epoch of singularity
formation, which is then fully hidden within a
black hole. There is no density gradient, and no shear.
On the other hand, if a density gradient is present at the
center, then the trapped surface development is delayed
via shear, and, depending on the “strength” of the density
gradient/shear at the center, this may expose the
singularity.

Our main result is that sufficiently strong shearing
effects in spherical collapsing dust delay the formation of
the apparent horizon, thereby exposing the strong gravity
regions to the outside world and leading to a (locally)
naked singularity. When shear decays rapidly near the
singularity, the situation is effectively like the shear-free
case, with a black hole end-state. An important point
is that naked singularities can develop in quite a natural
manner, very much within the standard framework of
general relativity, governed by shearing effects......

We have considered here spherical collapse. Very little
is known about nonspherical collapse, either analytically
or numerically, towards determining the outcome
in terms of black holes and naked singularities. However,
phenomena such as trapped surface formation and
apparent horizon are independent of any spacetime symmetries, and it is also clear that a naked singularity will
not develop in general unless there is a suitable delay
of the apparent horizon. This suggests that the shear
will continue to be pivotal in determining the final fate
of gravitational collapse, independently of any spacetime
symmetries.
Are those last few sentences taking a different view than "implausibly symmetric initial conditions" mentioned by PAllen. ?? I am surprised by that last sentence in particular.

But maybe what they mean is that given perfect symmetry, one still needs a density gradient, shear, for a naked singularity.
I wonder when this paper was published.

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Thanks, Naty -- the Penrose quote was exactly what I was looking for. That led me to some good info via googling; Penrose himself seems to have been the one who first formalized the definition:

Penrose, Gravitational radiation and gravitational collapse; Proceedings of the Symposium, Warsaw, 1973. Dordrecht, D. Reidel Publishing Co. pp. 82-91.

Basically you want to formalize a definition that describes the formation of a singularity by gravitational collapse from nonsingular initial conditions, such that signals can escape to infinity. He takes two shots at formalizing this, first with a global definition and then with a local one.

The global definition involves an absolute event horizon: in an asymptotically flat spacetime, the boundary of the set of events from which a timelike curve can get to i+. A naked singularity is one that is not on or outside the horizon. This definition has the problem that the big bang is considered a naked singularity, and the definition only applies to asymptotically flat spacetimes

The local definition is that a naked singularity is a timelike one, i.e., there exists an observer who can have it both in his past and in his future. ("Timelike" isn't automatically defined without reference to some observer, because the singularity isn't a point-set.) Basically the singularity is one that you can both anticipate and remember; metaphorically, it's OK if God creates you, and you may see God when you die, but you're not usually going to see God in person during your lifetime.

Another global definition I hit by googling: Rudnicki, 2006, http://arxiv.org/abs/gr-qc/0606007

There is no obvious way to prove a link between the global definition and the local one. They are connected, because they both cause Cauchy hypersurfaces not to exist. (Penrose references a theorem by Geroch, J Math Phys 11 (1970) 437.) Penrose remarks:

Thus, even a singularity inside a black hole might conceivably be 'naked' to some observer who is himself inside the black hole. However, in the normal picture of spherically symmetric collapse, such a situation does not occur. There is some indication also (Penrose and Simpson, 1973) that in a generic perturbed collapse this situation still does not occur.
I think the part about "the normal picture of spherically symmetric collapse" is basically just the result of Birkhoff's theorem.

So AFAICT we have various definitions of naked singularities. It still seems to be an open problem what is the most useful definition and how to relate the various definitions.

George Jones
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On a separate but somewhat related topic, can anyone help me understand the distinction between the weak and strong causality conditions (H&E pp. 190, 192)? I don't understand what their definition of the latter is trying to express, and I don't understand the example they give in a figure where only weak causality holds. Violating it would seem to mean that a nonspacelike curve would have to pass through a neighborhood twice, but I don't see that happening in the figure.
Just after you posted, I spent some time thinking about this, but I couldn't come with a non-spacelike curve that passed through the neighbourhood twice. I then forgot about this until yesterday, when, for some reason, it popped into my mind again. After thinking some more I finally found the curve, which I have put on the attached diagram. The curve can't be closed, so the weak causality condition is satisfied. This type of curve can intersect any arbitrarily small neighbourhood of p twice. Why didn't Hawking and Ellis put this curve on the diagram as, e.g., a dotted line?

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PeterDonis
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A naked singularity is one that is not on or outside the horizon.
Shouldn't this be "not on or inside the horizon"?

This definition has the problem that the big bang is considered a naked singularity, and the definition only applies to asymptotically flat spacetimes
But there is no spacetime with a big bang singularity that is asymptotically flat, is there? (Except for the obvious case of "big bang" solutions that are really Minkowski spacetime in disguise, like the Milne universe.)

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Shouldn't this be "not on or inside the horizon"?
Yep, thanks.

But there is no spacetime with a big bang singularity that is asymptotically flat, is there? (Except for the obvious case of "big bang" solutions that are really Minkowski spacetime in disguise, like the Milne universe.)
Hmm... What Penrose actually says is: "Consider, now, the set E consisting of all events from which a timelike curve (or a null curve -- it makes little difference) can be drawn into the future to infinity." He defines an absolute horizon as the boundary of E. I guess I'm confused about what this definition is really saying. He seemed to me to be referring to curves escaping to $i^+$ and $\mathscr{I}^+$. I don't see how one would apply this definition, for example, to a closed, matter-dominated cosmology with a Big Crunch. I guess a realistic cosmological model might have a Penrose diagram on which it would make sense to label an $i^+$. Anyway, my statement was clearly wrong.

After thinking some more I finally found the curve, which I have put on the attached diagram. The curve can't be closed, so the weak causality condition is satisfied. This type of curve can intersect any arbitrarily small neighbourhood of p twice. Why didn't Hawking and Ellis put this curve on the diagram as, e.g., a dotted line?
Cool, thanks, George! My copy of H&E is at work -- I'll puzzle over this on Monday.

PeterDonis
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Hmm... What Penrose actually says is: "Consider, now, the set E consisting of all events from which a timelike curve (or a null curve -- it makes little difference) can be drawn into the future to infinity." He defines an absolute horizon as the boundary of E.
This matches the definitions I'm familiar with (for example, I believe Hawking & Ellis uses a basically identical definition).

I guess I'm confused about what this definition is really saying. He seemed to me to be referring to curves escaping to $i^+$ and $\mathscr{I}^+$.
That's my understanding from the definitions I'm familiar with.

I don't see how one would apply this definition, for example, to a closed, matter-dominated cosmology with a Big Crunch. I guess a realistic cosmological model might have a Penrose diagram on which it would make sense to label an $i^+$.
I know I've seen Penrose diagrams of the various FRW spacetimes somewhere, but I can't dig them up right now. From what I remember, the closed cosmologies don't have any "infinity" at all: no $i^+$ or $\mathscr{I}^+$, no $i^-$ and $\mathscr{I}^-$, and no $i^0$. Their Penrose diagrams just look like a box, with the past singularity at the bottom, the future one at the top, and the left and right sides corresponding to "antipodal points" in the closed 3-sphere of each spacelike hypersurface in between the singularities.

The open cosmologies have $i^+$ and $\mathscr{I}^+$, but no $i^-$ or $\mathscr{I}^-$. Their Penrose diagrams look like a right triangle with the horizontal base being the big bang singularity, the vertical left side being the spatial origin, and the hypotenuse going up and to the left at 45 degrees, representing $\mathscr{I}^+$. I think the open ones have an $i^0$, which corresponds to the lower right vertex of the triangle (the top vertex, on the left, of course being $i^+$).

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