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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) [itex]\forall p,q\ J^+(p) \cap J^-(q)[/itex] 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.

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) [itex]\forall p,q\ J^+(p) \cap J^-(q)[/itex] 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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