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I'm currently trying to understand the notion of a naked singularity. After consulting books by Wald and Choquet-Bruhat, it seems that for a naked singularity one must have that the causal curves can leave the singularity and extend to infinity (that is, they will not hit the horizon). This looks like the notion of "geodesically complete" to me. However, doesn't the very fact that we have a naked singularity show that the spacetime is geodesically incomplete, i.e. the timeline geodesics cannot be infinitely extended?

I'm extremely confused because it seems that the spacetime is then geodesically complete and incomplete at the same time.

I would be very grateful if anyone could clarify this.

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    $\begingroup$ For the manifold to be geodesically complete, the geodesics have to be extendible in both directions. If they're emerging from a naked singularity, they're not infinitely extendible in the past direction. $\endgroup$ – twistor59 Apr 19 '13 at 12:13
  • $\begingroup$ thank you @twistor59 ! So, if I have a naked singularity, then only the future timelike geodesics are extendable. Then, this brings me to the question: is there a special way of defining a geodesic that emanates from a naked singularity? (so that I can apply the extendibility argument). Many thanks! $\endgroup$ – ConciseAndClear Apr 19 '13 at 12:21
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    $\begingroup$ I'm not sure - past incomplete but future complete i.e. ends up at $I^+$ (future timelike infinity) or ${\mathcal{I}}^+$ (future null infinity) I guess? $\endgroup$ – twistor59 Apr 19 '13 at 12:40
  • $\begingroup$ well, I agree, but I was more thinking of a methamatical/formula-type way of defining it. For me, the to show extendability will be simply to take the limit and see whether it lies on my manifold. The question being, to take the limit of what? $\endgroup$ – ConciseAndClear Apr 19 '13 at 12:43
  • $\begingroup$ Related: physics.stackexchange.com/q/60869/2451 $\endgroup$ – Qmechanic Apr 19 '13 at 13:00