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I'm trying to understand the notion of a naked singularity on a more mathematical level (intuitively, it's a singularity "one can see and poke with a stick", but I'm having troubles on how to actually show it).

Based on what (little) is written in Choquet-Bruhat's, a naked singularity is the one for which we can extend the outgoing time-like geodesics to infinity. Now, I was wondering, assuming I had a given solution, how would I "test" the nakedness of the singularity? A natural thing to do would be to write the solution in some null coordinates, but what then? (the idea of writing it in null coordinates came from this qustion Christodoulou's paper on naked singularities in inhomogeneous dust collapse)

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The formal definition of a naked singularity can be found in standard GR books (Wald, Hawking & Ellis, probably others). However, here's a slightly crude, rule of thumb way of checking whether your spacetime has a singularity and whether it is naked. There is a singularity if an affinely parametrized causal curve (past or future directed) reaches an infinite value for some curvature scalar in finite time (say Kretschmann scalar). If there are two curves, say A and B, such that A is future oriented, B is past oriented, B is in the future (domain of influence) of A, and both A and B end in a singularity, then the singularity is naked.

One can play around with other ways of checking for a singularity besides diverging curvature scalars. One can also require that instead of a single geodesic (or a single pair) there are geodesic congruences that all behave in the same way. These all correspond to more sophisticated ways of identifying singularities.

Specifically about null coordinates, they can help if your the coordinate curves parametrized by the null coordinate reach the singularity in the way described above.

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  • $\begingroup$ thanks @IgorKhavkine. I also wanted to know if there were any specific guidelines on how to define my geodesics for a given solution? So far, what I've seen in books (like Wald's) is a general statement, that we have to take a null causal curve and show that it's past-inextendable but cam be continued to the future. But this is a very abstract statement, i.e. I cannot work with it because I don't know how to define my geodesic explicitly, i.e. what kind of equation? (I know that a geodesic has to fulfill the geodesic equation, but I have to substitute (contd.) $\endgroup$
    – ConciseAndClear
    Commented May 3, 2013 at 10:51
  • $\begingroup$ (cont.)something concrete, nut just some abstract $\gamma(t)$, right? $\endgroup$
    – ConciseAndClear
    Commented May 3, 2013 at 10:52
  • $\begingroup$ Geodesics are defined by... the geodesic equation. Sounds like you need practice working with and solving the geodesic equation in coordinates in some simple examples. Wald is probably not the best book for that. On the other hand, Carroll, Landau & Lifshitz (vol 2), Misner & Thorn & Wheeler, Schutz, d'Inverno (roughly in decreasing order of sophistication) offer more worked examples, which should help you build up the experience you need. $\endgroup$
    – Igor Khavkine
    Commented May 3, 2013 at 16:39
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    $\begingroup$ At the risk of sounding a bit contrary, identifying naked singularities is not easy. Geroch has given an example of a geodesically complete spacetime that contains an incomplete future directed timelike curve. Therefore there is some kind of singular behavior going on. Also your manifold must be maximally extended and the relationship between maximal extension and curvature divergence is still a topic of research. There are also competing classification schemes that give different `results'. For one of the better definitions look at Krolak's weak cosmic censorship papers. $\endgroup$
    – Ben Whale
    Commented Nov 12, 2013 at 22:30
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    $\begingroup$ Hi, @BenWhale. Doesn't sound contrary to me. All valid points. $\endgroup$
    – Igor Khavkine
    Commented Nov 13, 2013 at 2:37

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