Geodesic completeness, the fact we can make the domain of the geodesic parametrized with respect an affine parameter the whole real line, is an important concept in GR. Especially, because the lack of it is the implication of Penrose and Hawking singularity theorems.

This raises the question if incompleteness of one type of geodesic implies incompleteness of the rest. In general this is not the case.

I've found examples of the following scenarios:

  • timelike complete, spacelike and null incomplete
  • spacelike complete, timelike and null incomplete
  • null complete, timelike and spacelike incomplete
  • timelike and null complete, spacelike incomplete
  • spacelike and null complete, timelike incomplete

Is it possible to construct a spacelike and timelike complete spacetime but null incomplete?

  • $\begingroup$ Discussions I've seen of this kind of thing actually talk about the characteristics of the singularity, not the characteristics of the incomplete geodesics. The main cases of interest are spacelike singularities (e.g., a black hole's singularity) and timelike singularities (e.g., cosmological ones, or naked singularities). Googling does turn up discussion of null singularities as well. I'm not convinced that the classification in terms of the characteristics of the complete and incomplete geodesics is actually useful or consistent. If so, then it would seem to have been unnecessary to [...] $\endgroup$ – user4552 Oct 16 '14 at 15:44
  • $\begingroup$ [...] create the definition of a naked singularity, which is pretty intricate and doesn't just refer to what types of geodesics hit it. $\endgroup$ – user4552 Oct 16 '14 at 15:45
  • $\begingroup$ @BenCrowell The term "parametrization of a geodesic" does not make any sense without being compatible with the ambient differentiable structure. I.e. the differentiable structure is fixed and the $\sim \mathbb{R}$ structure is unambiguously confirmed by finding a single $\mathbb{R}$-parametrization. It is a good definition. $\endgroup$ – Void Oct 16 '14 at 16:05
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    $\begingroup$ This paper claims to describe just such a spacetime, however it's behind a paywall. $\endgroup$ – John Rennie Oct 16 '14 at 16:34
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    $\begingroup$ I seem to remember a paper by Geroch, where this was stated as an open problem. But the paper was probably from the 70's. $\endgroup$ – MBN Oct 16 '14 at 17:55

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