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?