There are some theorems concerning whether a spacetime is geodesically connected (whether any two points $p, q \in M$ admit a geodesic connecting them) or not, ie , but all of these are concerned with either rather simplistic spacetimes, which would not really apply to our own universe, or have very specific conditions that seem perhaps too narrow.
Is there a theorem that our own universe would be geodesically connected? Throwing around a few "reasonable" conditions, that may imply spacetimes with the following properties :
- Globally hyperbolic
- Maximally extended
- Obeying some reasonable energy condition (ie the null energy condition or the generic condition)
- Bounded stress-energy tensor
It can be shown rather easily that this is false just admitting global hyperbolicity and the NEC (ie. the spacetime constructed by taking the union of two future light cone in Minkowski space), or by taking a Weyl transform of this spacetime so that the Riemann tensor diverges on its boundary, so that it is also maximally extended.
Is there a specific theorem for a plausible spacetime where it is geodesically connected, or is that simply wrong, or unknown?