The singularity theorem is as follows:
A spacetime $M$ necessarily contains incomplete, inextensible timelike or null geodesics if, in addition to Einstein's equations, these conditions hold:
- $M$ contains no closed timelike curves.
- Strong energy condition is satisfied.
- The manifold is general (i.e not too highly symmetric)
- The manifold contains a trapped surface.
For a spacetime to contains a singularity, do all these conditions have to necessarily hold simultaneously? Are there examples of metric where a some of these conditions hold and others don't, hence no singularity? Are there spacetimes where there are trapped surfaces but no singularity?
Or is it that these are sufficient condition, so even if one of these holds then it implies the existence of singularity?