# Singularity Theorem and its statements

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:

1. $M$ contains no closed timelike curves.
2. Strong energy condition is satisfied.
3. The manifold is general (i.e not too highly symmetric)
4. 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?

• Minkowski space with a conical singularity is an example where none of these conditions are true. – Slereah Apr 15 '18 at 8:06
• For a simple counterexample, Minkowski space has no closed timelike curves and obeys the SEC but has no singularities. de Sitter spacetime has no CTCs, obeys the SEC and the generic condition, but has no singularities. – Slereah Apr 15 '18 at 10:19
• I think an example with trapped surfaces can be done by gluing together two copies of Schwarzschild spacetime, which would break the strong energy condition – Slereah Apr 15 '18 at 10:32