Causal ordering on spacetimes

On any spacetime $(M,g)$ we can form the causal ordering $\leq$, where for any two points $a,b \in M$ we have that $a \leq b$ iff there exists some future directed, non-spacelike curve from $a$ to $b$. This causal ordering is transitive, irreflexive, antisymmetry and dense in general. However, causal oddities like CTC's manifest as violations of irreflexivity and antisymmetry.

I'm interested in the directedness of the causal order (unfortunate name, I know). Recall that a relation $\leq$ is directed iff for any two elements $a,b \in M$ there is some third element $c \in M$ such that both $a \leq c$ and $b \leq c$.

I can see that a Minkowksi space has a directed causal order. I have no proof, but the picture below is good enough to convince me. Clearly anything in the shaded area will serve as the witness for the condition. Note also that the condition is trivially met for timelike or lightlike-related points. I have heard that for non-flat spacetimes, directedness of the causal order is not guaranteed. Are there any concrete examples of this? And, moreover, is there some appropriate causality condition one can impose on the spacetime in order to guarantee directedness of the causal order?

• Chapter 6 branches out this question a little – user76568 Mar 9 '18 at 14:19
• A counterexample to this is to consider de Sitter space. Taking the Penrose diagram it's not too hard to show that there are light cones that never intersect. – Slereah Mar 9 '18 at 14:27
• Thank you for the comments. User76568, I've had a quick skim of that chapter and I can't see where it mentions the properties of the causal relation at all. Which part of the book were you referring to? And Slereah, thank you, is this also the case in other Malament-Hogarth spacetimes? – Doc Mar 10 '18 at 7:30
• If you have two black holes, then two events inside each will have nothing common in their futures. – MBN Mar 10 '18 at 13:09

Take two dimensional Minkowski space, and remove the line $\{ (x,t) | x=0, t \geq 0 \}$. Any event $t \geq 0$ will have its future lightcone restricted to either positive or negative $x$. Then picking any two points on either side of the singularity will produce future light cones that never intersect.