general relativity - In the most trivial spacetimes, is the existence of a null geodesic equivalent to horismos relations?

Take a globally hyperbolic topologically trivial spacetime $M \cong \mathbb{R} \times \Sigma$, $\Sigma \cong \mathbb{R}^{(n-1)}$. Given $p, q \in M$, such that there exists a future-directed null geodesic $\ell$ between $p$ and $q$, is this equivalent to the condition that $p \nearrow q$, an horismos relation ($q$ is on $p$'s lightcone), ie $p \leq q$ and $p \not \ll q$?

This is fairly obviously not true for say, totally vicious spacetimes, where $p \ll p$ for all points, ie every point has a closed timelike curve (there isn't even any horismos to be on), and for a globally hyperbolic example, the Minkowski cylinder $\Sigma = S$, where a null geodesic will connect to a point in $p$'s own lightcone after one turn. On the other hand, this is certainly true of Minkowski space, as well as any spacetime related to it by a Weyl transform.

This would be equivalent to proving that, if $q \in I^+(p)$, then there is no null geodesics linking $p$ to $q$ which, given the properties of globally hyperbolic spacetimes, means that there is a maximizing timelike geodesic linking the two points. If $q = \exp_p(v)$ for some $v$, this would be correct (since $\exp_p I^+(0, \mathbb{R}^n) = I^+(p, M)$), but that would be assuming that $\text{Im}(\exp_p) = M$ for such a spacetime, which I am not sure is correct even for such a benign example.

Is such a thing true and if so how to show it?

Answer

This is not true in general. As an example take a spherically symmetric (regular) ultracompact object's spacetime. That is, an spherical symmetric object that is compact enough to fit inside its own lightring, but without a horizon. The upshot is that we end up with a spacetime that is Schwarzschild outside some radius smaller than the lightring, and has some regular matter filled region inside such that the spacetime remains topologically trivial.

Expressed in Schwarzshild coordinates, a line with constant spacial coordinates on the lightring radius will be timelike. It is immediately clear that there are pairs of points on this line that are also connected by a null geodesic going around the lightring.