Proposition: Failure of global hyperbolicity

The regular product $M$ is not globally hyperbolic.

Proof. Fix $r_0>0$, an angle, and $z=0$. Let $p,q$ have that spatial position and times $-2r_0/c$ and $2r_0/c$. For $r_n\downarrow 0$, the events $(0,r_n,\varphi ,0)$ lie in $J^{+}(p)\cap J^{-}(q)$, by radial causal travel. They have no convergent subsequence in $M$. Thus this causal diamond is not compact.