Proposition: The full regular cone is not globally hyperbolic

The regular product (eq:cone) has a noncompact causal diamond.

Proof. Fix $r_0>0$, one angular coordinate, and $z=0$. Take events $p,q$ at that spatial point with 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 timelike travel. They have no convergent subsequence in $M$, because the apex is excluded.