Proposition: The regular product is not globally hyperbolic

The regular spacetime in (eq:host) is not globally hyperbolic.

Proof. Fix a radius $r_0>0$, an angle, and an axial coordinate. Choose events at that spatial point and times $T_{\ast }\mp 2r_0/c$. Points at time $T_{\ast }$, the same angle and axial coordinate, and radii $r_n\downarrow 0$, belong to the causal diamond between those events by radial causal curves. They have no convergent subsequence in the regular product. That diamond is noncompact.