Proposition: Non-global hyperbolicity

The regular spacetime $M$ is not globally hyperbolic.

Proof. Fix $r_0>0$, an angle, and $z=0$. Take two events at that spatial point with times $T_{\ast }\pm 2r_0/c$. Events at time $T_{\ast }$ and radii $r_n\downarrow 0$, at the same angle and axial coordinate, lie in their causal diamond by radial causal paths. They have no convergent subsequence in $M$. The diamond is therefore noncompact.