Proposition: Non-global hyperbolicity

The regular product $M_{\mu }$ is not globally hyperbolic.

Proof. Choose a regular spatial point of radius $r_0$, one angle, and $z=0$. Let $p,q$ have that spatial position and times $T_{\ast }\mp 2r_0/c$. Points at time $T_{\ast }$, the same angle and axial coordinate, and radii $r_n\downarrow 0$ lie in $J^{+}(p)\cap J^{-}(q)$ by radial causal curves. They have no convergent subsequence in the regular product. The causal diamond is noncompact.