Proposition: Why no global cone inverse is used

The regular product $({\mathcal{M}}^{\circ },G)$ is not globally hyperbolic. Consequently, the global Cauchy and Green operators used below are operators on $\hat{M}$, not global inverses on the regular cone.

Proof. Fix a radial ray, $r_0>0$, and $z=z_0$. Choose $a>r_0/c_{\mathrm{h}}$ and set

The points

lie in $J^{+}(p)\cap J^{-}(q)$: the radial trip can be made within the available time. They have no convergent subsequence in the regular product, because their only completion limit has $r=0$. Thus this causal diamond is not compact. Global hyperbolicity would require compact causal diamonds.