Proposition: Why a global transfer is unavailable

The regular product $(M,\bar{g})$ is not globally hyperbolic. Consequently, local flatness alone does not justify applying a global globally-hyperbolic Green-operator theorem to all of $M$.

Proof. Fix $r_0>0$, one angular coordinate, and $z=0$. Take

For any sequence $r_n\downarrow 0$, the events

belong to $J^{+}(p)\cap J^{-}(q)$. Radial travel has length $r_0-r_n<r_0$, while each available time interval is $2r_0/c$, so radial timelike routes exist.

The sequence has no convergent subsequence in $M$, because its only possible radial limit is the excluded apex. Thus this causal diamond is not compact. Global hyperbolicity requires compact causal diamonds.