Proposition: Local development, worldsheet, and non-GH host

The metric has signature $(-,+,+,+)$, and $\mathcal{W}$ is a two-dimensional worldsheet. On an injective angular chart,

gives $G=-\mathrm{d}{t}^2+\mathrm{d}{x}^2+\mathrm{d}{y}^2+\mathrm{d}{z}^2$. The regular product is not globally hyperbolic.

Proof. The signature and dimension are immediate, and differentiation gives the local development. For $r_0>0$, take in geometrized time

The points $x_n=(0,r_0/n,{\vartheta }_0,z_0)$, $n\ge 2$, belong to $J^{+}(p)\cap J^{-}(q)$ by radial causal curves. They have no convergent subsequence in the regular product, since their completion limit has $r=0$. The diamond is not compact.