Proposition: The regular product is not globally hyperbolic

The spacetime (eq:host) has a noncompact causal diamond.

Proof. Choose a regular spatial point of radius $r_{\ast }>0$, and two events there separated by more than $2r_{\ast }/c_{\mathrm{light}}$. At an intermediate time, points on the same radial ray with radii tending to zero lie in their causal diamond. They have no convergent subsequence in the regular product.