Let $\tilde{M}$ be a simply connected Lorentzian manifold and suppose that $\tilde{M}$ admits some Riemannian metric.
Question: What can be said about the relation between the geodesic completeness of $\tilde{M}$ as a Lorentzian manifold and the metric completion as a metric space via the Riemannian metric? In particular, if $\tilde{M}$ is already metrically complete, is it then also Lorentzian geodesically complete?
As every manifold admits a complete Riemannian metric (see here), you cannot deduce anything about the Lorentzian metric.
Even if every Riemannian metric on $\bar{M}$ is complete (so that $\bar{M}$ is necessarily compact), the Lorentzian metric may not be complete: the Clifton-Pohl torus is an example of a compact Lorentzian manifold which is not geodesically complete. This shows that the Hopf-Rinow Theorem does not generalise to Lorentzian manifolds.