Geometric Aharonov-Bohm phase effect around a black hole

In recent work, we have explored the novel possibility that the geodesic of a spacetime density flow in GR could be upgraded to an amplitude $\psi$ with density $|\psi|^2$. We show how this arises from generally-covariant quantum mechanics and the Klein-Gordon operator in a semiclassical analysis, and how it connects to the Raychaudhuri equations. In this setting, we demonstrate an Aharonov-Bohm type effect for the phase of $\psi$ in motion approaching a black hole.