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 $\mathrm{\mid }\psi {\mathrm{\mid }}^2$. We now show how exactly the relevant amplitude and velocity flow equations arise in an eikonal approximation of Stueckelberg’s proper-time quantum mechanics or Klein-Gordon flow. We also show how the divergence of the velocity field recovers the Raychaudhuri equations for relativistic fluids. In this setting, we demonstrate an Aharonov-Bohm type effect for the phase of $ψ& in motion approaching a black-hole.