Galactic Spacetime Solutions with a Varying Newton's Coupling

We find a new family of galactic metrics corresponding to flat rotation curves at the outer radii. These are vacuum solutions to a gravity theory where the Newton's coupling varies mildly in space. The effective `mass', whose origin is purely geometric, receives a negative non-baryonic contribution. The angle of deflection of a light ray propagating in this geometry is found to be diminished rather than enhanced compared to the Einsteinian bending, the effect being highly suppressed though. Hence, these spacetimes are observationally dintinguishable from other geometric or `dark matter' based alternatives invoked to explain mass discrepancies in galaxies.