Modelling the Rising Tails of Galaxy Rotation Curves

It is well known but under-appreciated in astrophysical applications, that it is possible for gravity to take on a life of its own in the form of Weyl-curvature-only metrics (note we are referring to the Weyl-only solutions of ordinary General Relativity, we are not considering Weyl conformal gravity or any other modified gravity theories), as numerous examples demonstrate the existence of gravitational fields not being sourced by any matter. In the weak field limit, such autonomous gravitational contents of our universe manifest as solutions to the homogeneous Poisson's equation. In this note, we tentatively explore the possibility that they may perhaps account for some phenomenologies commonly attributed to dark matter. Specifically, we show that a very simple solution of this kind exists that can be utilized to describe the rising tails seen in many galaxy rotation curves, which had been difficult to reconcile within the cold dark matter or modified Newtonian dynamics frameworks. This solution may also help explain the universal $\sim 1$Gyr rotation periods of galaxies in the local universe.