Exact Solutions with Asymptotically Flat Galactic Rotation Curves

Abstract

We present a new set of spacetime solutions for spiral galaxies corresponding to flat rotation curves at large distances. These correspond to anisotropic pressure, which could be sourced by either matter or geometry. As examples, we elucidate the galactic metrics corresponding to the non-ideal generalizations of `dust' and `radiation', respectively. Further, we find a dynamical realization of the Einstein cluster when the spatially averaged equation of state (w) happens to be non-zero but small. The correction to the Einsteinian bending angle for a light ray penetrating a large galactic halo is predicted to be: $(2+3w)πv^2/(1+3w)c^2$, v being the limiting rotational velocity. We infer a mild decline of the rotation curves ($ v'(r) ~ 1/r^2 $) at large radii for sufficiently luminous galaxies, as also indicated by some specific observations.