General Relativity and the Tully-Fisher Relation for Rotating Galaxies

Abstract

The flat limit of rotational velocity ( v φ ) approximately equal to the “edge”-velocity of a galaxy is related to the baryonic mass ( M B ) via the T-F relationship with n ≈ 4 . We explore the connection between mass and the limiting velocity in the framework of general relativity (GR) using the Weyl metric for axially-symmetric galaxies that are supported entirely by their rotational motion. While for small distances from the center, the Newtonian description is accurate as one moves beyond the (baryonic) edge of the galaxy, Lenz’s law and non-linearity of the gravitational field inherent in GR not only lead to a flat velocity (obviating its Keplerian fall), but also provide its tight log-log relationship with the enclosed (baryonic) mass.