Derivation of the Tully-Fisher law: Doubts about the necessity and existence of halo dark matter

For disk galaxies the fourth power of the circular velocity nu(c)(4) of stars around the core of the galaxy is proportional to the luminosity L, nu(c)(4) proportional to L (Tully-Fisher law). Since L is proportional to the mass M of the galaxy, it follows that nu(c)(4) proportional to M. Newtonian mechanics, however, yields nu(c)(4) = GM/r for a circular motion. In order to rectify this big difference, astronomers assume the existence of dark matter. We derive the equation of motion of a star moving in the central field of a galaxy and show that, for a circular motion, it yields a term of the form nu(c)(4) proportional to GMc/tau where G is Newton's gravitational constant, c is the speed of light, and tau is the Hubble time. This puts in doubt the existence of halo dark matter for galaxies.