A relativistic axisymmetric approach to the galactic rotation curves problem

It is known that galactic potentials can be kinematically linked to the observed red/blue shifts of the corresponding galactic rotation curves under a minimal set of assumptions (see [1] and [2] for details): i) that emitted photons come to us from stable timelike circular geodesic orbits of stars in a static spherically symmetric gravitational field, and ii) that these photons propagate to us along null geodesics. This relation can be established without appealing at all to a concrete theory of gravitational interaction. This kinematical spherically symmetric approach to the galactic rotation curves problem can be generalized to the stationary axisymmetric realm, which is precisely the symmetry that spiral galaxies possess [3]. Here we review the relativistic results obtained in the latter work. Namely, by making use of the most general stationary axisymmetric metric, we consider stable circular orbits of stars that emit signals which travel to a distant observer along null geodesics and express the galactic red/blue shifts in terms of three arbitrary metric functions, clarifying the contribution of the rotation as well as the dragging of the gravitational field. This stationary axisymmetric approach distinguishes between red and blue shifts emitted by circularly orbiting receding and approaching stars, respectively, even when they are considered with respect to the center of a spiral galaxy, indicating the need of precise measurements in order to confront predictions with observations. We also point out the difficulties one encounters in the attempt of determining the metric functions from observations and list some potential strategies to overcome them