On the galactic rotation curves problem within an axisymmetric approach
  In U. Nucamendi et al. Phys. Rev. D63 (2001) 125016 and K. Lake, Phys. Rev.
Lett. 92 (2004) 051101 it has been shown 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: the emitted
photons come from stable timelike circular geodesic orbits of stars in a static
spherically symmetric gravitational field, and propagate to us along null
geodesics. It is remarkable that this relation can be established without
appealing at all to a concrete theory of gravitational interaction. Here we
generalize this kinematical spherically symmetric approach to the galactic
rotation curves problem to the stationary axisymmetric realm since this is
precisely the symmetry that spiral galaxies possess. Thus, by making use of the
most general stationary axisymmetric metric, we also 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 possible strategies to overcome them.
