Geometric Conditions on the Type of Matter Determining the Flat Behavior of the Rotational Curves in Galaxies

In an arbitrary axisymmetric stationary spacetime, we determine the expression for the tangential velocity of test objects following a circular stable geodesic motion in the equatorial plane, as function of the metric coefficients. Next, we impose the condition, observed in large samples of disks galaxies, that the magnitude of such tangential velocity be radii independent in the dark matter dominated region, obtaining a constraint equation among the metric coefficients, and thus arriving to an iff (iff means: if and only if.) condition: The tangential velocity of test particles is radii independent iff the metric coefficients satisfied the mentioned constraint equation. Furthermore, for the static case, the constraint equation can be easily integrated, leaving the spacetime at the equatorial plane essentially with only one independent metric coefficient. With the geometry thus fixed, we compute the Einstein tensor and equate it to an arbitrary stress energy tensor, in order to determine the type of energy-matter which could produce such a geometry. Within an approximation, we deduce a constraint equation among the components of the stress energy tensor. We test in that constraint equation several well known types of matter, which have been proposed as dark matter candidates and are able to point for possible right ones. Finally, we also present the spherically symmetric static case and apply the mentioned procedure to perfect fluid stress energy tensor, recovering the Newtonian result as well as the one obtained in the axisymmetric case. We also present arguments on the need to use GR to study types of matter different than the dust one.