Curvature dependence of relativistic epicyclic frequencies in static,
  axially symmetric spacetimes
  The sum of squared epicyclic frequencies of nearly circular motion
($\omega_r^2+\omega_\theta^2$) in axially symmetric configurations of Newtonian
gravity is known to depend both on the matter density and on the angular
velocity profile of circular orbits. It was recently found that this sum goes
to zero at the photon orbits of Schwarzschild and Kerr spacetimes. However,
these are the only relativistic configurations for which such result exists in
the literature. Here, we extend the above formalism in order to describe the
analogous relation for geodesic motion in arbitrary static, axially symmetric,
asymptotically flat solutions of general relativity. The sum of squared
epicyclic frequencies is found to vanish at photon radii of vacuum solutions.
In the presence of matter, we obtain that $\omega_r^2+\omega_\theta^2>0$ for
perturbed timelike circular geodesics on the equatorial plane if the strong
energy condition holds for the matter-energy fluid of spacetime; in vacuum, the
allowed region for timelike circular geodesic motion is characterized by the
inequality above. The results presented here may be of use to shed light on
general issues concerning the stability of circular orbits once they approach
photon radii, mainly the ones corresponding to stable photon motion.
