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

Curvature dependence of relativistic epicyclic frequencies in static, axially symmetric spacetimes