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

## TL;DR

This paper generalizes the understanding of epicyclic frequency behavior near photon orbits in static, axially symmetric spacetimes within general relativity, highlighting conditions under which these frequencies vanish or remain positive.

## Contribution

It extends the formalism for epicyclic frequencies to arbitrary static, axially symmetric solutions, revealing how these frequencies behave at photon radii with and without matter presence.

## Key findings

- Sum of squared epicyclic frequencies vanishes at photon radii in vacuum solutions.
- In matter, the sum remains positive if the strong energy condition holds.
- Results inform stability analysis of circular orbits near photon radii.

## Abstract

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.