Epicyclic Frequencies for a Generic Ultracompact Object

TL;DR

This paper extends the analysis of ultracompact objects by examining both radial and vertical epicyclic frequencies, linking stability properties of light-rings and orbits to energy conditions in spacetime.

Contribution

It generalizes previous results by including vertical stability analysis and relates epicyclic frequencies to the Ricci tensor, connecting stability to energy conditions.

Findings

Vertical stability of light-rings determines vertical stability of orbits.

Relation between epicyclic frequencies and Ricci tensor established.

Objects with unstable light-rings violate energy conditions.

Abstract

Recently, it has been shown that the radial stability of a light-ring (LR) in a spacetime generated by a stationary, axisymmetric, asymptotically flat object with a $\mathbb{Z}_2$ symmetry determines the possibility and radial stability of timelike circular orbits (TCOs) around the LR. In this paper, we generalise this result by also considering the vertical (angular) stability of the orbits through the study of the radial and vertical epicyclic frequencies. We show that the vertical stability of the LR only determines the vertical stability of the TCOs around it. A relation between the sum of the squared epicyclic frequencies and the Ricci tensor is also provided. With such relation, we show that objects with radially and vertically unstable LRs (TCOs) violate the null (strong) energy condition.