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 generalize 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.

中文翻译：

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通用超紧凑物体的周转频率

最近，研究表明，光环 (LR) 在由静止的、轴对称的、渐近平坦的物体产生的时空中的径向稳定性${\mathbb{Z}}_2$对称性决定了围绕 LR 的类时圆轨道 (TCO) 的可能性和径向稳定性。在本文中，我们通过研究径向和垂直周转频率，还考虑了轨道的垂直（角）稳定性，从而推广了这一结果。我们表明，LR 的垂直稳定性仅决定其周围 TCO 的垂直稳定性。还提供了行星频率平方和与 Ricci 张量之间的关系。通过这种关系，我们表明具有径向和垂直不稳定 LR（TCO）的物体违反了零（强）能量条件。