# Existence and stability of circular orbits in general static and spherically symmetric spacetimes

## TL;DR

This paper establishes conditions for the existence and stability of circular orbits in static, spherically symmetric spacetimes, providing new criteria and applying them to various known metrics.

## Contribution

It introduces a fixed point method for existence conditions and Lyapunov exponents for stability, advancing understanding of orbit behavior in these spacetimes.

## Key findings

- Asymptotic flatness implies existence of circular orbits with negative Newtonian potential.
- Derived sufficient conditions for stability and instability of timelike and null circular orbits.
- Demonstrated applicability to SU(2) Yang-Mills-Einstein spacetimes and known metrics.

## Abstract

The existence and stability of circular orbits (CO) in static and spherically symmetric (SSS) spacetime are important because of their practical and potential usefulness. In this paper, using the fixed point method, we first prove a necessary and sufficient condition on the metric function for the existence of timelike COs in SSS spacetimes. After analyzing the asymptotic behavior of the metric, we then show that asymptotic flat SSS spacetime that corresponds to a negative Newtonian potential at large $r$ will always allow the existence of CO. The stability of the CO in a general SSS spacetime is then studied using the Lyapunov exponent method. Two sufficient conditions on the (in)stability of the COs are obtained. For null geodesics, a sufficient condition on the metric function for the (in)stability of null CO is also obtained. We then illustrate one powerful application of these results by showing that an SU(2) Yang-Mills-Einstein SSS spacetime whose metric function is not known, will allow the existence of timelike COs. We also used our results to assert the existence and (in)stabilities of a number of known SSS metrics.