# Existence and Stability of Circular Orbits in Time-Dependent Spherically Symmetric Spacetimes

## TL;DR

This paper develops a geometric framework to analyze the existence and stability of circular geodesics in time-dependent spherically symmetric spacetimes, applying it to various solutions including Schwarzschild-de Sitter and Kerr-de Sitter.

## Contribution

It introduces a novel geometric formalism for circular geodesics in dynamic spacetimes and analyzes their stability using a simple dynamical system approach.

## Key findings

- Existence of circular geodesics when J^2 > 0
- Stability requires (J^2)' > 0
- Application to Schwarzschild-de Sitter and Kerr-de Sitter solutions

## Abstract

For a general spherically four--dimensional metric the notion of "circularity" of a family of equatorial geodesic trajectories is defined in geometrical terms. The main object turns out to be the angular--momentum function $J$ obeying a consistency condition involving the mean extrinsic curvature of the submanifold containing the geodesics. The ana\-ly\-sis of linear stability is reduced to a simple dynamical system formally describing a damped harmonic oscillator. For static metrics the existence of such geodesics is given when $J^2 > 0$, and $(J^2)' > 0$ for stability. The formalism is then applied to the Schwarzschild--de Sitter solution, both in its static and in its time--dependent cosmological version, as well to the Kerr--de Sitter solution. In addition we present an approximate solution to a cosmological metric containing a massive source and solving the Einstein field equation for a massless scalar.