Stability of Circular Orbits in General Relativity: A Phase Space Analysis

TL;DR

This paper uses phase space analysis to study the stability of circular orbits in various spacetimes and media, revealing how stability depends on parameters like mass, rotation, and medium properties.

Contribution

It introduces a phase space method for analyzing orbit stability in general relativity and applies it to Schwarzschild, Schwarzschild-de Sitter, Kerr spacetimes, and optical media.

Findings

Stability of orbits in Schwarzschild spacetime can be characterized by a single parameter.

In Kerr spacetime, stability depends on the ratio of source rotation to particle angular momentum.

The method effectively analyzes motion in refractive media such as optical black holes.

Abstract

Phase space method provides a novel way for deducing qualitative features of nonlinear differential equations without actually solving them. The method is applied here for analyzing stability of circular orbits of test particles in various physically interesting environments. The approach is shown to work in a revealing way in Schwarzschild spacetime. All relevant conclusions about circular orbits in the Schwarzschild-de Sitter spacetime are shown to be remarkably encoded in a single parameter. The analysis in the rotating Kerr black hole readily exposes information as to how stability depends on the ratio of source rotation to particle angular momentum. As a wider application, it is exemplified how the analysis reveals useful information when applied to motion in a refractive medium, for instance, that of optical black holes.