Application of Sturm's theorem to marginal stable circular orbits of a test body in spherically symmetric and static spacetimes

TL;DR

This paper uses Sturm's theorem to analyze the stability of circular orbits in various spherically symmetric, static spacetimes, providing a mathematical framework to identify marginally stable orbits like the ISCO.

Contribution

It applies Sturm's theorem to determine MSCOs in different exact solutions of Einstein's and Weyl conformal gravity, offering a novel analytical approach.

Findings

Explicit application to Kottler spacetime (Schwarzschild-de Sitter)

Analysis of MSCOs in Weyl conformal gravity

Identification of conditions for stable and unstable orbits

Abstract

In terms of Sturm's theorem, we reexamine a marginal stable circular orbit (MSCO) such as the innermost stable circular orbit (ISCO) of a timelike geodesic in any spherically symmetric and static spacetime. MSCOs for some of exact solutions to the Einstein's equation are discussed. Strum's theorem is explicitly applied to the Kottler (often called Schwarzschild-de Sitter) spacetime. Moreover, we analyze MSCOs for a spherically symmetric, static and vacuum solution in Weyl conformal gravity.