A Variational Principle for the Axisymmetric Stability of Rotating Relativistic Stars

TL;DR

This paper introduces a variational principle to assess the axisymmetric stability of rotating relativistic stars, providing a new tool to analyze their stability without relying on the CFS instability mechanism.

Contribution

It generalizes Chandrasekhar's variational principle for spherical stars to rotating stars, offering a method to evaluate axisymmetric stability in relativistic stars.

Findings

Provides a lower bound on exponential growth rate of instabilities.

Extends variational principles from black hole stability analysis.

Enables stability testing without gravitational radiation considerations.

Abstract

It is well known that all rotating perfect fluid stars in general relativity are unstable to certain non-axisymmetric perturbations via the Chandrasekhar-Friedman-Schutz (CFS) instability. However, the mechanism of the CFS instability requires, in an essential way, the loss of angular momentum by gravitational radiation and, in many instances, it acts on too long a timescale to be physically/astrophysically relevant. It is therefore of interest to examine the stability of rotating, relativistic stars to axisymmetric perturbations, where the CFS instability does not occur. In this paper, we provide a Rayleigh-Ritz type variational principle for testing the stability of perfect fluid stars to axisymmetric perturbations, which generalizes to axisymmetric perturbations of rotating stars a variational principle given by Chandrasekhar for spherical perturbations of static, spherical stars. Our variational principle provides a lower bound to the rate of exponential growth in the case of instability. The derivation closely parallels the derivation of a recently obtained variational principle for analyzing the axisymmetric stability of black holes.