Evolution equations for the perturbations of slowly rotating relativistic stars

TL;DR

This paper derives new, simplified evolution equations for perturbations of slowly rotating relativistic stars using the ADM formalism, facilitating numerical simulations and revealing symmetries between polar and axial modes.

Contribution

The authors present a novel derivation of perturbation equations in a different gauge that simplifies numerical evolution and highlights symmetries, improving upon previous methods.

Findings

Derived hyperbolic first order evolution equations

Simplified numerical integration process

Revealed symmetry between polar and axial perturbations

Abstract

We present a new derivation of the equations governing the oscillations of slowly rotating relativistic stars. Previous investigations have been mostly carried out in the Regge-Wheeler gauge. However, in this gauge the process of linearizing the Einstein field equations leads to perturbation equations which as such cannot be used to perform numerical time evolutions. It is only through the tedious process of combining and rearranging the perturbation variables in a clever way that the system can be cast into a set of hyperbolic first order equations, which is then well suited for the numerical integration. The equations remain quite lengthy, and we therefore rederive the perturbation equations in a different gauge, which has been first proposed by Battiston et al. (1970). Using the ADM formalism, one is immediately lead to a first order hyperbolic evolution system, which is remarkably simple and can be numerically integrated without many further manipulations. Moreover, the symmetry between the polar and the axial equations becomes directly apparent.