Regularizing the r-mode problem for nonbarotropic relativistic stars

TL;DR

This paper develops a regularized approach to analyze r-modes in relativistic nonbarotropic stars, demonstrating the existence of a single r-mode per mode combination and discussing implications for gravitational wave emission.

Contribution

It introduces a regularization method for the singular differential equation in relativistic r-mode analysis, extending previous results to more realistic stellar models.

Findings

Existence of a single r-mode for each (l,m) mode combination.

Regularization via higher order terms enables practical solutions.

Results support earlier findings for uniform density stars.

Abstract

We present results for r-modes of relativistic nonbarotropic stars. We show that the main differential equation, which is formally singular at lowest order in the slow-rotation expansion, can be regularized if one considers the initial value problem rather than the normal mode problem. However, a more physically motivated way to regularize the problem is to include higher order terms. This allows us to develop a practical approach for solving the problem and we provide results that support earlier conclusions obtained for uniform density stars. In particular, we show that there will exist a single r-mode for each permissible combination of $l$ and $m$. We discuss these results and provide some caveats regarding their usefulness for estimates of gravitational-radiation reaction timescales. The close connection between the seemingly singular relativistic r-mode problem and issues arising because of the presence of corotation points in differentially rotating stars is also clarified.