The rotational modes of relativistic stars: Analytic results

TL;DR

This paper analytically investigates the rotational modes of relativistic stars, revealing the absence of pure r-modes in stars with a common equation of state and computing post-Newtonian corrections for specific stellar models.

Contribution

It provides the first analytic results on the spectrum of rotational modes in relativistic stars, especially highlighting the non-existence of pure r-modes under certain conditions.

Findings

Pure r-modes do not exist in stars with a shared equation of state.

Post-Newtonian corrections to l=m r-modes are computed for uniform density stars.

The mode spectrum is highly sensitive to the star's equation of state.

Abstract

We study the r-modes and rotational ``hybrid'' modes of relativistic stars. As in Newtonian gravity, the spectrum of low-frequency rotational modes is highly sensitive to the stellar equation of state. If the star and its perturbations obey the same one-parameter equation of state (as with isentropic stars), there exist {\it no pure r-modes at all} - no modes whose limit, for a star with zero angular velocity, is an axial-parity perturbation. Rotating stars of this kind similarly have no pure g-modes, no modes whose spherical limit is a perturbation with polar parity and vanishing perturbed pressure and density. We compute the post-Newtonian corrections to the $l=m$ r-modes of isentropic and non-isentropic uniform density stars.