Quasi-normal Modes and Instabilities of Rotating Relativistic Stars

Abstract

We derive the basic equations suitable for the coupled fluid--spacetime eigenvalue problem from the second-order action in full general relativity and accurately determine quasi-normal modes of a variety of rapidly rotating stars with outgoing-wave boundary conditions. For MIT-bag strange stars, the spectrum realizes Chandrasekhar's instability picture, with distinct Chandrasekhar--Friedman--Schutz and Jacobi-like neutral conditions and the transition to dynamical bar-mode growth. For neutron stars, we determine the $f$-, $r$-, and other inertial-mode spectra and their radiative damping or growth times. Interactions between $f$- and inertial modes reveal avoided crossings and exchanges for radiative damping or growth rates. We locate quasiradial collapse thresholds through a static stability analysis and independently verify them by tracking the fundamental $F$-mode frequency to zero, clarifying that the central density at the onset of the dynamical instability is always larger than that of the secular instability for uniform rotation. Our formalism and numerical implementation can also resolve highly damped $w$-modes.