Normal modes of relativistic stars in slow rotation limit

Abstract

Rotational shifts of normal frequencies in polytropic stellar models are calculated within the framework of general relativity. The stellar rotation is assumed to be slow and the first-order rotational effects are included to the eigenfrequencies of the nonrotating stars. Unlike in Newtonian theory, the normal frequencies of nonradial oscillations of slowly rotating configurations are complex numbers. The real and imaginary parts correspond to the oscillatory and damping rates of the mode. The frequency and decay rate of the corotating f-mode increase with the increase of the stellar angular velocity. On the other hand, those of the counterrotating mode decrease. The counterrotating mode eventually becomes unstable beyond a critical angular velocity. The critical angular velocity cannot be determined exactly but can be estimated by extrapolating the behavior of the normal mode of a slowly rotating star. The critical value is smaller for larger azimuthal wavenumber and harder equation of state. The relativistic effects also tend to diminish the critical value.