R-mode oscillations of rapidly rotating barotropic stars in general relativity: Analysis by the relativistic Cowling approximation

We develop a numerical scheme for obtaining the $r$-mode oscillations of rapidly rotating relativistic stars. In the present scheme, we neglect all metric perturbations and only take account of the dynamics of the fluid in the background spacetime of the unperturbed star (the relativistic Cowling approximation). We also assume the star is barotropic, i.e. neutrally stable against convection under the assumption of adiabatic oscillations. Our numerical scheme is based on the Yoshida-Eriguchi formulation for the analysis of the general relativistic f-mode oscillations in the Cowling approximation and a general relativistic generalization of the Karino-Yoshida-Yoshida-Eriguchi's numerical scheme for obtaining oscillations of rapidly rotating Newtonian stars. By this new numerical scheme, frequencies of the $r$-mode oscillations are obtained as functions of the ratio of the rotational energy to the absolute value of the gravitational energy $T/|W|$ along sequences of polytropic equilibrium stars whose ratio of the pressure to the total energy density at the center of the star and the polytropic index are kept constant. It is found that the dimensionless oscillation frequency $\sigma/\Omega$ is a linearly decreasing function of $T/|W|$, where $\sigma$ and $\Omega$ are the oscillation frequency and the angular velocity of the star measured in an inertial frame at spatial infinity. We also find that oscillation frequencies of the $r$-modes are highly dependent on the relativistic factor $M/R$ of the star as already found in previous studies in which the slow rotation approximation has been used. Here $M$ and $R$ denote the mass and radius of the star, respectively.