On the r-mode spectrum of relativistic stars in the low frequency approximation

The axial modes for barotropic relativistic rotating neutron stars with uniform angular velocity are studied, using the slow rotation formalism together with the low frequency approximation, first investigated by Kojima. The time independent form of the equations leads to a singular eigenvalue problem, which admits a continuous spectrum. We show that is nevertheless always possible to find discrete mode solutions (the r mode). However, under certain conditions related to the equation of state and the compactness of the stellar model, the associated velocity perturbation is divergent, hence these solutions have to be discarded as being unphysical. Thus, in these cases there exists only the continuous part of the spectrum, but no r mode. We corroborate our results by explicitly integrating the time dependent equations. For stellar models admitting a physical r-mode solution, it can indeed be excited by arbitrary initial data. For those models admitting only an unphysical mode solution, the evolutions do not show any tendency to oscillate with the respective frequency. Instead, the fluid shows a steady oscillation with its characteristic position dependent frequency, whereas the metric perturbation eventually damps out.