Relativistic r-modes and Shear viscosity: regularizing the continuous spectrum

Within a fully relativistic framework, we derive and solve numerically the perturbation equations of relativistic stars, including the stresses produced by a non-vanishing shear viscosity in the stress-energy tensor. With this approach, the real and imaginary parts of the frequency of the modes are consistently obtained. We find that, approaching the inviscid limit from the finite viscosity case, the continuous spectrum is regularized and we can calculate the quasi-normal modes for stellar models that do not admit solutions at first order in perturbation theory when the coupling between the polar and axial perturbations is neglected. The viscous damping time is found to agree within factor 2 with the usual estimate obtained by using the eigenfunctions of the inviscid limit and some approximation for the energy dissipation integrals. We find that the frequencies and viscous damping times for relativistic $r-$modes lie between the Newtonian and Cowling results. We compare the results obtained with homogeneous, polytropic and realistic equations of state and find that the frequencies depend only on the rotation rate and on the compactness parameter (M/R), being almost independent of the equation of state. Our numerical results for realistic neutron stars give viscous damping times with the same dependence on mass and radius as previously estimated, but systematically larger of about 60%.