Elastic stars in general relativity: II. Radial perturbations

Abstract

We study radial perturbations of general relativistic stars with elastic matter sources. We find that these perturbations are governed by a second-order differential equation which, along with the boundary conditions, defines a Sturm–Liouville-type problem that determines the eigenfrequencies. Although some complications arise compared to the perfect fluid case, leading us to consider a generalization of the standard form of the Sturm–Liouville equation, the main results of Sturm–Liouville theory remain unaltered. As an important consequence we conclude that the mass–radius curve for a one-parameter sequence of regular equilibrium models belonging to some particular equation of state can be used in the same well-known way as in the perfect fluid case, at least if the energy density and the tangential pressure of the background solutions are continuous. In particular, we find that the fundamental mode frequency has a zero for the maximum mass stars of the models with solid crusts considered in paper I of this series.