Elastic Stars in General Relativity: II. Radial perturbations

TL;DR

This paper analyzes radial oscillations of relativistic stars with elastic matter, extending Sturm-Liouville theory to determine eigenfrequencies and stability, and confirms the applicability of mass-radius relations for such models.

Contribution

It generalizes the Sturm-Liouville framework to elastic matter in relativistic stars, enabling stability analysis similar to perfect fluid models.

Findings

Eigenfrequencies are determined by a generalized Sturm-Liouville problem.

Mass-radius curves remain valid for elastic stars with continuous energy density and tangential pressure.

Fundamental mode frequency vanishes at maximum mass stars with solid crusts.

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 generalisation 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.