Static Solutions of the Einstein Equations for Spherically Symmetric Elastic Bodies

TL;DR

This paper investigates static, spherically symmetric elastic bodies within Einstein's theory, formulating the problem as a Fuchsian ODE system and proving existence and regularity of solutions under general conditions.

Contribution

It introduces a novel formulation of Einstein's equations for elastic bodies as a Fuchsian ODE system and establishes regularity results for solutions, including quadratic strain relations.

Findings

Solutions exist and are regular under general constitutive relations.

Solutions remain regular up to the boundary for quadratic strain models.

The ODE system provides a new approach to studying elastic bodies in general relativity.

Abstract

The paper is concerned with the Einstein equations for a spherically symmetric static distribution of anisotropic matter. The equations are cast into a system of Fuchsian type ODE for certain scalar invariants of the strain. And then the existence and regularity of this ODE is studied under general constitutive relation. In the case the constitutive relation is given by a quadratic form of strain, it is also shown that the solutions stay regular up to the boundary of the material ball.