Newtonian and General Relativistic Models of Spherical Shells - II

TL;DR

This paper develops analytical Newtonian and General Relativistic models of spherical shells with various thicknesses and structures, analyzing their stability and properties using elementary functions and spherical metrics.

Contribution

It introduces new families of spherical shell models with finite and infinite thickness, including double shells, and extends them into General Relativity with anisotropic matter.

Findings

All models are analytically expressed in elementary functions.

All tested shell configurations are stable against radial perturbations.

The relativistic models feature anisotropic matter distributions.

Abstract

A family of potential-density pairs that represent spherical shells with finite thickness is obtained from the superposition of spheres with finite radii. Other families of shells with infinite thickness with a central hole are obtained by inversion transformations of spheres and of the finite shells. We also present a family of double shells with finite thickness. All potential-density pairs are analytical and can be stated in terms of elementary functions. For the above-mentioned structures, we study the circular orbits of test particles and their stability with respect to radial perturbations. All examples presented are found to be stable. A particular isotropic form of a metric in spherical coordinates is used to construct a General Relativistic version of the Newtonian families of spheres and shells. The matter of these structures is anisotropic, and the degree of anisotropy is a function of the radius.