Energetic Stability of the Solutions of the Einstein Field Equations for Spherically Symmetric Liquid Shells

TL;DR

This paper analyzes the energetic stability of spherically symmetric liquid shell solutions in General Relativity, identifying stable configurations and relating boundary parameters to gravitational binding energies.

Contribution

It introduces a new energetic stability criterion for these solutions and reduces the parameter space to stable configurations, including a subset of shell solutions.

Findings

Certain solutions are energetically stable with non-zero internal radii.

The interior Schwarzschild solution is shown to be maximally unstable.

A subset of shell solutions are confirmed to be energetically stable.

Abstract

We interpret the exact solutions previously obtained for spherically symmetric shells of liquid fluid in General Relativity in terms of the energies involved. We show that a certain parameter that was introduced into the solutions by the interface boundary conditions is related to the binding energies of the gravitational systems. We then use this fact in order to discuss the energetic stability of those solutions. We include in the stability discussion the well-known interior Schwarzschild solution for a liquid sphere, which can be obtained as a specific limit of the solutions that we previously obtained for the liquid shells. We show that this solution turns out to be a maximally unstable one, from the energetic point of view discussed here. We also perform a numerical exploration of the energetic stability criterion of the solutions, and show that indeed there is a particular subset of the solutions which are energetically stable. All these solutions have the form of shells with non-vanishing internal radii. This reduces the original three-parameter family of solutions to a two-parameter family of energetically stable solutions.