Equilibrium and stability of thin spherical shells in Newtonian and relativistic gravity

TL;DR

This paper analyzes the equilibrium and stability of thin spherical shells in Newtonian and relativistic gravity, revealing a simple algebraic link between maximum mass configurations and dynamical instability.

Contribution

It demonstrates a straightforward algebraic proof of the link between maximum mass and instability for thin shells, simplifying the understanding compared to fluid models.

Findings

Maximum mass configurations indicate the onset of instability.

Simple algebraic proof established for thin-shell models.

Insights applicable to both Newtonian and relativistic gravity.

Abstract

We consider thin spherical shells of matter in both Newtonian gravity and general relativity, and examine their equilibrium configurations and dynamical stability. Thin-shell models are admittedly a poor substitute for realistic stellar models. But the simplicity of the equations that govern their dynamics, compared with the much more complicated mechanics of a self-gravitating fluid, allows us to deliver, in a very direct and easy manner, powerful insights regarding their equilibria and stability. We explore, in particular, the link between the existence of a maximum mass along a sequence of equilibrium configurations and the onset of dynamical instability. Such a link is well-established in the case of fluid bodies in both Newtonian gravity and general relativity, but the demonstration of this link is both subtle and difficult. The proof is very simple, however, in the case of thin shells, and it is constructed with nothing more than straightforward algebra and a little calculus.