Convection and cracking stability of spheres in General Relativity

TL;DR

This paper investigates the stability of relativistic spheres against convection and cracking, providing criteria for instability and analyzing the effects of density profiles and pressure reactions.

Contribution

It introduces a simple criterion for convection instability and explores the interplay between convection and cracking in relativistic spheres, considering both isotropic and anisotropic matter.

Findings

Density profiles with decreasing and concave density are stable against convection if sound velocity decreases outward.

Isotropic models can be unstable to cracking if pressure reaction is neglected.

Considering pressure reaction can eliminate cracking instabilities in both isotropic and anisotropic models.

Abstract

In the present paper we consider convection and cracking instabilities as well as their interplay. We develop a simple criterion to identify equations of state unstable to convection, and explore the influence of buoyancy on cracking (or overturning) for isotropic and anisotropic relativistic spheres. We show that a density profile $\rho(r)$, monotonous, decreasing and concave , i.e. $\rho' < 0$ and $\rho'' < 0$, will be stable against convection, if the radial sound velocity monotonically decreases outward. We also studied the cracking instability scenarios and found that isotropic models can be unstable, when the reaction of the pressure gradient is neglected, i.e. $\delta \mathcal{R}_p = 0$; but if it is considered, the instabilities may vanish and this result is valid, for both isotropic and anisotropic matter distributions.