Convection and cracking stability of spheres in General Relativity

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. ${\rho }^{\mathrm{\prime }}<0$ and ${\rho }^{\mathrm{\prime }\mathrm{\prime }}<0$, 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.