Dynamical instability and the expansion-free condition

TL;DR

This paper investigates the conditions under which a collapsing anisotropic fluid sphere remains stable or becomes unstable, emphasizing the roles of pressure anisotropy and density profile over fluid stiffness in both Newtonian and post-Newtonian regimes.

Contribution

It demonstrates that the adiabatic index does not influence instability in the expansion-free collapse, highlighting the importance of pressure anisotropy and density distribution.

Findings

Instability range is governed by pressure anisotropy.

Density profile influences collapse stability.

Fluid stiffness has no effect on instability in this context.

Abstract

We study the dynamical instability of a spherically symmetric anisotropic fluid which collapses adiabatically under the condition of vanishing expansion scalar. The Newtonian and post Newtonian regimes are considered in detail. It is shown that within those two approximations the adiabatic index $\Gamma_1$, measuring the fluid stiffness, does not play any role. Instead, the range of instability is determined by the anisotropy of the fluid pressures and the radial profile of the energy density, independently of its stiffness, in a way which is fully consistent with results previously obtained from the study on the Tolman mass.