On viability of isentropic perfect fluid collapse with a linear equation of state

TL;DR

This paper investigates the conditions under which perfect fluid collapse with a linear equation of state can occur in general relativity, highlighting the mathematical restrictions and exploring the effects of perturbations on collapse outcomes.

Contribution

It demonstrates that consistent solutions for perfect fluid collapse with a linear EoS are generally unavailable unless specific symmetries reduce the equations to ODEs or special conditions are met.

Findings

Compatible solutions are rare and require additional symmetries.

Perturbations in mass profiles influence the visibility of singularities.

Without linear EoS constraints, PDE compatibility issues do not arise.

Abstract

The gravitational collapse of a barotropic perfect fluid having the Equation of State (EoS) $p=k\rho$, where $k$ is constant, is studied here in the framework of general relativity. We examine the restrictions on the Misner-Sharp mass function, because of the introduction of such an EoS, in terms of the compatibility of a certain pair of quasi-linear partial differential equations, obtained from Einstein's field equations. We find that except when this system of PDEs reduces to ODEs because of additional symmetries imposed on the spacetimes, or when they become compatible with each other in some special situations, consistent solution to perfect fluid collapse with linear EoS is not available. The end state of collapse with no such constraint of EoS has also been investigated. Since considering arbitrary pressures in a collapsing cloud to study its end state is difficult as this requires information about the dynamics of collapse not known in general, we consider small perturbations to mass profiles corresponding to inhomogeneous dust collapse. This, in turn, provides small pressure perturbations to the otherwise pressureless fluid. The dependence of visibility or otherwise of the singularity on initial conditions of collapse, in the presence or absence of such perturbations is studied numerically. As long as no linear EoS is imposed on the matter field, no incompatibility issue between its corresponding pair of PDEs arise, unlike the case when $k$ is restricted to be a constant.