HYDROSTATIC EQUILIBRIUM OF INSULAR, STATIC, SPHERICALLY SYMMETRIC, PERFECT FLUID SOLUTIONS IN GENERAL RELATIVITY

Abstract

An analysis of insular solutions of Einstein's field equations for static, spherically symmetric, source mass, on the basis of exterior Schwarzschild solution is presented. Following the analysis, we demonstrate that the regular solutions governed by a self-bound (that is, the surface density does not vanish together with pressure) equation of state (EOS) or density variation cannot exist in the state of hydrostatic equilibrium, because the source mass which belongs to them, does not represent the "actual mass" appears in the exterior Schwarzschild solution. The only configuration which could exist in this regard is governed by the homogeneous density distribution (i.e. the interior Schwarzschild solution). Other structures which naturally fulfill the requirement of the source mass, set up by exterior Schwarzschild solution (and, therefore, can exist in hydrostatic equilibrium) are either governed by gravitationally-bound regular solutions (i.e. the surface density also vanishes together with pressure), or self-bound singular solutions (i.e. the pressure and density both become infinity at the centre).