Complete solution of the Einstein field equations for a spherical shell of truly incompressible liquid

Abstract

We present the solution of the Einstein field equations, in the static and spherically symmetric case, for an incompressible fluid, that has constant proper energy density at each and every point of the volume where it exists, according to a set of local observers who are stationary with respect to the fluid at each point. In the general case the fluid exists within a spherically symmetric shell with an inner vacuum-matter interface at a radial position \(r_{1}\) and an outer matter-vacuum interface at a radial position \(r_{2}\) in the Schwarzschild coordinate system. Therefore, in the general case there is an inner vacuum region with a repulsive singularity at the origin, just like in all other similar shell solutions. We present the parameter plane of the problem, and show that there are limits of solutions that approach the configuration of black holes, with the formation of an event horizon at the radial position \(r_{2}\).