Exact Solution of the Einstein Field Equations for a Spherical Shell of Fluid Matter

TL;DR

This paper derives an exact analytical solution to Einstein's field equations for a spherical shell of fluid matter with constant density, revealing a non-flat interior spacetime and a stabilizing gravitational field.

Contribution

It provides a new exact solution for a spherical shell of constant density matter, including analytical and numerical descriptions of the metric and pressure profiles.

Findings

The solution features a singularity at the origin with zero energy density.

The interior spacetime is non-flat, indicating a non-trivial gravitational field.

The gravitational field stabilizes the matter configuration by pulling particles back into the shell.

Abstract

We determine the exact solution of the Einstein field equations for the case of a spherically symmetric shell of liquid matter, characterized by an energy density which is constant with the Schwarzschild radial coordinate $r$ between two values $r_{1}$ and $r_{2}$. The solution is given in three regions, one being the well-known analytical Schwarzschild solution in the outer vacuum region, one being determined analytically in the inner vacuum region, and one being determined mostly analytically but partially numerically, within the matter region. The solutions for the temporal coefficient of the metric and for the pressure within this region are given in terms of a non-elementary but fairly straightforward real integral. We show that in this solution there is a singularity at the origin, and give the parameters of that singularity in terms of the geometrical and physical parameters of the shell. This does not correspond to an infinite concentration of matter, but in fact to zero energy density at the center. It does, however, imply that the spacetime within the spherical cavity is not flat, so that there is a non-trivial gravitational field there, in contrast with Newtonian gravitation. This gravitational field has the effect of stabilizing the geometrical configuration of the matter, since any particle of the matter that wanders out into the vacuum regions tends to be brought back to the bulk of the matter by the gravitational field.