Are there any models with homogeneous energy density?

TL;DR

This paper proves that in spherical symmetry, the only static solution with homogeneous energy density is Schwarzschild, and no dynamic solutions with this property exist, highlighting the uniqueness of such static models.

Contribution

It introduces a tetrad-based method to analyze homogeneous energy density in spherical symmetry and establishes the uniqueness of the Schwarzschild solution under these conditions.

Findings

Schwarzschild is the only static homogeneous energy density solution.

No spherically symmetric dynamic solutions with homogeneous density exist.

Certain conditions like isotropy and conformal flatness are equivalent and lead to static solutions.

Abstract

By applying a recent method --based on a tetrad formalism in General Relativity and the orthogonal splitting of the Riemann tensor-- to the simple spherical static case, we found that the only static solution with homogeneous energy density is the Schwarzschild solution and that there are no spherically symmetric dynamic solutions consistent with the homogeneous energy density assumption. Finally, a circular equivalence is shown among the most frequent conditions considered in the spherical symmetric case: homogeneous density, isotropy in pressures, conformally flatness and shear-free conditions. We demonstrate that, due to the regularity conditions at the center of the matter distribution, the imposition of two conditions necessarily leads to the static case.