  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.
