Interior Schwarzschild metric

The interior Schwarzschild metric is the exact static solution for a spherical fluid of constant density. If the star has mass $M$ and radius $R$, its pressure is

The Interior Schwarzschild metric has central pressureIt diverges when $M/R=4/9$, the limiting compactness in Buchdahl's theorem.

The dominant energy condition for the Interior Schwarzschild metric reduces to its largest pressure satisfying $p_c\le \rho$. Substitution of the exact central pressure gives $\sqrt{1-2M/R}\ge 1/2$, hence $M/R\le 3/8$. Equality has finite central pressure and is allowed by the non-strict condition.