Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 54 / 1 / b / ii / Solution

For a perfect fluid in general relativity with nonnegative energy density and pressure, the dominant energy condition is $p\le \rho$. Pressure decreases outward, so it is enough to test the central pressure. Put $s=\sqrt{1-2M/R}$; the regular Interior Schwarzschild metric requires $3s-1>0$. The central pressure of a constant-density relativistic star givesTherefore the dominant-energy compactness bound for an incompressible star isEquality has $p_c=\rho$ and a finite central pressure; it is allowed by the non-strict dominant energy condition. This is stronger than the regular-star Buchdahl's theorem limit $M/R<4/9$.