= Solution
For a <perfect fluid in general relativity> with nonnegative <energy density> and <pressure>, the <dominant energy condition> is $p\leq\rho$. <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> gives
$$
\frac{p_c}{\rho}=\frac{1-s}{3s-1}\leq1\quad\Longleftrightarrow\quad1-s\leq3s-1\quad\Longleftrightarrow\quad s\geq\frac12.
$$
Therefore the <dominant-energy compactness bound for an incompressible star> is
$$
\boxed{\frac MR\leq\frac38.}
$$
Equality 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 $M/R<4/9$.
Back to article page
