Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 311 / 1 / a / i / Solution

Outside the star, $p=\rho =0$. The first Tolman–Oppenheimer–Volkoff equation gives $m(r)=M$, and the second givesAsymptotic flatness fixes the integration constant, so $e^{2\Phi }=1-2M/r$. The exterior line element is therefore the Schwarzschild metric with mass $M=m(R)$.