Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 54 / 1 / a / i / Solution

In the vacuum exterior, the Tolman–Oppenheimer–Volkoff equation gives $m^{\prime }=0$, so $m(r)=M$. The remaining metric function obeysThus $e^{2\Phi }=C(1-2M/r)$. A constant rescaling of the time coordinate removes $C$; equivalently, asymptotic flatness fixes $C=1$ when time is normalized at infinity. Consequently the exterior is the Schwarzschild metric:The mass $M=m(R)$ includes the full perfect fluid in general relativity interior. No equation of state beyond the surface enters this vacuum result.