Outside the star, $p=\rho=0$. The first <Tolman–Oppenheimer–Volkoff equation> gives $m(r)=M$, and the second gives
$$
\Phi'(r)=\frac{M}{r(r-2M)}
=\frac12\frac d{dr}\log\left(1-\frac{2M}{r}\right).
$$
Asymptotic flatness fixes the integration constant, so $e^{2\Phi}=1-2M/r$. The exterior line element is therefore the <Schwarzschild metric> with mass $M=m(R)$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-311/1/a/i/solution