Since $v=t+r_*$ and $dr_*=f^{-1}dr$,
$$
dt=dv-f^{-1}dr.
$$
Substitution into the metric gives
$$
\begin{aligned}
ds^2
&=-f(dv-f^{-1}dr)^2+f^{-1}dr^2+r^2d\Omega^2\\
&=\boxed{-f\,dv^2+2\,dv\,dr+r^2d\Omega^2}.
\end{aligned}
$$
These are <Ingoing Eddington-Finkelstein coordinates>. Every coefficient is finite at $r=2M$, and the determinant of the $v,r$ block is $-1$. Thus the apparent singularity there in Schwarzschild coordinates is a coordinate singularity; the metric extends smoothly across the <Schwarzschild event horizon>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 37E / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-3/37e/b/solution