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

Since $v=t+r_{\ast }$ and $d{r}_{\ast }=f^{-1}dr$,Substitution into the metric givesThese 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.