Solution
= Solution
Let $F=1-2M/r$, define the tortoise coordinate by $dr_*/dr=F^{-1}$, and set the advanced time $v=t+r_*$. Since $dt=dv-F^{-1}dr$, substitution gives
$$
ds^2=-Fdv^2+2\,dv\,dr+r^2dΩ^2.
$$
These are <Ingoing Eddington-Finkelstein coordinates>; all coefficients are regular at the <Schwarzschild event horizon>[event horizon] $r=2M$.