 Solution

Let $F=1-2M/r$, define the tortoise coordinate by $d{r}_{\ast }/dr=F^{-1}$, and set the advanced time $v=t+r_{\ast }$. Since $dt=dv-F^{-1}dr$, substitution givesThese are Ingoing Eddington-Finkelstein coordinates; all coefficients are regular at the event horizon $r=2M$.