 Ingoing Kerr-Schild coordinates for Schwarzschild spacetime

With $f=1-2\mu /r$, the time coordinate $t^{\prime }=t+(2\mu /c)\log |r/(2\mu )-1|$ gives $d{s}^2=f{c}^{2}d{t}^{\prime 2}-(4\mu c/r)d{t}^{\prime }dr-(1+2\mu /r)d{r}^2-r^{2}d{\Omega }^2$. Its radial metric tensor block has determinant $-c^2$, so it extends regularly across the future Schwarzschild event horizon. The advanced null time of ingoing Eddington-Finkelstein coordinates is $v=t^{\prime }+r/c$, not $t^{\prime }$ itself. Radial null directions satisfy $dr/(cd{t}^{\prime })=-1$ or $(1-2\mu /r)/(1+2\mu /r)$.