 Solution

Write $r_s=2GM/c^2$, $f=1-r_s/r$ and $r_{\ast }=r+r_s\ln |r/r_s-1|$, so $d{r}_{\ast }/dr=f^{-1}$. The ingoing Eddington-Finkelstein coordinates use $v=t+r_{\ast }/c$ to label ingoing radial light fronts. Substituting $dt=dv-dr/(cf)$ into the Schwarzschild metric givesThe radial block has determinant $-c^2$ even at $r=r_s$; its coefficients have no horizon divergence. The apparent singularity of Schwarzschild time at the future horizon is therefore a coordinate failure. The physical curvature singularity at $r=0$ remains.

The radial null trajectories in ingoing Eddington-Finkelstein coordinates haveThe ingoing family crosses the future horizon smoothly. The outgoing family has zero outward radial coordinate speed at the horizon and decreasing radius inside it; both future null directions then lead toward smaller radius. These advanced coordinates cover the exterior and future black-hole horizon, rather than simultaneously regularizing the past white-hole horizon. Their null time coordinate is not a clock reading of a stationary observer.