= Solution
Write $r_s=2GM/c^2$, $f=1-r_s/r$ and $r_*=r+r_s\ln|r/r_s-1|$, so $dr_*/dr=f^{-1}$. The <ingoing Eddington-Finkelstein coordinates> use $v=t+r_*/c$ to label ingoing radial light fronts. Substituting $dt=dv-dr/(cf)$ into the <Schwarzschild metric> gives
$$
\boxed{ds^2=fc^2dv^2-2c\,dv\,dr-r^2d\Omega^2}.
$$
The 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> have
$$
\boxed{v=\text{constant}\quad\text{(ingoing)},\qquad
\frac{dr}{dv}=\frac c2\left(1-\frac{r_s}r\right)\quad\text{(outgoing)}}.
$$
The 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.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2013/paper-1-advanced-lightcones.png]
{title=Original future light-cone schematic in regular advanced Schwarzschild coordinates, using vertical coordinate cv minus r}
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