= Solution
Write $r_s=2GM/c^2$ and $f=1-r_s/r$. The <Schwarzschild tortoise coordinate> satisfies $dr_*/dr=1/f$ and can be chosen as $r_*=r+r_s\ln|r/r_s-1|$. Define the advanced time $v=t+r_*/c$. Substituting $dt=dv-dr/(cf)$ into the <Schwarzschild metric> gives the <ingoing Eddington-Finkelstein coordinates> form
$$
\boxed{ds^2=f c^2dv^2-2c\,dv\,dr-r^2d\Omega^2.}
$$
The coefficients are regular at $r=r_s$ and the radial-block determinant is $-c^2$, so the exterior Schwarzschild-time divergence is a coordinate singularity at the future <event horizon>. The physical curvature singularity at $r=0$ remains.
The two families of <radial null trajectories in ingoing Eddington-Finkelstein coordinates> obey
$$
\boxed{v=\mathrm{constant}\quad\text{(ingoing)},\qquad
\frac{dr}{dv}=\frac c2\left(1-\frac{r_s}{r}\right)\quad\text{(outgoing)}.}
$$
Outside the <event horizon>, the outgoing family increases $r$; at the horizon it generates the horizon itself; inside, even the outgoing future-directed family decreases $r$. An infaller crosses the future horizon at finite $v$ and finite <proper time>. The chart makes this causal geometry visible without mistaking Schwarzschild coordinate time for the infaller's clock.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2017-advanced-null-rays.png]
{title=Radial null rays in advanced <Eddington-Finkelstein coordinates>}
{description=Original plot of $v=\mathrm{constant}$ ingoing rays and $dr/dv=(1-r_s/r)/2$ outgoing rays in dimensionless units. Arrows indicate the future direction. Both families move toward smaller <areal radius> inside the <Schwarzschild event horizon>; this is an uncompactified coordinate diagram.}
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