= Solution
Write $f(r)=1-2M/r$. Radial null paths satisfy $dt/dr=\pm f^{-1}$, whose integral uses the <Schwarzschild tortoise coordinate> $r_*=r+2M\log|r/(2M)-1|$. The incoming and outgoing families are therefore
$$
\boxed{t=v-r_*,\qquad t=u+r_*}
$$
with constants $v,u$. For an exterior outgoing ray, put $r/(2M)-1=\varepsilon$. As $t\to-\infty$, $t=u+2M(1+\varepsilon)+2M\log\varepsilon$, so $\varepsilon\sim C e^{t/(2M)}$ for a positive ray-dependent $C$. As $t\to\infty$, the logarithm is smaller than the linear term, giving $r\sim t$.
In <Ingoing Eddington-Finkelstein coordinates>, $v=t+r_*$, so $dt=dv-dr/f$. Substitution cancels the singular $dr^2/f$ term and gives
$$
\boxed{ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega^2.}
$$
This metric is regular at $r=2M$. In the requested chart $\widehat t=v-r$, the incoming rays are $\widehat t=v_0-r$ and the outgoing rays satisfy
$$
\widehat t=u+r+4M\log|r/(2M)-1|,\qquad
\frac{dr}{d\widehat t}=\frac{r-2M}{r+2M}.
$$
Incoming rays move inward everywhere. Outgoing rays move outward outside the horizon, remain at constant radius on it, and also move inward inside it.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-4-null-rays.png]
{title=Radial null trajectories in a regular ingoing Schwarzschild chart, with future directions and the event horizon marked}
The line $r=2M$ is the <Schwarzschild event horizon>, a null boundary of the future black-hole region. It is a coordinate singularity of Schwarzschild time, not a curvature singularity; the curvature singularity is at $r=0$. Signals from the future black-hole interior cannot cross it outward.
Finally parametrize any future-directed timelike observer in that interior by <proper time> $s$. Normalization in the given metric gives
$$
\frac{\dot r^2}{2M/r-1}=1+(2M/r-1)\dot t^2+r^2(\dot\theta^2+\sin^2\theta\dot\phi^2)\geq1.
$$
The future direction in the ingoing chart has $\dot r<0$, so $-\dot r\geq\sqrt{2M/r-1}$. The <proper-time bound inside a Schwarzschild black hole> is
$$
\boxed{\Delta s\leq\int_0^{r_0}\sqrt{\frac r{2M-r}}\,dr
=2M\arcsin\sqrt{\frac{r_0}{2M}}-\sqrt{r_0(2M-r_0)}<\pi M.}
$$
Thus a future-continued observer must reach the singular boundary in finite <proper time>, regardless of acceleration or angular motion. The black-hole future orientation is essential; a past-directed path or a future path in a white-hole interior is a different statement.
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