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Write $f(r)=1-2M/r$. Radial null paths satisfy $dt/dr=\pm f^{-1}$, whose integral uses the Schwarzschild tortoise coordinate $r_{\ast }=r+2M\log |r/(2M)-1|$. The incoming and outgoing families are thereforewith constants $v,u$. For an exterior outgoing ray, put $r/(2M)-1=\epsilon$. As $t\to -\infty$, $t=u+2M(1+\epsilon )+2M\log \epsilon$, so $\epsilon \sim C{e}^{t/(2M)}$ 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, $v=t+r_{\ast }$, so $dt=dv-dr/f$. Substitution cancels the singular $d{r}^2/f$ term and givesThis metric is regular at $r=2M$. In the requested chart $\hat{t}=v-r$, the incoming rays are $\hat{t}=v_0-r$ and the outgoing rays satisfyIncoming rays move inward everywhere. Outgoing rays move outward outside the horizon, remain at constant radius on it, and also move inward inside it.

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 givesThe future direction in the ingoing chart has $\dot{r}<0$, so $-\dot{r}\ge \sqrt{2M/r-1}$. The proper-time bound inside a Schwarzschild black hole isThus 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.