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For radial light rays, the transformed metric givesThe two families of radial null trajectories in ingoing Eddington-Finkelstein coordinates are thereforefor ingoing rays andfor outgoing rays.

Now set $t_{\ast }=v-r$. Along ingoing rays,Along outgoing rays,and integration givesThus the ingoing rays are straight lines of slope $-1$ in the $r$-$t_{\ast }$ plane. Outgoing rays have positive slope outside $r=2M$, become vertical as they approach the horizon, and have negative slope inside it; the horizon itself is the limiting outgoing null ray.

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.

Write $r_s=2GM/c^2$ and $f=1-r_s/r$. The Schwarzschild tortoise coordinate satisfies $d{r}_{\ast }/dr=1/f$ and can be chosen as $r_{\ast }=r+r_s\ln |r/r_s-1|$. Define the advanced time $v=t+r_{\ast }/c$. Substituting $dt=dv-dr/(cf)$ into the Schwarzschild metric gives the ingoing Eddington-Finkelstein coordinates formThe 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 obeyOutside 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.