 Outgoing Eddington-Finkelstein coordinates

The retarded coordinate $u=t-r_{\ast }$ puts the radial Schwarzschild metric into $d{s}^2=-fd{u}^2-2dudr+r^{2}d{\Omega }^2$, with $f=1-2M/r$. Outgoing radial null geodesics have constant $u$. These coordinates are regular at the past horizon; an infaller crossing the future horizon has $u\to +\infty$.