# Ingoing Eddington-Finkelstein coordinates The Schwarzschild black hole can be written in ingoing Eddington-Finkelstein coordinates as$ds^2=−\left(1−\frac{2GM}{r}\right)dv^2+2dvdr+r^2d\Omega^2.$Ingoing null rays follow constant-$v$ lines in this coordinate system, a property often desired. For a more general spherically symmetric and static black hole, a similar ingoing Eddington-Finkelstein coordinate system can be used:$ds^2=−f(r)dv^2+2dvdr+h(r)d\Omega^2,$where the functions $f(r)$ and $h(r)$ depend on the theory and what matter fields are present. In asymptotically AdS spacetimes, $f(r)\to r^2$ and $h(r)\to r^2$ (AdS length set to unity) for large $r$; in asymptotically flat spacetimes, $f(r)\to 1$ and $h(r)\to r^2$. ## Symmetry Since lines of constant $r$ are the same in either Schwarzschild or iEF coordinates, the orbits of $\partial_v$ are exactly those of $\partial_t$. So $\partial_v$ is the KVF. ## Outgoing Similarly, we also have the outgoing EF coordinates: $d s^2=-\left(1-\frac{2 G M}{r}\right) d u^2-2 d u d r+r^2 d \Omega^2.$ ## Coordinate transformation For a metric given in Schwarzschild-like coordinates,$ds^2=-f(r)dt^2+\frac{dr^2}{f(r)}+h(r)d\Omega^{2},$we need to make the following coordinate transformation:$dt=dv-\frac{dr}{f(r)}.$Substituting $dt$ into the metric gives the metric in ingoing EF coordinates.