Eddington–Finkelstein Coordinates

In general relativity Eddington–Finkelstein coordinates are named for Arthur Stanley Eddington and David Finkelstein, even though neither ever wrote down these coordinates or the metric in these coordinates. They seem to have been given this name by Misner, Thorne, and Wheeler in their book Gravitation (book). They are a pair of coordinate systems for a Schwarzschild geometry which are adapted to radial null geodesics (i.e. the worldlines of photons moving directly towards or away from the central mass). The outward (inward) traveling radial light rays define the surfaces of constant "time" while the radial coordinate is the usual area coordinate so that the surfaces of rotation symmetry have an area of 4π. One advantage of this coordinate system is that it shows that the apparent singularity at the Schwarzschild radius is only a coordinate singularity and not a true physical singularity. (While this was recognized by Finkelstein, it was not (or at least not commented on) by Eddington, whose primary purpose was to compare and contrast the spherically symmetric solutions in Whitehead's theory of gravitation and Einstein's.)

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Eddington–Finkelstein Coordinates - Metric
... The ingoing Eddington–Finkelstein coordinates are obtained by replacing the coordinate t with the new coordinate ... The metric in these coordinates can be written where is the standard metric on a unit radius two sphere ... Likewise, the outgoing Eddington–Finkelstein coordinates are obtained by replacing t with the null coordinate ...