Schwarzschild Coordinates - Coordinate Singularities

Coordinate Singularities

Looking back at the coordinate ranges above, note that the coordinate singularity at marks the location of the North pole of one of our static nested spheres, while marks the location of the South pole. Just as for an ordinary polar spherical chart on E3, for topological reasons we cannot obtain continuous coordinates on the entire sphere; we must choose some longitude (a great circle) to act as the prime meridian and cut this out of the chart. The result is that we cut out a closed half plane from each spatial hyperslice including the axis and a half plane extending from that axis.

When we said above that is a Killing vector field, we omitted the pedantic but important qualifier that we are thinking of as a cyclic coordinate, and indeed thinking of our three spacelike Killing vectors as acting on round spheres.

Possibly, of course, or, in which case we must also excise the region outside some ball, or inside some ball, from the domain of our chart. This happens whenever f or g blow up at some value of the Schwarzschild radial coordinate r.