Brinkmann coordinates
Brinkmann coordinates are a particular coordinate system for a spacetime belonging to the family of pp-wave metrics. They are named for Hans Brinkmann. In terms of these coordinates, the metric tensor can be written as
where, the coordinate vector field dual to the covector field, is a null vector field. Indeed, geometrically speaking, it is a null geodesic congruence with vanishing optical scalars. Physically speaking, it serves as the wave vector defining the direction of propagation for the pp-wave.
The coordinate vector field can be spacelike, null, or timelike at a given event in the spacetime, depending upon the sign of at that event. The coordinate vector fields are both spacelike vector fields. Each surface can be thought of as a wavefront.
In discussions of exact solutions to the Einstein field equation, many authors fail to specify the intended range of the coordinate variables. Here we should take
to allow for the possibility that the pp-wave develops a null curvature singularity.
