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 formula_2, the coordinate vector field dual to the covector field formula_3, 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 formula_4 can be spacelike, null, or timelike at a given event  in the spacetime, depending upon the sign of formula_5 at that event. The coordinate vector fields formula_6 are both spacelike vector fields. Each surface formula_7 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 formula_8. Here we should take formula_9 to allow for the possibility that the pp-wave develops a null curvature singularity.