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. References Coordinate charts in general relativity