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Brinkmann coordinates
Brinkmann coordinates (named for Hans Brinkmann) are a particular coordinate system for a spacetime belonging to the family of pp-wave metrics. In terms of these coordinates, the metric tensor can be written as
\( ds^2 \, = H(u,x,y) du^2 + 2 du dv + dx^2 + dy^2 \)
where \( \partial_{v}, the coordinate vector field dual to the covector field dv, 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 \( \partial_{u} \) can be spacelike, null, or timelike at a given event in the spacetime, depending upon the sign of H(u,x,y) at that event. The coordinate vector fields \( \partial_{x}, \partial_{y} \) are both spacelike vector fields. Each surface \( u=u_{0}, v=v_{0} \) 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 u,v,x,y . Here we should take
\( -\infty < v,x,y < \infty, u_{0} < u < u_{1} \)
to allow for the possibility that the pp-wave develops a null curvature singularity.
References
Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard (2003). Exact Solutions of Einstein's Field Equations. Cambridge: Cambridge University Press. ISBN 0-521-46136-7.
H. W. Brinkmann (1925). "Einstein spaces which are mapped conformally on each other". Math. Ann. 18: 119. doi:10.1007/BF01208647.
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