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}+2dudv+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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