Brinkmann coordinates

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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

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.


  • 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.