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

${\displaystyle ds^{2}=H(u,x,y)du^{2}+2dudv+dx^{2}+dy^{2}}$.

Note that ${\displaystyle \partial _{v}}$, the coordinate vector field dual to the covector field ${\displaystyle dv}$, 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 ${\displaystyle \partial _{u}}$ can be spacelike, null, or timelike at a given event in the spacetime, depending upon the sign of ${\displaystyle H(u,x,y)}$ at that event. The coordinate vector fields ${\displaystyle \partial _{x},\partial _{y}}$ are both spacelike vector fields. Each surface ${\displaystyle u=u_{0},v=v_{0}}$ 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 ${\displaystyle u,v,x,y}$.^[citation needed] Here we should take

${\displaystyle -\infty <v,x,y<\infty ,u_{0}<u<u_{1}}$

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: 119–145. doi:10.1007/BF01208647. S2CID 121619009.