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

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

Note that $\partial _{v}$, 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: 119–145. doi:10.1007/BF01208647. S2CID 121619009.