Brinkmann coordinates

Brinkmann coordinates 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

to allow for the possibility that the pp-wave develops a null curvature singularity.

References

*cite book | author=Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard | title=Exact Solutions of Einstein's Field Equations | location=Cambridge | publisher=Cambridge University Press | year=2003 | id=ISBN 0-521-46136-7
*cite journal | author=H. W. Brinkmann | title=Einstein spaces which are mapped conformally on each other | journal=Math. Ann. | year=1925 | volume=18 | pages = 119 | doi=10.1007/BF01208647