Physics: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

[math]\displaystyle{ ds^2 = H(u,x,y) du^2 + 2 du dv + dx^2 + dy^2 }[/math]

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

[math]\displaystyle{ -\infty \lt v,x,y \lt \infty, u_{0} \lt u \lt u_{1} }[/math]

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.