Brinkmann coordinates information

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 $u,v,x,y$ .^{[citation needed]} 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.

Brinkmann coordinates information

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