Self-energy

Brinkmann coordinates (named for Hans Brinkmann) 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

$d{s}^2=H(u,x,y)d{u}^2+2dudv+d{x}^2+d{y}^2$

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

$-\mathrm{\infty }<v,x,y<\mathrm{\infty },u_0<u<u_1$

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

References

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