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Brinkmann coordinates

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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 + 2 d u d v + d x 2 + d y 2 {\displaystyle ds^{2}\,=H(u,x,y)du^{2}+2dudv+dx^{2}+dy^{2}}

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

− ∞ < v , x , y < ∞ , u 0 < u < u 1 {\displaystyle -\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.{{cite book}}: CS1 maint: multiple names: authors list (link)
  • H. W. Brinkmann (1925). "Einstein spaces which are mapped conformally on each other". Math. Ann. 18: 119. doi:10.1007/BF01208647.


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