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{{Expert-subject|Science|date=November 2008}} {{Expert-subject|Science|date=November 2008}}


'''Brinkmann coordinates''' are a particular ] for a ] belonging to the family of ]. In terms of these coordinates, the ] can be written as '''Brinkmann coordinates''' (named for Hans Brinkmann) are a particular ] for a ] belonging to the family of ]. In terms of these coordinates, the ] can be written as


:<math>ds^2 \, = H(u,x,y) du^2 + 2 du dv + dx^2 + dy^2</math> :<math>ds^2 \, = H(u,x,y) du^2 + 2 du dv + dx^2 + dy^2</math>

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This article needs attention from an expert in Science. Please add a reason or a talk parameter to this template to explain the issue with the article. WikiProject Science may be able to help recruit an expert. (November 2008)

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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