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{{technical}} {{technical|date=April 2020}}
'''Brinkmann coordinates''' are a particular ] for a ] belonging to the family of ]. They are named for ]. In terms of these coordinates, the ] can be written as '''Brinkmann coordinates''' are a particular ] for a ] belonging to the family of ]. They are named for ]. 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>.


where <math>\partial_{v}</math>, the ] field dual to the ] field <math>dv</math>, is a ] field. Indeed, geometrically speaking, it is a ] with vanishing ]. Physically speaking, it serves as the ] defining the direction of ] for the pp-wave. Note that <math>\partial_{v}</math>, the ] field dual to the ] field <math>dv</math>, is a ] field. Indeed, geometrically speaking, it is a ] with vanishing ]. Physically speaking, it serves as the ] defining the direction of ] for the pp-wave.


The coordinate vector field <math>\partial_{u}</math> can be spacelike, null, or timelike at a given ] in the ], depending upon the sign of <math>H(u,x,y)</math> at that event. The coordinate vector fields <math>\partial_{x}, \partial_{y}</math> are both ] fields. Each surface <math>u=u_{0}, v=v_{0}</math> can be thought of as a ]. The coordinate vector field <math>\partial_{u}</math> can be spacelike, null, or timelike at a given ] in the ], depending upon the sign of <math>H(u,x,y)</math> at that event. The coordinate vector fields <math>\partial_{x}, \partial_{y}</math> are both ] fields. Each surface <math>u=u_{0}, v=v_{0}</math> can be thought of as a ].


In discussions of ] to the ], many authors fail to specify the intended ]{{disambig needed}} of the ] ] <math> u,v,x,y </math>.{{cn}} Here we should take In discussions of ] to the ], many authors fail to specify the intended ] of the ] ] <math> u,v,x,y </math>.{{cn|date=April 2020}} Here we should take


<math>-\infty < v,x,y < \infty, u_{0} < u < u_{1}</math> <math>-\infty < v,x,y < \infty, u_{0} < u < u_{1}</math>
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==References== ==References==


*{{cite book |author1=Stephani, Hans |author2=Kramer, Dietrich |author3=MacCallum, Malcolm |author4=Hoenselaers, Cornelius |author5=Herlt, Eduard |lastauthoramp=yes | title=Exact Solutions of Einstein's Field Equations | location=Cambridge | publisher=] | year=2003 | isbn=0-521-46136-7}} *{{cite book |author1=Stephani, Hans |author2=Kramer, Dietrich |author3=MacCallum, Malcolm |author4=Hoenselaers, Cornelius |author5=Herlt, Eduard |name-list-style=amp | title=Exact Solutions of Einstein's Field Equations | location=Cambridge | publisher=] | year=2003 | isbn=0-521-46136-7}}
*{{cite journal | author=H. W. Brinkmann | title=Einstein spaces which are mapped conformally on each other | journal=Math. Ann. | year=1925 | volume=18 | pages = 119 | doi=10.1007/BF01208647}} *{{cite journal | author=H. W. Brinkmann | title=Einstein spaces which are mapped conformally on each other | journal=Math. Ann. | year=1925 | volume=18 | pages = 119–145 | doi=10.1007/BF01208647| s2cid=121619009 }}


] ]
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{{relativity-stub}} {{relativity-stub}}
{{math-physics-stub}}

Latest revision as of 07:44, 5 April 2024

This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (April 2020) (Learn how and when to remove this message)

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

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}} .

Note that ∂ 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


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