Brinkmann coordinates: Difference between revisions

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	'''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 ].

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

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 $u,v,x,y$ . 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.

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.
H. W. Brinkmann (1925). "Einstein spaces which are mapped conformally on each other". Math. Ann. 18: 119–145. doi:10.1007/BF01208647. S2CID 121619009.

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