Brinkmann coordinates: Difference between revisions

Browse history interactively Next edit →Content deleted Content addedVisual WikitextInline

Revision as of 04:36, 24 May 2005 editHillman (talk \| contribs)11,881 editsNo edit summary		Revision as of 04:50, 24 May 2005 edit undoHillman (talk \| contribs)11,881 editsNo edit summaryNext edit →
Line 1:		Line 1:
	] are a particular ] for a ] belonging to the family of ]. In terms of these coordinates, the ] can be written		] are a particular ] for a ] belonging to the family of ]. In terms of these coordinates, the ] can be written

	<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>

	Here, <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.		Here, <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 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 ] of the ] ] <math> u,v,x,y </math>. Here we should take
	The coordinate vector fields <math>\partial_{x}, \partial_{y}</math> are both ] fields. The surfaces <math>u=u_{0}, v=v_{0}</math> can be thought of as defining ]. In the special case of ], these are each ] to an ordinary ]; in general, they might have nonzero ].

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

			to allow for the possibility that our pp-wave develops a ].

			==References==

			{{Book reference \| Author=Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard \| Title=Exact Solutions of Einstein's Field Equations \| Publisher=Cambridge: Cambridge University Press \| Year=2003 \| ID=ISBN 0-521-46136-7}}

Revision as of 04:50, 24 May 2005

Brinkmann coordinates 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

$ds^{2}=H(u,x,y)du^{2}+2dudv+dx^{2}+dy^{2},$

Here, $\partial _{v}$ , 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 our pp-wave develops a null curvature singularity.

References

. ISBN 0-521-46136-7. {{cite book}}: Missing or empty |title= (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Publisher= ignored (|publisher= suggested) (help); Unknown parameter |Title= ignored (|title= suggested) (help); Unknown parameter |Year= ignored (|year= suggested) (help)