- Brinkmann coordinates
Brinkmann coordinates are a particular
coordinate system for aspacetime belonging to the family ofpp-wave metrics . In terms of these coordinates, themetric tensor can be written as:
where , the
coordinate vector field dual to thecovector field , is a null vector field. Indeed, geometrically speaking, it is anull geodesic congruence with vanishingoptical scalars . Physically speaking, it serves as thewave vector defining the direction of propagation for the pp-wave.The coordinate vector field can be spacelike, null, or timelike at a given event in the
spacetime , depending upon the sign of at that event. The coordinate vector fields are both spacelike vector fields. Each surface can be thought of as awavefront .In discussions of
exact solutions to theEinstein field equation , many authors fail to specify the intendedrange of thecoordinate variables . Here we should taketo allow for the possibility that the pp-wave develops a
null curvature singularity .References
*cite book | author=Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard | title=Exact Solutions of Einstein's Field Equations | location=Cambridge | publisher=Cambridge University Press | year=2003 | id=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
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