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The ] are a family of ] of ]. They represent wavelike disturbances in the ] of ] which propagate at the ]. In terms of ], the ] defining a pp-wave spacetime can be written The ] are a family of ] of ]. They represent wavelike disturbances in the ] of ] which propagate at the ]. In terms of ], the ] defining a pp-wave spacetime can be written


<math>ds^2=H(u, x, y)du^2+2dudv+dx^2+dy^2</math> <math>ds^2=H(u, x, y)du^2+2dudv+dx^2+dy^2</math>
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==References== ==References==


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


{{physics-stub}} {{physics-stub}}

Revision as of 04:02, 24 May 2005

The pp-wave metrics are a family of exact solutions of Einstein's field equation. They represent wavelike disturbances in the curvature of spacetime which propagate at the speed of light. In terms of Brinkmann coordinates, the line element defining a pp-wave spacetime can be written

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}}

To obtain a null dust solution, we may choose H ( u , x , y ) {\displaystyle H(u,x,y)} to be any smooth function. If we require H ( u , x , y ) {\displaystyle H(u,x,y)} > to be a harmonic function (that is, a solution of the Laplace equation in the variables x , y {\displaystyle x,y} ), then we obtain a vacuum solution.

An important class of pp-waves are the Baldwin/Jeffery plane waves, which are obtained by choosing

H ( u , x , y ) = a ( u ) ( x 2 − y 2 ) + 2 b ( u ) x y + c ( u ) ( x 2 + y 2 ) {\displaystyle H(u,x,y)=a(u)(x^{2}-y^{2})+2b(u)xy+c(u)(x^{2}+y^{2})}

Here, if c ( u ) {\displaystyle c(u)} vanishes, we have the plane gravitational waves.

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)

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