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Solving a metric in vacuum

  1. Dec 2, 2011 #1
    Hello,

    I'm having problems solving this problem I got in class.
    I want to learn the concept and how to approach the solution.

    Here it is:

    Consider the metric

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

    What form must the function H have for this metric to represent a plane gravitational wave propagating in vacuum?




    This is how I'm approaching the problem:

    first I let the metric be g_{σβ}=[1,0,0,0;0,1,0,0;0,0,2H,-1;0,0,-1,0]

    and I know that in vacuum the Einstein equation in far outside the source's field leads to
    T^{σβ} = 0
    and I know that a metric representing a plane gravitational wave propagating in vacuum is
    g_{σβ=[1,0,0,0;0,1,0,0;0,0,0,-1;0,0,-1,0]

    coming from the metric ds^2=-dudv+dx^2+dy^2

    Then I don't know how to put the pieces together. I was thinking relating the Einstein equation in terms of the ricci tensor and solving for the Christoffel symbols.

    Not sure. I need to understand better the problem and how to approach this problem.
    Thanks.
     
  2. jcsd
  3. Dec 3, 2011 #2
    Can't you LaTeX this?

    If the metric has a -dudv term then I think the matrix should have -0.5 in dudv and dvdu positions
     
  4. Dec 4, 2011 #3
    the metric is

    [itex]ds=dx^2+dy^2-dudv+2H(x,y,u)du^2 [/itex]

    I know that in vacuum
    [itex]T^{σβ} = 0 [/itex]
     
  5. Dec 5, 2011 #4
    I think you should derive the vacuum field equations.

    Consider the trace of the EFE. This should allow you to prove that [itex]R_{\mu \nu}=0[/itex]

    So it looks like you're going to have to work out a bunch of Christoffel symbols. Although it looks as though everything will vanish except for the situations when you have to take a derivative of [itex]g_{uu}[/itex]

    Hopefully....
     
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