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Components of metric perturbation in TT gauge

  1. Apr 25, 2009 #1
    1. The problem statement, all variables and given/known data

    I'm just working through a text book and there's a line in which I'm clearly missing something. What I want to do is show that from:
    [tex] \bar{h^{TT}_{\mu \nu}} = A^{TT}_{\mu \nu} cos(\omega (t-z)) [\tex]
    to
    [tex] h^{TT}_{\mu \nu} = B^{TT}_{\mu \nu} cos(\omega (t-z)) [\tex]


    2. Relevant equations

    [tex]\bar{h}_{\mu \nu} = h_{\mu \nu} - \frac{1}{2} \eta_{\mu \nu} h[\tex]
    [tex]h = h^{\alpha}_{\alpha} = \eta^{\alpha \beta} h_{\alpha \beta}[\tex]


    3. The attempt at a solution

    There's just one line where I have an issue, i think.

    I've used the above equations to say:

    [tex] \bar{h} = \eta^{\mu \nu}\bar{h}_{\mu \nu}[\tex]
    then substituted the equation for [tex]\bar{h}_{\mu \nu}[\tex] to get:

    [tex] \bar{h} = h - \frac{1}{2} \eta^{\mu \nu} \eta_{\mu \nu} h[\tex]

    which is fine, but for it to work this last line must equal [tex]-h[\tex], I think and I'm not sure i see where that comes from.

    thanks.

    ftj
     
  2. jcsd
  3. Apr 25, 2009 #2
    Your eta is the Minkowski metric, right?

    eta_{alpha beta} eta^{mu nu} = delta_{alpha}^{mu} delta_{beta}^{nu}

    Take it away.
     
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