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I Help with derivation of linearized Einstein field equations

  1. Oct 25, 2016 #1
    Hi all -
    I am trying to follow a derivation of the above. At some point I need to find gαβ for
    gαβ = ηαβ + hαβ
    with |hαβ|<<1
    I am stuck. The text says
    gαβ = ηαβ - hαβ
    but I cannot figure out why. Can anybody help?
  2. jcsd
  3. Oct 25, 2016 #2


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    Well, you want both expressions to be each-other inverses, up to linear order in the metric perturbation h. So what do you get?
  4. Oct 25, 2016 #3


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    Take ##g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}## and ##g^{\alpha\beta}=\eta^{\alpha\beta}-h^{\alpha\beta}## and multiply them together and what do you get?
  5. Oct 25, 2016 #4
    I get an identity as long as ##h_{\alpha\beta} h^{\alpha\beta}## is <<1
  6. Oct 25, 2016 #5


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    Voila!. The quadratic term is discarded. Depending on the source, ##h_{\alpha\beta}## is called the Pauli-Fierz field. It was discovered by Pauli and Fierz as far back as 1939 that the only Lagrange action (hence field equations) describing a spin 2 field is necessarily the linearized Hilbert-Einstein action.
  7. Oct 26, 2016 #6
    Thank you guys!
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