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Gauge Transformations in GR

  1. Nov 14, 2013 #1
    I need some help with a derivation in GR.

    The linearized field equation in GR is:

    [tex]G_{ab}^{(1)} = - \frac{1}{2}{\partial ^c}{\partial _c}{{\bar \gamma }_{ab}} + {\partial ^c}{\partial _{(b}}{{\bar \gamma }_{a)c}} - \frac{1}{2}{\eta _{ab}}{\partial ^c}{\partial ^d}{{\bar \gamma }_{cd}} = 8\pi {T_{ab}}[/tex]

    How would I use the gauge transformation

    [tex]{\gamma _{ab}} \to {\gamma _{ab}} + {\partial _b}{\xi _a} + {\partial _a}{\xi _b}[/tex]

    to simplify the linearized equation to:

    [tex]{\partial ^c}{\partial _c}{{\bar \gamma }_{ab}} = - 16\pi {T_{ab}}[/tex]?

    EDIT: I should also mention:

    [tex]{{\bar \gamma }_{ab}} = {\gamma _{ab}} - \frac{1}{2}{\eta _{ab}}\gamma [/tex]

    [tex]\gamma = \gamma _a^a[/tex]

    and

    [tex]{g_{ab}} = {\eta _{ab}} + {\gamma _{ab}}[/tex]
     
  2. jcsd
  3. Nov 16, 2013 #2
    Nvm. I figured it out.
     
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