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Varying the action with respect to metric

  1. Apr 19, 2015 #1
    1. The problem statement, all variables and given/known data
    i want to find the variation of this action with respect to ## g^{\mu\nu}## , where ##N_\mu(x^\nu)## is unit time like four velocity and ##\phi## is scalar field.
    ##I_{total}=I_{BD}+I_{N}##
    ##
    I_{BD}=\frac{1}{16\pi}\int dx^4\sqrt{g}\left\{\phi R-\frac{\omega}{\phi}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi\right\}##
    ##
    I_N=\frac{1}{16\pi}\int
    dx^4\sqrt{g}\{\zeta(x^{\nu})(g^{\mu\nu}N_{\mu}N_{\nu}+1)+2\phi F_{\mu\nu}F^{\mu\nu}-\phi N_\mu
    N^{\nu}(2F^{\mu\lambda}\Omega_{\nu\lambda}+
    F^{\mu\lambda}F_{\nu\lambda}+\Omega^{\mu\lambda}\Omega_{\nu\lambda}-2R_{\mu}^{\nu}+\frac{2\omega}{\phi^2}\nabla_{\mu}\phi\nabla^{\nu}\phi)\}##
    where
    ##F_{\mu\nu}=2(\nabla_{\mu}N_{\nu}-\nabla_{\nu}N_{\mu})##
    ##\Omega_{\mu\nu}=2(\nabla_{\mu}N_{\nu}+\nabla_{\nu}N_{\mu})##





    2. Relevant equations
    i know the variation of Brans-Dicke and its field equations , but i have no idea about ##\delta I_N## with respect to ## \delta g^{\mu\nu}##.

    3. The attempt at a solution

    ##\delta \sqrt{-g}=\frac{-1}{2}\sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu}##
    and ##\delta R=(R_{\mu\nu}+g_{\mu\nu}\Box-\nabla_\mu \nabla_\nu)\delta g^{\mu\nu}##
     
    Last edited: Apr 19, 2015
  2. jcsd
  3. Apr 19, 2015 #2

    Orodruin

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    Did you enter the problem exactly as stated? Your expression is adding terms with different free indices.
     
  4. Apr 19, 2015 #3
    No, this is part of action which i have problem with that
     
  5. Apr 19, 2015 #4

    Orodruin

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    The point is that your expression, as it stands, makes absolutely no sense. You cannot add tensors of different type ...
     
  6. Apr 19, 2015 #5
    now i will edit post with the entire action.
     
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