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Tensor variation wrt metric

  1. May 15, 2015 #1
    1. The problem statement, all variables and given/known data
    I'm just wondering if i'm doing this calculation correct?
    eta and f are both tensors

    2. Relevant equations


    3. The attempt at a solution
    [tex]\frac{\delta \left ( \gamma_{3}f{_{\lambda}}^{k}f{_{k}}^{\sigma}f{_{\sigma}}^{\lambda} \right )}{\delta f^{\mu\nu}}=\frac{\delta\left (\gamma_{3} f^{\epsilon k}\eta_{\lambda\epsilon}f^{\rho\sigma}\eta_{k\rho}f^{\omega\lambda}\eta_{\sigma\omega} \right ) }{\delta f^{\mu\nu}}\\
    =\gamma_{3}\left ( \delta_{\mu}^{\epsilon}\delta_{\nu}^{k}f^{\rho\sigma}f^{\omega\lambda}+\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}f^{\epsilon k}f^{\omega\lambda}+\delta_{\mu}^{\omega}\delta_{\nu}^{\lambda}f^{\epsilon k}f^{\rho\sigma} \right )\times\left ( \eta_{\lambda\epsilon}\eta_{k\rho}\eta_{\sigma\omega} \right )\\
    =\gamma_{3}\left ( f{_{\nu}}^{\sigma}f{_{\sigma}}^{\lambda}\eta_{\lambda\mu}+f{_{\nu}}^{\lambda}f{_{\lambda}}^{k}\eta_{k\mu}+f{_{\nu}}^{k}f{_{k}}^{\sigma}\eta_{\sigma\mu} \right )\\
    =3\gamma_{3} f{_{\nu}}^{\sigma}f{_{\sigma}}^{\lambda}\eta_{\lambda\mu}[/tex]
     
  2. jcsd
  3. May 15, 2015 #2

    fzero

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    It looks ok if this is a 1st order formalism where the metric is being expanded around the flat metric: ##g_{\mu\nu}=\eta_{\mu\nu} + f_{\mu\nu}##. If it is something else, it may or may not be correct.
     
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