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Raising indices in curved space

  1. May 1, 2009 #1
    In curved space, can I raise an index on a tensor that is being differentiated? Ie, is the following true?

    [tex]g^{\mu\lambda}\partial^\nu(F_{\mu\nu})=\partial^\nu(F^\lambda_\nu)[/tex]
     
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  3. May 1, 2009 #2

    HallsofIvy

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    Only if the metric tensor is independent of [itex]x_{\nu}[/itex].
     
  4. May 1, 2009 #3
    This is troubling. I'm trying to obtain Maxwell's Equations from the Lagrangian, and since we have indices both up and down ie [itex]F^{\mu\nu}F_{\mu\nu}[/itex], varying with respect to [itex]\delta A_\lambda[/itex] inevitably introduces metrics:

    [tex] \delta F^{\mu\nu}=\partial^\mu g^{\nu\lambda}\delta A_\lambda - \partial^\nu g^{\mu\lambda}\delta A_\lambda[/tex]

    Integrating a term:

    [tex] \int d^4x\sqrt{-g}(\delta F^{\mu\nu} \delta F_{\mu\nu}) [/tex]
    [tex] = \int d^4x\sqrt{-g}\left[\partial^\mu(g^{\nu\lambda}\delta A_\lambda)F_{\mu\nu} - \partial^\nu(g^{\mu\lambda}\delta A_\lambda)F_{\mu\nu}\right] [/tex]
    [tex] = \int d^4x\sqrt{-g}(g^{\mu\lambda} \partial^\nu F_{\mu\nu} - g^{\nu\lambda} \partial^\mu F_{\mu\nu})\delta A_\lambda[/tex]

    And now I have these terms which are confusing me.
     
  5. May 1, 2009 #4

    Mentz114

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    What happens if you vary

    [tex]g_{\mu b}g_{\nu a}F^{ab}F^{\mu\nu}[/tex]

    instead ?
     
    Last edited: May 1, 2009
  6. May 1, 2009 #5
    I will try that, but I suspect it is equivalent.
    If I had used covariant derivatives the whole time, there would be no problem. But here
    http://en.wikipedia.org/wiki/Maxwell's_equations_in_curved_spacetime#Summary
    it states that the extra terms introduced by using covariant derivatives would cancel out. So maybe my original method is fine.
    Of course, I was stupid to overlook look this fact:
    [tex]\delta F^{\mu\nu} F_{\mu\nu}=\delta F_{\mu\nu} F^{\mu\nu}[/tex]
     
  7. May 1, 2009 #6

    Mentz114

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    This may interest you.

    "Maxwell’s Equations In a Gravitational Field" by Andrew E. Blechman
     

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