- #1

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[tex]g^{\mu\lambda}\partial^\nu(F_{\mu\nu})=\partial^\nu(F^\lambda_\nu)[/tex]

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- Thread starter bdforbes
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- #1

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[tex]g^{\mu\lambda}\partial^\nu(F_{\mu\nu})=\partial^\nu(F^\lambda_\nu)[/tex]

- #2

HallsofIvy

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Only if the metric tensor is independent of [itex]x_{\nu}[/itex].

- #3

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[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.

- #4

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What happens if you vary

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

instead ?

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

instead ?

Last edited:

- #5

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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]

- #6

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