Raising indices in curved space

  • Thread starter bdforbes
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  • #1
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Main Question or Discussion Point

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]
 

Answers and Replies

  • #2
HallsofIvy
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Only if the metric tensor is independent of [itex]x_{\nu}[/itex].
 
  • #3
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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.
 
  • #4
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What happens if you vary

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

instead ?
 
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  • #5
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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]
 
  • #6
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This may interest you.

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

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