Raising indices in curved space

[bdforbes]

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

$$g^{\mu\lambda}\partial^\nu(F_{\mu\nu})=\partial^\nu(F^\lambda_\nu)$$

 

[HallsofIvy]

Only if the metric tensor is independent of $x_{\nu}$.

 

[bdforbes]

This is troubling. I'm trying to obtain Maxwell's Equations from the Lagrangian, and since we have indices both up and down ie $F^{\mu\nu}F_{\mu\nu}$, varying with respect to $\delta A_\lambda$ inevitably introduces metrics:

$$\delta F^{\mu\nu}=\partial^\mu g^{\nu\lambda}\delta A_\lambda - \partial^\nu g^{\mu\lambda}\delta A_\lambda$$

Integrating a term:

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

And now I have these terms which are confusing me.

 

[Mentz114]

What happens if you vary

$$g_{\mu b}g_{\nu a}F^{ab}F^{\mu\nu}$$

instead ?

 

[bdforbes]

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:
$$\delta F^{\mu\nu} F_{\mu\nu}=\delta F_{\mu\nu} F^{\mu\nu}$$

 

[Mentz114]

This may interest you.

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