Manipulation of Tensors I have an expression:
$\eta^{\mu \nu} F_{\alpha \beta, \nu} F^{\alpha \beta}$
Where $\eta^{\mu \nu}$ is the Minkowski metric, F is an antisymmetric tensor, and the comma on the middle tensor denotes a derivative. I am not going to pretend like I have a lot of experience manipulating tensors, but I am trying to raise the co-variant derivative so it is a contravariant derivative, and change it from $\nu$ to $\mu$. Here is what I've accomplished:
$\eta^{\mu \nu} F_{\alpha \beta, \nu} F^{\alpha \beta} = \eta^{\mu \nu} F_{\alpha \beta, \nu} \eta^{\mu \nu} \eta_{\mu \nu} F^{\alpha \beta} = \eta^{\mu \nu} F_{\alpha \beta}^{,\mu} \eta_{\mu \nu} F^{\alpha \beta}$
I am unsure where to proceed from here, but essentially what I'd like to do is be able to get rid of the $\eta's$. Any help is appreciated
 A: I will start again, the trouble I spotted right away with your initial calculation is that you have used $\mu$ as a dummy index of summation whereas it is apparently free from the initial expression:
$$  \eta^{\mu \nu} F_{\alpha \beta, \nu} F^{\alpha \beta} $$
Ok, so, to raise the derivative index, you can just use the existing metric in the expression above: poof it's gone and we have:
$$   F_{\alpha \beta}^{ \ \ \ , \mu} F^{\alpha \beta} $$
Now, as the metric here is constant we can just as well write:
$$   F_{\alpha \beta} F^{\alpha \beta, \mu} $$
To see why the above is true, notice,
$$ F_{\alpha \beta}  = \eta_{\alpha \gamma}\eta_{\beta \sigma} F^{\gamma \sigma}$$
and
$$ F_{\alpha \beta}^{ \ \ \ , \mu} = \partial^{\mu}F_{\alpha \beta} = \partial^{\mu} \left( \eta_{\alpha \gamma}\eta_{\beta \sigma} F^{\gamma \sigma}\right) = \eta_{\alpha \gamma}\eta_{\beta \sigma}\partial^{\mu} F^{\gamma \sigma}$$
hence,
$$ F_{\alpha \beta}^{ \ \ \ , \mu} F^{\alpha \beta} = 
\eta_{\alpha \gamma}\eta_{\beta \sigma}\partial^{\mu} (F^{\gamma \sigma}) F^{\alpha \beta} = 
\partial^{\mu} (F^{\gamma \sigma}) \eta_{\gamma \alpha}\eta_{ \sigma\beta}F^{\alpha \beta}  = \partial^{\mu} (F^{\gamma \sigma}) F_{\gamma \sigma} = F^{\alpha \beta, \mu} F_{\alpha \beta}.$$
In the last step I switched the dummy variables of summation back to $\alpha,\beta$. Note I also used the symmetry of the minkowski metric in the middle step.
Well, I hope this helps. Let me know if you need further elaboration.
