Can index swapping be applied to relativistic Lagrangian equations?

[bananabandana]

Homework Statement

Show that

$$ \mathcal{L} = -\frac{1}{4}F^{\mu \nu}F_{\mu \nu} = - \frac{1}{2}\partial^{\mu}A^{\nu}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}) $$

Where $$ F^{\mu \nu} = \partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu} $$

Homework Equations

The Attempt at a Solution

$$ \mathcal{L} = -\frac{1}{4} F^{\mu \nu}F_{\mu \nu} = -\frac{1}{4}(\partial^{\mu}A^{\nu} - \partial^{\nu}A^{\mu})(\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}) $$
Which expands out to:
$$ -\frac{1}{4} \bigg( \partial^{\mu}A^{\nu}\partial_{\mu}A_{\nu} - \partial^{\mu}A^{\nu}\partial_{\nu}A_{\mu} -\partial^{\nu}A^{\mu}\partial_{\mu}A_{\nu}+\partial^{\nu}A^{\mu}\partial_{\nu}A_{\mu} \bigg) $$

So if I just exchange indices on half of the terms, and then take out a factor, I get to the result I want... question is, how am I allowed to do that??

 

[Orodruin]

It does not matter what you call summation indices, in general
$$
\sum_\mu V_\mu W^\mu = \sum_\nu V_\nu W^\nu.
$$
The only difference in your expression is that the summation convention is being used and therefore we do not write out the sums explicitly but always sum over repeated indices.