Originally posted by turin
Wouldn't this be just as complicated?

you be the judge
Another thing: How do you know that this is going to give you the dot product? I mean, I can see that going throught the calculations would show this, but, how do you know to start with the LeviCivita contraction in the first place? It seems kind of outoftheblue to me. Is there some theorem or something?

let s have a look:
the LeviCivita tensor will select all the terms with even permutations of 1234 with a plus sign, and all the terms with odd permutations with a minus sign
[tex]
\begin{multline*}
F^{\mu\nu}F^{\rho\sigma}\epsilon_{\mu\nu\rho\sigma}\\
=4(F^{01}F^{23}+F^{02}F^{13}+F^{03}F^{12})=4\mathbf{E}\cdot\mathbf{B}
\end{multline*}
[/tex]
and that s all i need to do, it is guaranteed to be lorentz invariant.