How to raise indices on the electromagnetic tensor How do you transform between the electromagnetic tensors $F_{\mu\nu}$ and $F^{\mu\nu}$?
$$
F_{\mu \nu}=
\begin{pmatrix}
0 & E_x & E_y & E_z \\
-E_x & 0 & -B_z & B_y \\
-E_y & B_z & 0 & -B_x \\
-E_z & -B_y & B_x & 0
\end{pmatrix},\\ \ F^{\mu \nu} = 
\begin{pmatrix}
0 & -E_x & -E_y & -E_z \\
E_x & 0 & -B_z & B_y \\
E_y & B_z & 0 & -B_x \\
E_z & -B_y & B_x & 0
\end{pmatrix}
$$
In other words, what do you do to $F_{\mu\nu}$ to get $F^{\mu\nu}$?
 A: Index raising and lowering is defined through the metric, in this case the flat space metric (Minkowski)
$$
g^{\mu\nu} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{pmatrix}$$
We raise an index by aplying the metric to a tensor, like this $A^\mu=g^{\mu\nu}A_\nu$. Now, if you want to raise two index you need to operate with the metric twice.
$$F^{\mu\nu}=g^{\mu\alpha}g^{\beta\nu}F_{\alpha\beta}$$
In a more formal language lowering and raising indices is a way to construct isomorphisms between covariant and contravariant tensorial spaces. We use the metric tensor because it help us to map basis vectors $e_i$ to dual basis vector $\beta^i$.
A: I see now, as with transforming one-forms to/from vectors, you apply the metric. Because you want to make two subscripts superscripts, apply it twice. So, with 
$$
g^{\mu\nu} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{pmatrix}
$$
$$
F^{\mu\nu} = g^{\mu\nu}F_{\mu\nu}g^{\mu\nu}
$$
