How do I properly construct the electromagnetic tensor in curved space-time? I have my curved spacetime metric $(+,-,-,-)$ and my magnetic vector potential $A$. I tried two ways but not sure which is right (if there is one).
First way:
Compute the magnetic field $B$ from the curl of the magnetic vector potential $A$: $$ \mathbf{B} = \nabla \times \mathbf{A}. $$
Place the resulting components directly in the contravariant electromagnetic tensor definition in cylindrical coordinates: $$ F^{\mu\nu} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & B^z \\ 0 & 0 & 0 & -B^r \\ 0 & -B^z & B^r & 0 \end{pmatrix}. $$
Second way:
Define the electromagnetic four-potential ($\phi$ is zero in my problem): $$ A^\alpha = (\phi, \mathbf{A}). $$
Lower the four-potential index by contracting it with my covariant metric tensor.
Compute the electromagnetic field components with the formula $$ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. $$ I replaced the ordinary derivatives with covariant derivatives.
Raise the indexes of this covariant electromagnetic field tensor to compare with the first way.
The problem is that I can't seem to have the same results with both methods, which says pretty clearly that I am doing something wrong. Is there something fundamentally wrong in taking these steps?