Maybe it's a very basic physics question but I'm confused and not sure if I should ask my question here. From this paper, I have the potential vector
$$ A=A_t d t+A_\phi(d \phi-\omega d t) $$
where
$$ \begin{aligned} & A_t=-\frac{Q}{r}+\frac{3}{4} Q B^2 r\left(1+f \cos ^2 \theta\right), \\ & A_\phi=\frac{2}{B}-H^{-1}\left[\frac{2}{B}+\frac{1}{2} B\left(r^2 \sin ^2 \theta+3 Q^2 \cos ^2 \theta\right)\right] \end{aligned} $$
and I want to calculate the components of the Faraday tensor. To do this, I took the potential vector in the following way
$$ A_{\mu} = (A_t - \omega A_{\phi}, 0, 0, A_{\phi}) $$
$$ f = 1 - \frac{2 M}{r} + \frac{Q^{2}}{r^{2}} $$
and in Mathematica I wrote
Clear["Global`*"] coords = {t, r, \[Theta], \[Phi]}; At = -(Q/r) + 3/4*Q*B^2*r*(1 + f*(Cos[\[Theta]])^2) /. {f -> 1 - (2*M)/r + Q^2/r^2}; Aphi = 2/B - (H)^-1*(2/B + 1/2*B*(r^2*(Sin[\[Theta]])^2 + 3*Q^2*(Cos[\[Theta]])^2)); A = {At - Aphi *\[Omega], 0, 0, Aphi}; F = Table[ D[A[[j]], coords[[k]]] - D[A[[k]], coords[[j]]], {j, 1, 4}, {k, 1, 4}]; And the items I get are not what appears in the article. For example, according to the article, it should have the following components (Eq. 8, 9, 10 and 11)
$$ \begin{aligned} & F_{r t}=-\frac{Q}{r^2}+\frac{3}{4} Q B^2\left(1+\left(1-\frac{Q^2}{r^2}\right) \cos ^2 \theta\right) \\ & F_{\theta t}=-\frac{3}{2} Q B^2 r\left(1-\frac{2 M}{r}+\frac{Q^2}{r^2}\right) \sin \theta \cos \theta \end{aligned} $$
I do not write the $F_{r \phi}$ and $F_{\theta \phi}$ components because they are very extensive.
