Is there only one convention to define the electromagnetic field tensor?

Question

I know that the electromagnetic field tensor depends on which metric is used. For example wikipedia uses the $(+---)$ sign convention, but in the Griffiths we have the $(-+++)$ sign convention.

That's why on wikipedia, to define $F_{\mu \nu}$ independently of the metric tensor, they define $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu.$

However, in my electrodynamic course we define $F_{\mu\nu} = - \partial_\mu A_\nu + \partial_\nu A_\mu$ so all the signs are flipped. For example using the $(-+++)$ sign convention (and $c=1$) we get:

$F_{\mu\nu}=\left( \begin{array}{cccc} 0 & E_x & E_y & E_z \\ -E_x & 0 & -B_z & B_y \\ -E_y & B_z & 0 & -B_x \\ -E_z & -B_y & B_x & 0 \\ \end{array} \right)$

This matrix is the same as on wikipedia, and the opposite as the one in the Griffiths (all the signs flipped).

I haven't seen anyone else use the definition $F_{\mu\nu} = - \partial_\mu A_\nu + \partial_\nu A_\mu$ for the electromagnetic field tensor, so I am wondering if there is something wrong in my course, or if there is no problem by defining the tensor in this way.

Answer 1

As long as you keep track of that sign in your formulas, there is no issue. All formulas in which the field tensor appears will simply have a sign difference. The fact most references use the same convention is probably the same we write $2\pi$ instead of $\tau$: someone choose that convention a lot of time ago and it would get messy if we used different conventions.