Maxwell equations from Euler-Lagrange equation: I keep obtaining the wrong equation

Question

I'm deriving the Maxwell equations from this Lagrangian:

$$ \mathscr{L} \, = \, -\frac{1}{4} F^{\mu \nu}F_{\mu \nu} + J^\mu A_\nu \tag{1}$$

My signature is $$(+ - - -)\tag{2}$$ and

$$ F^{\mu \nu} \, = \,\left(\begin{matrix}0 & -E_x & -E_{y} & -E_{z} \\ E_{x} & 0 & -B_{z} & B_{y} \\ E_y & B_{z} & 0 & -B_x \\ E_z & -B_{y} & -B_{x} & 0\end{matrix}\right)\tag{3} $$

My procedure is almost exactly the same as this one: https://physics.stackexchange.com/a/14854/121554

But he has a $+\frac{1}{4}F^{\mu \nu}F_{\mu \nu}$ in the lagrangian.

So, while he obtains the right equation $$\partial_\mu F^{\mu \nu}\, = \, J^\nu,\tag{4}$$ I carry a minus sign till the end and my final equation is

$$ J^\nu \,=\, -\partial_\mu F^{\mu \nu}\, = \, \partial_\mu F^{\nu \mu}\, ; \tag{5} $$

Which is clearly wrong if you write it down explicitely in function of the fields, the charge and the currents.

Is my lagrangian wrong for my metric and my definition of the elctromagnetic field tensor? We spent some time talking about that lagrangian at lesson and the professor gave a lot of importance to that minus sign in order to have positive kinetic term. Am I missing something?

I can write down all my calculations if requested but they are basically the same of the link I provided above.

Answer 1

I suspect the error is in your source term: with reference to Jackson's "Classical electrodynamics", the correct Lagrangian density is $$ {\cal L}=-\frac{1}{16\pi} F_{\alpha\beta}F^{\alpha\beta}-\frac{1}{c}J_\alpha A^\alpha\, , $$ which differ from yours by a sign in the source term. (The other factors $1/16\pi$ and $1/c$ are linked to the use of Gaussian units.) The article you link to also have the same sign for both terms in the Lagrangian density.