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With much courtesy I ask a simple explanation to be able to obtain a minus sign missing from the compact form of Maxwell's equations:

$$\boxed{\square \overleftrightarrow F=\mu_0 \boldsymbol{\mathcal{J}}} \tag{1}$$ where the current density quadrivector is given by the relation $\boldsymbol{\mathcal{J}}=(\bar J, ic\rho)$. The tensor (or quadritensor) of the electromagnetic field is given by $$F_{\mu\nu}:=\frac{\partial \mathcal{A}_{\nu}}{\partial X_{\mu}}-\frac{\partial \mathcal{A}_{\mu}}{\partial X_\nu} \tag{2}$$ calculated using the four-potential $\boldsymbol{\mathcal{A}}=\left(\bar{A}, \dfrac ic \varphi\right)$.

It is known $F_{\mu\nu}=-F_{\nu\mu}$ and with several steps (which I am not reporting) I have proved that: \begin{equation} \sum^4_{\nu=1}\frac{\partial F_{\mu\nu}}{\partial X_\nu}=\mu_0\mathcal{J}_\mu,\quad \mu=1,2,3,4. \tag{3} \end{equation} Hence $(3) \iff (1)$. Obviously if I exchange the subscripts of the tensor of the electromagnetic field I easily get the minus sign. Infact being $$F_{\mu\nu}=-F_{\nu\mu} \tag{4}:$$ \begin{equation} \sum^4_{\mu=1}\frac{\partial F_{\nu\mu}}{\partial X_\mu}=-\mu_0\mathcal{J}_\nu,\quad \nu=1,2,3,4. \tag{5} \end{equation} or $$\square \overleftrightarrow F=-\mu_0 \boldsymbol{\mathcal{J}} \tag{6}$$

If all this is correct, why should I consider $F_{\nu\mu}$ instead of $F_{\mu\nu}$? (see the last image). Thank you all.

I should obtain instead of the $(1)$ this: $$\boxed{\square \overleftrightarrow F=-\mu_0 \boldsymbol{\mathcal{J}}} \tag{*}$$

I add any screenshots for my notation and some notes (in Italian language). Excuse me everybody.

enter image description here

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    $\begingroup$ If (3) is correct, then (5) must be wrong (for $\mu_0\mathcal{J}_\nu\ne0$) since the equations differ only by an exchange of the indices, i.e. a relabeling of $\mu\leftrightarrow\nu$ and a sign. $\endgroup$ – user178876 Jul 11 at 21:10
  • $\begingroup$ @marmot Hello very kind :-). There are a lot of passages that I have not reported but possibly when you make the partial derivative of the tensor of the electromagnetic field compared to the second index tick a minus sign. Obviously all this is in my "book". I am not giving an explanation. Thank you for your comment. $\endgroup$ – Sebastiano Jul 11 at 21:14
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    $\begingroup$ I have successfully erased all memory of $\mu_0$ but equation (3) can be derived very simply from the equations of motion for $A_\mu$. And by exchanging $\mu$ and $\nu$ one gets (5) up to the sign, whose origin I do not understand. if you use the antisymmetry of $F_{\mu\nu}$, then you get an equation with a minus, but where the derivative acts on the first index of $F$. $\endgroup$ – user178876 Jul 11 at 21:22
  • $\begingroup$ @marmot I agree totally with you. For to obtain the (5) I have used the (4) with the minus sign. Infact, in this case, the derivative acts on the first index of $F$. Infact my question is: Why I must use the antisymmetry of $F$? If my question it is not very clear you are authorized to edit. $\endgroup$ – Sebastiano Jul 11 at 21:28
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    $\begingroup$ Oh, please, do not use the $x_4 = ict$. It is so 1909, and we are in 2019... $\endgroup$ – DanielC Jul 11 at 22:03
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Starting from your equation (3) we have \begin{equation} \mu_0\mathcal{J}_\mu = \sum^4_{\mu=1}\frac{\partial F_{\mu\nu}}{\partial X_\nu} = -\sum^4_{\mu=1}\frac{\partial F_{\nu\mu}}{\partial X_\nu} \end{equation} therefore your equation (5) should read \begin{equation} \sum^4_{\mu=1}\frac{\partial F_{\nu\mu}}{\partial X_\nu} = - \mu_0\mathcal{J}_\mu \end{equation} which can also be written (after renaming indices) \begin{equation} \sum^4_{\mu=1}\frac{\partial F_{\mu\nu}}{\partial X_\mu} = - \mu_0\mathcal{J}_\nu \end{equation} therefore your equation (5) has a sign error.

But perhaps what you meant to say is that you do not obtain (5) from (3) but rather you just assert that (5) is what you would get if you started out from a different equation in the first place.

I think the problem here may be that you are not aware of an assumption built into the index-free notation you have adopted in (1), concerning on which index of $F$ the differential operator is being employed. There must be a convention, otherwise the equation is ambiguous because $F$ is not symmetric.

The general point is that you cannot expect $\partial_\mu F^{\mu b}$ to be equal to $\partial_\mu F^{b \mu}$ when $F^{ab} \ne F^{ba}$, so any equation involving such quantities has to pay attention to whether the derivative is taken on the first or the second index.

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  • $\begingroup$ I really hope that the site of physicists is as human as possible. $\endgroup$ – Sebastiano Jul 16 at 22:00
  • $\begingroup$ @Sebastiano Sorry I had no time to give you further individual help. $\endgroup$ – Andrew Steane Jul 16 at 22:16
  • $\begingroup$ Ok, don't worry. I accept your comment. $\endgroup$ – Sebastiano Jul 16 at 22:19
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In order to derive the inhomogeneous Maxwell equations, you do not need to use the antisymmetry of the field strength tensor. They follow from the equations of motion for $A_\mu$. If you start with $$ \mathscr{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-e J^\mu A_\mu\;,$$ the Euler-Lagrange equations give you $$ \partial_\mu F^{\mu\nu}=e J^\nu$$ up to a total derivative term that can be gauged or argued away. If you relabel $\mu\leftrightarrow\nu$, you get $$ \partial_\nu F^{\nu\mu}=e J^\mu\;.$$ It might very well be that our conventions differ because I do not work with these $\mu_0$ oddities. That is, all the above assumes natural units, in which $c=1$. In any case, you do not get a sign when renaming the summation indices.

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  • $\begingroup$ You used the lagrangian and the CGS :-(. I have understood something :-) but the minus sign how do I make it appear? $\endgroup$ – Sebastiano Jul 11 at 21:54
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    $\begingroup$ @Sebastiano I do not know. There are a lot of convention-dependent signs, such as the sign of e (which is negative in my conventions) or the signature of the metric (I am using $(+,-,-,-)$). All I can say is that you cannot get it by relabelling, Buona notte! $\endgroup$ – user178876 Jul 11 at 21:57
  • $\begingroup$ Thank you so much for your wonderful and wonderful cooperation. It's my day here and I don't know if you're sleeping. If it were so good to rest. My most sincere greetings. $\endgroup$ – Sebastiano Jul 12 at 9:01
  • $\begingroup$ Nooooooooooooo, why are you canceling from TeX.SE? Don't tell me it's a joke. You have and will have my support fully. I don't really want to believe it, you are part of my friends and not only for the help you gave me. I don't want to think at all that you are leaving. Actually, I am writing to you with the heart to think about it. I have had to face all kinds of offenses but friends first of all. You are an important person. $\endgroup$ – Sebastiano Jul 16 at 21:59
  • $\begingroup$ No, no, I don't really understand. Why would they suspend you? :-( I don't see anything at all that you've done badly. Please stay. Why do good and competent people always have to flee? I don't want to believe you're leaving. $\endgroup$ – Sebastiano Jul 16 at 22:04

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