I was looking at my special relativity notes and when covering EM, I was wondering if the following is true?

Namely, given the four-potential $A_{\mu}$, by definition it is that: $F_{\mu \nu} = -A_{\mu,\nu} + A_{\nu,\mu}$.

Q: I am wondering if it is also that $A_{\mu,\nu} + A_{\nu,\mu} = 0$?

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    $\begingroup$ Wouldn't that require that $A_{\mu,\mu}=0$? $\endgroup$ – Alfred Centauri May 31 at 18:34

A simple counterexample is if you have $A_{\mu} = (\phi(\vec{x}), \vec{0})$, where $\phi$ is the electrostatic potential. Then $\partial_t A_x +\partial_x A_t = \partial_x \phi(\vec{x}) = E_x \neq 0$, in general.

That is one non-zero component of $A_{\mu, \nu} + A_{\nu, \mu}$, hence $A_{\mu, \nu} + A_{\nu, \mu} \neq 0$, in general.

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  • $\begingroup$ Hi. Thanks for the answer. But, can the vector potential actually be zero as you have set it? $\endgroup$ – Thomas Moore May 31 at 16:51
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    $\begingroup$ It can, but not in general. Perhaps I have misunderstood your question? Perhaps you can clarify more precisely what it is that you would like to know. $\endgroup$ – Stratiev May 31 at 17:41

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