# Electromagnetic four-potential manipulation

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 $$A_{\mu,\nu} + A_{\nu,\mu} = 0$$?

• Wouldn't that require that $A_{\mu,\mu}=0$? – 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.