# Properties of the Faraday tensor for constant fields

I'm doing a special relativity past exam paper and have got caught up with something that I hope someone can help me with!

I have to show that for constant fields, the magnitude of A, the acceleration 4-vector, is constant.

Given that the 4-force in the presence of electric and magnetic fields is $f^{\mu}=eF^{\mu\nu}U_{\nu}$ we can use $A^{\mu}=\frac{f^{\mu}}{m}$ to get $\frac{d|A^2|}{d\tau}=2A_\mu\frac{dA^\mu}{d\tau}=2\frac{e}{m}F^{\mu\nu}A_{\mu}A_{\nu}$. Now apparently this last expression equals zero but I cant work out or find any justification for this, can anyone help?

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$F^{\mu \nu}$ is antisymmetric, and $A_\mu A_\nu$ is symmetric in its indices.