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In Dirac's paper: Classical theory of radiating electrons, he decides to raise and lower the indices on the same object multiple times:

\begin{align*} \frac{\partial{A_{\mu}}}{\partial{x_{\mu}}} &=0\tag{3} \\ F^{{\mu}{\nu}}&= \frac{\partial{A^{\nu}}}{\partial{x_{\mu}}}- \frac{\partial{A^{\mu}}}{\partial{x_{\nu}}}\tag{6}\\ F^{{\mu}{\nu}}_{rad}&= F^{{\mu}{\nu}}_{ret} - F^{{\mu}{\nu}}_{adv}\tag{11}\\ F_{{\mu}{\nu}{\;rad}}&= \frac{4e}{3}(\frac{d^3z_{\mu}}{ds^3}\frac{dz_{\nu}}{ds}- \frac{d^3z_{\nu}}{ds^3}\frac{dz_{\mu}}{ds}) \tag{12}\\ 4\pi T_{{\mu}{\rho}}&= F_{{\mu}{\nu}}F^{{\nu}}_{\rho}+ \frac{1}{4}g_{{\mu}{\rho}}F_{{\alpha}{\beta}}F^{{\alpha}{\beta}}\tag{14}\\ \end{align*}

There's no way of knowing for certain why Dirac did this, so I've asked what some of the reasons are within physics generally.

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The decision whether to write a tensor equation using contravariant free indices, or covariant free indices, or a mixture of both, has no physical significance and is thus completely arbitrary.

For example,

$$F^{\mu\nu}=\partial^\mu A^\nu-\partial^\nu A^\mu$$

and

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$$

both express exactly the same physics, so it doesn’t matter which one you write. Each equation implies the other.

Because of this, most physicists do not bother being particularly consistent in their free index placement.

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