We're busy doing a GR course, and index notation has always been something that confuses me. In particular, is there a difference between the following, and if so, what is it?
$A^\mu_\nu$; $A^\mu{}_\nu$ and $A_\nu{}^\mu$
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$\begingroup$ Possible duplicates: Difference between slanted indices on a tensor and links therein. $\endgroup$– Qmechanic ♦Commented Aug 13, 2021 at 18:25
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Let's say we start with a general (not symmetric or anti-symmetric) tensor $A_{\mu\nu}$. Note $A_{01} \neq A_{10}$ (and similarly for other combinations). Then \begin{equation} A^\mu{}_\nu = g^{\mu\rho} A_{\rho\nu} \neq g^{\mu\rho}A_{\nu\rho} = A_\nu{}^\mu \end{equation} which you could see by explicitly writing out both sides of the not equals sign for a specific value of $\mu$ and $\nu$ (say $\mu=\nu=0$).
If $A_{\mu\nu}$ is symmetric however, in the sense that $A_{\mu\nu}=A_{\nu\mu}$, then $A^\mu{}_\nu = A_\nu{}^\mu$. In that case, out of sheer laziness, some people write $A^\mu_\nu$ instead of one of $A^\mu{}_\nu$ or $A_\nu{}^\mu$. However the notation $A^\mu_\nu$ is just a shorthand and is meaningless if $A$ is not a symmetric tensor.
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7$\begingroup$ To add on, writing ambiguous notation like $A^\mu_\nu$ is bad practice and highly discouraged. $\endgroup$ Commented Aug 13, 2021 at 11:55