Contravariant and covariant indices

[spookyfish]

When we write contravariant and covariant indices, for example for the Lorentz transformation, does it matter if we write $\Lambda^\mu\,_\nu$ or $\Lambda^\mu_\nu$?
i.e. if the $\nu$ index is to the right of the $\mu$ or they are at the same place with respect to left-right?

 

[WannabeNewton]

https://www.physicsforums.com/showthread.php?t=695850

 

[Fredrik]

Lorentz transformations are linear operators on $\mathbb R^4$ (or $\mathbb R^2$ or $\mathbb R^3$). So they can be represented by matrices. (See the https://www.physicsforums.com/showthread.php?t=694922 about matrix representations of linear transformations). I will not make any notational distinction between a linear operator and its matrix representation with respect to the standard basis.

Let $\Lambda$ be an arbitrary Lorentz transformation. By definition of Lorentz transformation, we have $\Lambda^T\eta\Lambda=\eta$. This implies that $\Lambda^{-1}=\eta^{-1}\Lambda^T\eta$. Let's use the notational convention that for all matrices X, we denote the entry on row $\mu$, column $\nu$ by $X^\mu{}_\nu$. If we use this convention, the definition of matrix multiplication, our formula for $\Lambda^{-1}$ and the convention that every index that appears twice is summed over, we get
$$(\Lambda^{-1})^\mu{}_\nu = (\eta^{-1})^\mu{}_\rho (\Lambda^T)^\rho{}_\sigma \eta^\sigma{}_\nu = (\eta^{-1})^\mu{}_\rho \Lambda^\sigma{}_\rho \eta^\sigma{}_\nu.$$ This is where things get funny. It's conventional to write $\eta_{\mu\nu}$ instead of $\eta^\mu{}_\nu$, and $\eta^{\mu\nu}$ instead of $(\eta^{-1})^\mu{}_\nu$. If we use this convention, we have
$$(\Lambda^{-1})^\mu{}_\nu = \eta^{\mu\rho} \Lambda^\sigma{}_\rho \eta_{\sigma\nu}.$$ Now if we also use the convention that $\eta^{\mu\nu}$ raises indices and $\eta_{\mu\nu}$ lowers them, we end up with
$$(\Lambda^{-1})^\mu{}_\nu = \Lambda_\nu{}^\mu.$$ So if $\Lambda$ isn't the identity transformation, we have
$$\Lambda_\nu{}^\mu = (\Lambda^{-1})^\mu{}_\nu \neq \Lambda^\mu{}_\nu.$$ As you can see, the inequality is a result of the definitions of $\eta_{\mu\nu}$ and $\eta^{\mu\nu}$, so if you use a notational convention that denotes these things by something else, or doesn't use these things to raise and lower indices, it may be OK to write $\Lambda^\mu_\nu$.