# Raising and lowering indices

I read multiple sources about raising and lowering indices and Einstein summary notation but am having problems doing it.

Here is an example. I am trying to derive Maxwell's inhomogeneous equations in the Minkowski vacuum $$$$\partial_\mu F^{\mu\nu} = 0$$$$ using the alternative form in curved space: $$$$\frac{1}{\sqrt{-g}}\partial_\mu\Bigg[(\sqrt{-g\,}g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}\Bigg] = 0$$$$ Substituting $$g^{\phi\theta} = \eta^{\phi\theta} = \text{diag} (+,-,-,-)$$: $$$$\partial_\mu F_{\alpha\beta} = 0$$$$

What is the proper way to convert this to $$\partial_\mu F^{\mu\nu} = 0$$ or $$\partial_\alpha F^{\alpha\beta} = 0$$ or $$\partial_\lambda F^{\lambda\kappa} = 0$$? I keep getting a mess.

edit:

I am at a level of trying to make sure I am doing the mechanics properly. I think in the back of my mind I knew I could contract indices to get your alternative form. Here is my messy approach: $$$$\eta_{\mu\phi} \: \partial^\phi F_\alpha \, ^\rho \: \eta_{\beta\rho} = 0$$$$ $$$$\eta_{\mu\phi} \: \eta^{\phi\sigma} \: \partial_\sigma F_\alpha \, ^\rho \: \eta_{\beta\rho} = 0$$$$ $$$$\eta_{\mu\phi} \: \eta^{\phi\sigma} \: \partial_\sigma \: \eta_{\lambda\alpha} \: F^{\lambda\rho} \: \eta_{\beta\rho} = 0$$$$ Substituting $$\eta^{\phi\theta} = \text{diag} (+,-,-,-)$$: $$$$\partial_\sigma F^{\lambda\rho} = 0$$$$ Then I would keep raising and lowering and substituting until I get the preferred form, if ever. I assume I am missing the proper approach.

• How did you mange to lose the sums over $\mu$, $\alpha$ and $\beta$? Do you understand how $A_\mu \delta^\mu_\nu=A_\nu$? Nov 7, 2020 at 13:11

Multiplication by a $$g$$ tensor is what's usually meant by 'raising' or 'lowering' indices. Depending on how you structure your axioms, it is possible to take $$g^{\mu\alpha}F_{\alpha\beta} = F^\alpha_\beta$$ as a definition of what $$F$$ with an upper index means.
Generally, $$F$$ is defined as a covariant (bottom indices) tensor. This has various nice properties, chief among which being that it also makes $$F$$ a 2-form, which is a notion that is invariant under coordinate changes. The presence of a metric makes it possible to convert covariant objects to contravariant ones, so the top line could be equivalently written as
$$\frac{1}{\sqrt{-g}} \partial_\mu \left[\sqrt{-g} F^{\mu\nu} \right]$$ The only reason it is not written like that is to make the metric tensor dependence explicit, while leaving $$F_{\mu\nu}$$ in its 'natural' state.