This question is an exact duplicate of:

I am trying to understand the order of the indices when raising or lowering tensors.

For example, the electromagnetic tensor: $$F^{\alpha \beta} = \begin{bmatrix} 0 & -\frac{E_{x}}{c} & -\frac{E_{y}}{c} & -\frac{E_{z}}{c} \\ \frac{E_{x}}{c} & 0 & -B_{z} & B_{y} \\ \frac{E_{y}}{c} & B_{z} & 0 & -B_{x} \\ \frac{E_{z}}{c} & -B_{y} & B_{x} & 0 \\ \end{bmatrix} $$ When lowering the indices I have seen $$F_{\alpha \beta} = \eta_{\alpha \mu} \eta_{\beta v} F^{\mu v}$$ and $$F_{\alpha \beta} = \eta_{\alpha \mu} \eta_{v \beta} F^{\mu v}$$

Which is the right order of the indices on the second $\eta$? Does the index you want always go first or does it go in the index spot where you want it to end up after contraction?


marked as duplicate by Qmechanic May 6 at 18:18

This question was marked as an exact duplicate of an existing question.


The order of the indices on the Minkowski tensor doesn’t matter because it is symmetric. Personally, I prefer the first example where the contracted index is close to what it is contracting with.


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