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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?

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marked as duplicate by Qmechanic May 6 at 18:18

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

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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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