I know that the Kronecker delta is used to raise and lower indices, but I am not certain if the order in which it is placed in an equation matters, what I mean is, for example:

Are the following two any different?

$$\delta^\mu _\nu x_\mu\hspace{5mm} and \hspace{5mm}x_\mu \delta^\mu _\nu $$

I know the first one would give me $x_\nu$ but I don't know if changing the order they are written in would give me the same answer.

Would there be any difference in a situation in which instead of using a Kronecker delta, a metric tensor, $\eta_{\mu\nu}$, was used?


No, the order doesn't matter. In general, things with indices are just components of tensors, so they are just numbers, so they commute (as far as the indices are of concern)

  • $\begingroup$ In addition, note that if the metric tensor is $\mathbf{\eta}$, then $\eta^\mu{}_\nu = \delta^\mu{}_\nu$: what you are using is the metric tensor in fact. $\endgroup$
    – user107153
    Apr 8 '20 at 13:53

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