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To my knowledge, 3 or more indices may not appear in a given term, as I've found in a video produced by "Faculty of Khan": enter image description here

However, on page 30, Sean Carroll writes:enter image description here

As obvious, the indices 0 and 1 are repeated 3 times in the RHS of the underlined equation.

I am aware that Carroll keeps an errata for Spacetime and Geometry but I was not able to find information regarding this, so I'm not sure if there is some implied meaning that I'm missing.

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There is nothing wrong with what Carroll wrote, which is why it is not in his errata.

Khan is talking about symbolic contracted indices, which must occur in pairs. You sum over their possible values (typically 0, 1, 2, and 3). In Carroll’s equations the indices already have explicit values and are not being contracted. It doesn’t make sense to “contract over 0 and 1” because you can’t assign values to them.

Carroll’s equation follows from a correct double contraction,

$$F^{\mu\nu}=\eta^{\mu\alpha}\eta^{\nu\beta}F_{\alpha\beta},$$

when you set $\mu=0$ and $\nu=1$, write out the 16-term double sum over the contracted indices $\alpha$ and $\beta$, and use the fact that all the off-diagonal elements of the Minkowski metric and its inverse vanish. After doing this a few times, you can do it in your head.

It would be instructive to understand why contracting a tensor over a pair of indices produces another tensor, but that is beyond the scope of this particular question.

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  • $\begingroup$ Thanks for your answer sir! I did indeed misinterpret 0 as a varying index, rather than referring to a specific entry. With the method that you outlined I was indeed able to verify the equality. (I can't wait to be able to do this in my head as writing out the terms took almost an hour) $\endgroup$ – Ken Wang Jul 14 at 5:10

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