Here is exercise 19.5

Note that the sign change depends on the Lorentzian signature of the metric because



but



so



i.e. the sign change is caused by the determinant of the metric.

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Note that the comment in the caption to figure 19.1 that

is a potential source of confusion here. Index reversal has nothing to do with it, and in fact this is wrong, since the reversal of 4 indices is an even permutation that does not cause a sign change anyway.

See my post in the "Errors, typos, and corrections" forum here.