Converting covariant objects into non-covariant

Question

I need to rewrite expressions of the type $(\partial_\nu A_\mu)(\partial^\mu A^\nu)$ from the "covariant" form, to non-covariant form (so with roman indices). Here the greek indices run from 0 to 3 and the roman indices from 1 to 3. My attempt at it looks like $$(\partial_0A_0)(\partial_0A_0)-(\partial_0A_j)(\partial_0A_j)-(\partial_iA_0)(\partial_iA_0)+(\partial_iA_j)(\partial_i A_j)$$

I am not sure if the signs are correct. Do "space indices" carry the minus even if the greek letter index is downstairs, so that everything turns out positive? An example is the expression $(\partial_0A_\mu)(\partial_0A^\mu)$. Would it be written non-covariantly as $(\partial_0A_0)(\partial_0A_0)+(\partial_0A_i)(\partial_0A_i)$ or with the minus sign as $(\partial_0A_0)(\partial_0A_0)-(\partial_0A_i)(\partial_0A_i)$?

The minus sign comes from the Minkowski metric $\eta=\text{diag}(+1,-1,-1,-1)$ (note the convention) which you are using to relate the "upper indices" to the "lower indices". An expression like $X_\mu Y^\mu$ evaluates as $X_0Y^0+X_1Y^1+X_2Y^2+X_3Y^3$ but this is equal to $X_0Y_0-X_1Y_1-X_2Y_2-X_3Y_3$ if you want to lower the indices. This is the difference between $X^\mu Y_\mu$ and $\eta^{\mu\nu}X_\mu Y_\nu$. — Charlie, Apr 28 '21 at 21:58
Thank you for your reply @Charlie. That point is clear. I edited my question, to make it more clear what exactly I'm asking. — Polarized photon, Apr 28 '21 at 22:03
Related : Squaring the E&M (Maxwell) field strength tensor. See equation (a) and its proof in my answer. — Frobenius, Apr 28 '21 at 22:17

score 1 · Accepted Answer · answered Apr 29 '21 at 00:04

Your attempt is correct. If one wants to compute $$ (\partial_0 A_\mu) (\partial_0 A^\mu) = \eta^{\mu \nu} (\partial_0 A_\mu) (\partial_0 A_\nu) $$ it is indeed \begin{align} \eta^{\mu \nu} (\partial_0 A_\mu)(\partial_0 A_\nu) &= \eta^{00} (\partial_0 A_0)(\partial_0 A_0) + \eta^{ij} (\partial_0 A_i)(\partial_0 A_j) \\ &= +1 (\partial_0 A_0)(\partial_0 A_0) - \sum_i (\partial_0 A_i)(\partial_0 A_i) \end{align} where one can remove the summation sign if they wish.

Converting covariant objects into non-covariant

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