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Given

$$\Gamma ^l_{ik}A^k_l$$

I want to lower/raise index $l$, I can insert $\delta ^m_m=g_{ml}g^{ml}$

Q1, is $\delta ^m_m=g_{ml}g^{ml}=4$?

Q2, after inserting, it becomes:

$$\Gamma ^l_{ik}A^k_l$$ $$=\Gamma ^l_{ik}g_{ml}g^{ml}A^k_l/4$$ $$=\Gamma _{mik}A^{km}/4$$

right? I guess I made some mistakes here. The result should not be divided by 4.

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  • $\begingroup$ Answer to question 1 is yes in four dimensions. Concerning the second question, is $\Gamma^l_{ik}$ Christoffel symbol? $\endgroup$ – Apogee Oct 16 '16 at 15:29
  • $\begingroup$ Q1: You are right, $\delta^m_m = 4$, in four dimensions, assuming the index $m$ runs over all four dimensions. This should be enough to tell you that you can't just insert $\delta_m^m$ into the middle of an expression --- this amounts to multiplying it by 4, which in general changes the thing you've got! As a further point, note that in the second line of Q2, you have four instances of the dummy index $l$, which is illegal! $\endgroup$ – gj255 Oct 16 '16 at 15:32
  • $\begingroup$ Thank you! Yes, $Γ^l_{ik}$ Christoffel symbol. Can you give me a hint how to proof $\Gamma ^l_{ik}A^k_l=\Gamma _{mik}A^{km}$? Or are they equal? $\endgroup$ – HYW Oct 16 '16 at 15:38
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When you raise/lower indices using the metric, you are not simply "multiplying by 1," i.e. inserting $(1/4){\delta_m}^m$. For instance, if you start with a vector field $X^\mu$, then lowering the index with the metric turns it into a one-form $X_\mu = g_{\mu\nu}X^\nu$. These objects are not the same, so you cannot get from one to the other just by multiplying by 1.

In your question, you want to both raise and lower a given index to go from $${\Gamma^k}_{ij} {A_k}^j$$ to $$\Gamma_{kij}A^{kj}.$$ To do this, we apply the rule for raising and lowering $k$ one step at a time: $$\Gamma_{kij} = g_{kl}{\Gamma^l}_{ij},$$ and $$A^{kj} = g^{km}{A_m}^j.$$ Combining these gives $$\Gamma_{kij}A^{kj} = g_{kl}g^{km}{\Gamma^l}_{ij}{A_m}^j = {\delta_l}^m {\Gamma^l}_{ij}{A_m}^j = {\Gamma^l}_{ij}{A_l}^j.$$ The key step is to not use the same letter to label the dummy indices in the two contractions with $g$.

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