general relativity - Raising and Lowering Indices Question

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.

Answer

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 $g$.