# Contraction of Christoffel symbol and metric tensor

How can I prove this contraction of Christoffel symbol with metric tensor? $$g^{k\ell} \Gamma^i_{\ \ k\ell} = \frac{-1}{\sqrt{|g|}}\frac{\partial\left(\sqrt{|g|}g^{ik}\right)}{\partial x^k}$$ I know the relation for the Christoffel symbol contracted with itself and this one is similar, but I cannot find the clue.

I start from the definition of gamma: $$g^{k\ell} \Gamma^i_{\ \ k\ell} = \frac{1}{2}g^{kl}g^{ij}(\partial_k g_{jl} + \partial_l g_{jk} - \partial_j g_{kl}) = \frac{1}{2}g^{ij}(2g^{kl}\partial_k g_{jl} - g^{kl}\partial_j g_{kl})$$ Now I can see that I can use the relation for derivative of det(g) in the second term in bracket, but don't know what to do with the first term.

• Hi! It'd be useful to see what calculations you've tried so far and where you've struggled. The method is almost identical to the other case physics.stackexchange.com/questions/309535/… Commented Feb 19, 2021 at 12:33
• Welcome to Physics Stack Exchange! Note that we use MathJax to typeset mathematics; you can find a good tutorial here. I've transcribed your image, but in future you should typeset it yourself. (As a good starting point, you can often copy-paste directly from Wikipedia and then trim the results.) Commented Feb 19, 2021 at 12:37
• That said, please take a minute to read our guidelines for homework and exercise questions as well as check-my-work questions. We intend our questions to be potentially useful to a broader set of users than just the one asking, and we prefer conceptual questions over those just asking for a specific computation. Commented Feb 19, 2021 at 12:38
• Work out the RHS first. Do you know how to take the derivative of a determinant? Commented Feb 19, 2021 at 13:05

The most important point about this computation is to use the formula for the derivative of the metric determinant $$\frac{\partial_i g}{g} = g^{jk} \partial_i g_{jk}$$ The derivation of this identity can be found in the answer to this question. You can then derive the relationship between $$g^{ij}{}_{,k}$$ and $$g_{ij,k}$$ by taking a derivative of $$\delta^i{}_{j} = g^{ik}g_{kj}$$. Finally, you take the formula for the Christoffel symbols in terms of metric derivatives and after some algebra you get the result!