Contraction of Christoffel symbol and metric tensor

Question

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.

Answer 1

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!