homework and exercises - Contracted Christoffel symbol in BSSN formulation

Can someone kindly please help and explain how can obtain this equation?

$$ \Gamma^i =\gamma^{jk} \Gamma^i_{jk}= - \frac{1}{\sqrt{|\gamma}|}\partial_l \Big(\sqrt{|\gamma|}~ \gamma^{il} \Big) $$

where $\Gamma^i = ~^{(3)}\Gamma^i$ are the 3D Christoffel symbols and $\gamma_{ij}$ is the spatial metric tensor. The source is the book "Numerical Relativity" (Baumgarte, Shapiro), Eq. 4.40. (in this book the equation uses the 4D metric and the 4D Christoffel symbols, but in the Baumgarte video lectures he uses the above equation.) Here BSSN is the Baumgarte-Shapiro-Shibata-Nakamura formulation.

Thanks

Answer

I will assume that $\gamma_{ij}$ has signature $(1,1,1)$, so that we don't need the absolute value for $\gamma$ in your expression. To calculate the RHS you need to use the product rule and Jacobi's formula for calculating the derivative of a determinant $$ \partial_l \gamma = \gamma \, \gamma^{ij} \, \partial_l \gamma_{ij} .$$ Using this you should get $$ - \frac{1}{\sqrt{\gamma}}\partial_l \Big(\sqrt{\gamma}\, \gamma^{il} \Big) = \, ... \, = - \frac{1}{\sqrt{\gamma}} \big(\sqrt{\gamma} \, \partial_l \gamma^{il} + \gamma^{il} \frac{1}{2} \sqrt{\gamma} \gamma^{jk} \partial_l \gamma_{jk} \big) .$$

Now you need to calculate the derivative of the inverse of the metric, you can do this using $\gamma^{ij} \gamma_{jk} = \delta^i_k.$ This should get you to $$ - \frac{1}{\sqrt{\gamma}}\partial_l \Big(\sqrt{\gamma}\, \gamma^{il} \Big) = \big( \gamma^{ij} \gamma^{kl} \partial_l \gamma_{jk} - \frac{1}{2} \gamma^{il} \gamma^{jk} \partial_l \gamma_{jk} \big) .$$ If you now calculate the LHS using the definition and the symmetry of the Christoffel symbols you should get the desired equality (be aware of matching the dummy indices).