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In Carroll's Spacetime and Geometry (§3.2) I found $$\nabla_\mu V^\mu =\partial_\mu V^\mu + \Gamma^\mu_{\mu\sigma}V^\sigma =\frac{1}{\sqrt{\left|g\right|}}\partial_\mu \left(\sqrt{\left|g\right|}V^\mu\right),$$ where $g\equiv\mathrm{det}\left(g_{\alpha\beta}\right)$. Using the definition, I tried to compute $$\Gamma^\mu_{\mu\sigma} = \frac{1}{2}g^{\mu\rho}\left(\partial_\mu g_{\sigma\rho}+\partial_\sigma g_{\rho\mu}-\partial_\rho g_{\mu\sigma}\right)=\frac{1}{2}\left(\partial_\sigma+g^{\mu\rho}\partial_\sigma g_{\rho\mu}-\partial_\sigma\right).$$ I don't know what I should do next. Carroll says that $$\Gamma^\mu_{\mu\sigma}=\frac{1}{\sqrt{\left|g\right|}}\partial_\sigma\sqrt{\left|g\right|},$$ but I can't see why.


marked as duplicate by Qmechanic Feb 1 at 12:56

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