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I'm reading Sean Carroll's Space Time and Geometry and haven't figure out how equation4.64is derived, where he is in the process of deriving Einstein's equation from Hilbert action.

Given there is a variation of the metric,

[itex]g_{\mu\nu} \rightarrow g_{\mu\nu} + \delta g_{\mu\nu}[/itex],

The corresponding variation of Christoffel connection is:

[itex]\delta\Gamma^{\sigma}_{\mu\nu} = -1/2[g_{\lambda\mu}\nabla_{\nu}(\delta g^{\lambda\sigma})+g_{\lambda\nu}\nabla_{\mu}(

\delta g^{\lambda\sigma}) - g_{\mu\alpha}g_{\nu\beta}\nabla^{\sigma}(\delta g^{\alpha\beta})] [/itex]

The first thing I don't understand is where the covariant derivatives come from. Because the Christoffel connection is defined through partial derivative of metric.

Can anyone tell me how to derive this equation explicitly? Thanks very much.

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# A question about variation of Christoffel connection

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