# Linearized Einstein Field Equations

PLuz
Hi everyone,

Say that one can separate the metric of a space time in a background metric and a small perturbation such that $g_{\alpha \beta}=g'_{\alpha \beta}+h_{\alpha \beta}$, where $g'_{\alpha \beta}$ is the background metric and $h_{\alpha \beta}$ the perturbation.

Computing the christoffel symbols one would get, to first order in the perturbation: $$\Gamma^\alpha_{\beta \gamma}=\Gamma'^\alpha_{\beta \gamma}+\frac{1}{2}(h^{\alpha}_{\beta,\gamma}+h^{ \alpha }_{\gamma,\beta}-h_{\beta \gamma}\hspace{.2mm}^{,\alpha}),$$

right?
Then why, in http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html reference, in the text right after Eq.19.23, $C^\alpha_{\beta \gamma}=\frac{1}{2}(h^{\alpha}_{\beta;\gamma}+h^{ \alpha }_{\gamma;\beta}-h_{\beta \gamma}\hspace{.2mm}^{;\alpha})$, is written with covariant derivatives?

Thank you

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