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## Main Question or Discussion Point

Hi everyone,

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

Computing the christoffel symbols one would get, to first order in the perturbation: [tex]\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}),[/tex]

right?

Then why, in this reference, in the text right after Eq.19.23, [itex]C^\alpha_{\beta \gamma}=\frac{1}{2}(h^{\alpha}_{\beta;\gamma}+h^{ \alpha }_{\gamma;\beta}-h_{\beta \gamma}\hspace{.2mm}^{;\alpha})[/itex], is written with covariant derivatives?

Thank you

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

Computing the christoffel symbols one would get, to first order in the perturbation: [tex]\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}),[/tex]

right?

Then why, in this reference, in the text right after Eq.19.23, [itex]C^\alpha_{\beta \gamma}=\frac{1}{2}(h^{\alpha}_{\beta;\gamma}+h^{ \alpha }_{\gamma;\beta}-h_{\beta \gamma}\hspace{.2mm}^{;\alpha})[/itex], is written with covariant derivatives?

Thank you

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