Linearized Levi-Civita connection
= Linearized Levi-Civita connection
About a Cartesian flat background,
$$
\Gamma^\alpha_{\beta\gamma}
=\frac12\eta^{\alpha\rho}
(\partial_\beta h_{\gamma\rho}+\partial_\gamma h_{\beta\rho}-\partial_\rho h_{\beta\gamma})+O(h^2).
$$