Differences of Christoffel Symbols

Question

In textbooks on general relativity, it is stated that if a metric $g$ is given by the sum of a background metric $g_B$ and a perturbation $h$ ie. $g_{ij} = g_{Bij} + h_{ij}$, then the difference of the Christoffel symbols for the background metric and the perturbing piece can be written as a tensor which is equal (at first order perturbations) to

$$\frac{1}{2}g_{B}^{cd}(\partial_a h_{bd} + \partial_b h_{ad} - \partial_d h_{ab}) .$$

However, in some situations, the background metric is flat so that the Christoffel symbols for the background metric $g_B$ vanish.

Does this mean that the above expression gives the Christoffel symbols for the actual full perturbed metric $g$, since the difference is just a difference away from zero?

Answer 1

Yes, this expression gives the perturbed Christoffel symbols in flat coordinates. If you move to arbitrary coordinates, the quantity you gave transforms as a tensor, while the Christoffel symbols, both perturbed and unperturbed have a different transformation law. In arbitrary coordinates, the Christoffel symbols of the flat metric are not zero.