For a spacetime that is nearly flat, the metric $g_{a b}$ can be expressed in the form

$$g_{a b}=\eta_{a b}+h_{a b}$$

where $\eta_{a b}$ is a flat metric (not necessarily diagonal) with constant components, and the components of $h_{a b}$ and their derivatives are small. Show that

$$2 R_{b d} \approx h_{d}^{a}, b a+h_{b}^{a}, d a-h_{a, b d}^{a}-h_{b d, a c} \eta^{a c}$$

where indices are raised and lowered using $\eta_{a b}$.

[You may assume that $\left.R_{b c d}^{a}=\Gamma_{b d, c}^{a}-\Gamma_{b c, d}^{a}+\Gamma_{c e}^{a} \Gamma_{d b}^{e}-\Gamma_{d e}^{a} \Gamma_{c b}^{e} .\right]$

For the line element

$$d s^{2}=2 d u d v+d x^{2}+d y^{2}+H(u, x, y) d u^{2},$$

where $H$ and its derivatives are small, show that the linearised vacuum field equations reduce to $\nabla^{2} H=0$, where $\nabla^{2}$ is the two-dimensional Laplacian operator in $x$ and $y$.