Write $h_{ab}=\delta g_{ab}$ and $h=g^{ab}h_{ab}$. Standard <metric variation> gives
$$
\boxed{\delta\sqrt{-g}=\frac12\sqrt{-g}\,h},
\qquad
\boxed{\delta g^{ab}=-h^{ab}},
$$
and variation of metric compatibility gives
$$
\boxed{\delta\Gamma^a{}_{bc}
=\frac12g^{ad}
(\nabla_bh_{cd}+\nabla_ch_{bd}-\nabla_dh_{bc})}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 309 / 4 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-309/4/a/i/solution