 Solution

Let $q^i=e^i-\frac{1}{2}{h}^i{}_j{e}^j$, so $p^0=ϵ/a^2$ and $p^i=(ϵ/a^2)q^i$. The orthonormal tetrad gives ${\delta }_{ij}{e}^i{e}^j=1$. To first order,Hence the time-index Christoffel symbols contract toThe $\mathcal{H}{h}_{ij}$ term cancels the perturbation in the coordinate direction norm. This cancellation is why the local tetrad direction must not be mistaken for the coordinate direction.

Along the photon null geodesic, $d/d\lambda =p^{0}d/d\eta$, andThe time component of the geodesic equation now gives the photon energy redshift from a tensor metric perturbation:The derivative on the left follows the ray; the dot on $h$ is a partial conformal time derivative. The background redshift has already been removed by using comoving energy. The calculation only needs ${\Gamma }^0{}_{00}$ and ${\Gamma }^0{}_{ij}$; the unrelated printed ${\Gamma }^i{}_{j0}$ should contain $\frac{1}{2}{\dot{h}}^i{}_j$, rather than the full derivative.