Photon energy redshift from a vector metric perturbation ($\frac{d\ln ϵ}{d\eta }=-e^i{\dot{B}}_i$)

For the metric in vector cosmological perturbation, an orthonormal tetrad is $E_0=a^{-1}{\partial }_{\eta }$, $E_i=a^{-1}(B_i{\partial }_{\eta }+{\partial }_i)$ to first order. A photon has $p^0=(ϵ/a^2)(1+B_i{e}^i)$ and $p^i=(ϵ/a^2)e^i$, so the covariant component is $p_0=-ϵ+O(B^2)$. The covariant geodesic equation gives $d{p}_0/d\eta =(2p^0{)}^{-1}{\partial }_{\eta }{g}_{\alpha \beta }{p}^{\alpha }{p}^{\beta }=ϵe^i{\dot{B}}_i$. The derivative of the scale factor cancels by the null vector condition, proving the redshift formula.