= Photon energy redshift from a vector metric perturbation
{title2=$\frac{d\ln\epsilon}{d\eta}=-e^i\dot B_i$}
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 $p^0=(\epsilon/a^2)(1+B_ie^i)$ and $p^i=(\epsilon/a^2)e^i$, so the covariant component is $p_0=-\epsilon+O(B^2)$. The covariant <geodesic equation> gives $dp_0/d\eta=(2p^0)^{-1}\partial_\eta g_{\alpha\beta}p^\alpha p^\beta=\epsilon e^i\dot B_i$. The <derivative> of the <scale factor> cancels by the <null vector> condition, proving the redshift formula.
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