= Synchronous photon momentum redshift
{title2=$q'=-\tfrac12q h'_{ij}n^in^j$}
For a photon in $ds^2=a^2[-d\tau^2+(\delta_{ij}+h_{ij})dx^idx^j]$, the <null geodesic> condition cancels the homogeneous $2\mathcal H p^0$ term in the derivative of $q=a^2p^0$. The remaining first-order <geodesic equation> gives the displayed momentum change. Its direction is constant at zeroth order, so first-order redshift and brightness sources can be evaluated on the unperturbed ray; direction deflection multiplied by an already first-order anisotropy is second order.
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