Distance duality relation ($D_L=(1+z{)}^2{D}_A$)

In metric light propagation with photon conservation and the usual reciprocity assumptions, luminosity distance and angular diameter distance satisfy $D_L=(1+z{)}^2{D}_A$. Comparing received flux with apparent angular area then gives the bolometric cosmological surface-brightness dimming factor $(1+z{)}^{-4}$. Absorption, photon conversion or an inconsistent definition of emitted/observed bands must be treated separately.