= Vacuum conservation of specific intensity
In a transparent, stationary Euclidean vacuum, the <radiative transfer equation> has zero absorption and emission, so $dI_\nu/ds=0$. Equivalently, two surface elements exchange beam power $I_\nu\cos\theta_1\cos\theta_2\,dA_1dA_2d\nu/s^2$; reversing the description of the same beam cancels its geometric factors. The observed <radiative flux> of an unresolved source still falls as its apparent <solid angle>, proportional to $s^{-2}$. Across gravitational or cosmological frequency shifts the invariant is $I_\nu/\nu^3$, not generally $I_\nu$.
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