 Vacuum conservation of specific intensity

In a transparent, stationary Euclidean vacuum, the radiative transfer equation has zero absorption and emission, so $d{I}_{\nu }/ds=0$. Equivalently, two surface elements exchange beam power $I_{\nu }\cos {\theta }_1\cos {\theta }_{2}d{A}_{1}d{A}_{2}d\nu /s^2$; 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 $I_{\nu }$.