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Consider a narrow bundle of rays between two surface elements $d{A}_1,d{A}_2$ separated by $s$ in a transparent, stationary Euclidean vacuum. Let the ray make angles ${\theta }_1,{\theta }_2$ to the respective surface normals. The receiving element subtends solid angle $d{\Omega }_1=\cos {\theta }_{2}d{A}_2/s^2$ at the emitter. By the definition of specific intensity, the beam power in frequency interval $d\nu$ isAt the receiving end the source subtends $d{\Omega }_2=\cos {\theta }_{1}d{A}_1/s^2$, so the identical power isIn the absence of absorption, emission or frequency shifts, conservation of energy equates these expressions:This vacuum conservation of specific intensity also follows directly from the radiative transfer equation $d{I}_{\nu }/ds=0$. The apparent solid angle of an unresolved source shrinks as $s^{-2}$, so its integrated observed radiative flux still follows the inverse-square law. Specific intensity and unresolved flux are different quantities.

The vacuum and frequency assumptions are essential: absorption and emission change specific intensity, and gravitational or cosmological redshift conserves $I_{\nu }/{\nu }^3$ along a ray instead of $I_{\nu }$ itself. A fixed telescope aperture observing an unresolved object also cannot infer constant total received power from intensity conservation.