Blackbody Spectrum Preservation Under Frequency-Independent Redshift: An Explicit Geodesic Transport Verification

The preservation of a Planck blackbody spectrum under cosmological redshift is commonly taken as evidence for metric expansion. In previous work, this preservation was shown to follow directly from Liouville invariance under collisionless photon transport, provided the redshift is frequency independent. However, that argument relies on the compact form of Liouville's theorem and does not explicitly display the underlying geodesic transport steps. In this note, an explicit derivation is given using geodesic photon transport and phase-space conservation alone. Starting from the collisionless Boltzmann equation, a uniform redshift mapping p → p/R(t) is introduced as a kinematic assumption, without invoking any spacetime dynamics. The transport equation is solved by characteristics, yielding f (p, t) = f 0 (pR). For an initial Planck distribution, this mapping preserves the blackbody form exactly, with temperature scaling as T (t) = T 0 /R(t). The photon number density and energy density are then derived explicitly, scaling as n γ ∝ R −3 and ρ γ ∝ R −4 , respectively, and no chemical potential is generated. The result provides a step-by-step geodesic verification of blackbody spectrum preservation under frequency-independent redshift, clarifying that this property follows from transport kinematics rather than from metric expansion itself.