Directional Density of Gases Under FitzGerald–Lorentz Contraction: Statistical Relaxation and Predictions for Optical Interferometry

In Lorentz Ether Theory, FitzGerald–Lorentz contraction would normally drive a gas toward the same effective contracted equilibrium as its solid container, restoring SR/LET equivalence. For an unbound gas, however, this equilibration need not be instantaneous. A single molecule–molecule collision should not be assumed to be the ideal collision of two perfectly contracted geometric ellipsoids. At the molecular scale the collision is a stochastic scattering event involving electronic structure, internal states, impact parameters, and quantum uncertainty. The contracted-ellipsoid picture may therefore emerge only as a small statistical bias after a very large number of collisions. We propose that the relevant relaxation is a statistical collisional relaxation. If the effective directional bias per collision is χ, then the number of independent collisions required for the contracted directional statistics to emerge is of order Nₛtat ~ χ⁻². With a molecular collision frequency νcoll, this gives τₛtat ~ 1/ (χ² νcoll). For a weak bias χ ~ 10⁻⁶ and a typical atmospheric collision frequency νcoll ~ 10¹⁰ s⁻¹, the relaxation time is of order 10² s. This is long compared with a continuously rotating apparatus with periods of seconds to tens of seconds. In that regime the gas cannot fully track the changing FitzGerald direction, whereas the solid container and optical reference components track it essentially instantaneously. This mismatch creates an optical signal. In the full-mismatch limit the predicted fringe shift is ΔNfull = k (L/2λ) (n − 1) (v/c) ², where k is the number of same-sign passes. More generally the observed amplitude is multiplied by a relaxation factor depending on 2Ω·τₛtat, where Ω is the mechanical rotation rate. The model predicts a second-harmonic signal, null results for solid dielectrics, and a characteristic dependence on gas pressure and rotation frequency.