On the Observer-Frame Interpretation of Radial Coordinates in Gravitational Potentials

The discrepancy between galactic rotation curves and visible baryonic masspersists despite empirical scaling relations like the Radial Acceleration Relation(RAR) and Baryonic Tully-Fisher Relation (BTFR). We explore a phenomenologicalframework where this discrepancy arises from the geometric misinterpretationof observables. Inspired by Painlevé-Gullstrand coordinates, we model thevacuum as a radially infalling compliant medium that induces an apparent compressionof radial coordinates for distant observers, the "Mezzi effect". AssumingNewtonian dynamics govern an undistorted "true frame", we developed a discreteshell reconstruction method parameterized by a single universal complianceconstant, tested against photometric and kinematic data from 175 late typegalaxies in the SPARC database. This single parameter model yields universalscaling relations of Σtrue/Σobs∝(Rtrue/Robs)−0.5and Mobs/M. And reproduces observed rotation curves (RMS residual ∼ 34km/s). Thegeometric projection recovers the empirical RAR and shifts the BTFR slopefrom ∼ 2.8in the true frame to ∼ 3.7in the observer frame, and eliminatingthe normalization offset. Furthermore, The Mezzi scale factor ζ governsmass and lensing corrections via distinct power laws: Mtrue/Mtrue and αtrue/αobs∝ζ−1.26, revealing that geometric scaling affects dynamical massmore strongly than lensing mass. These results indicate that geometric projectioneffects may offer a viable phenomenological explanation for galactic dynamicswhile remaining consistent with both Newtonian gravity and weak field generalrelativity. For reproducibility, the code used for this analysis is publicly availableat https://github.com/Brahim-Benaissa/Zeta