The Evolving Faber-Jackson Relation: A Unifying Framework for Galaxy Ages and the Baryonic Tully-Fisher Connection

The baryonic Tully-Fisher relation (BTFR) and Faber-Jackson relation (FJR) represent fundamental scaling laws linking the baryonic mass of galaxies to their kinematics, yet their physical origin and apparent offsets between different galaxy populations have remained enigmatic. Here we present a unified theoretical framework demonstrating that both relations emerge from a common acceleration scale of order $10^{-10}m/s^2$ and evolve with cosmic time through a common exponential kernel. We derive the evolving FJR directly from the evolving BTFR within the Nexus Paradigm of quantum gravity, showing that the normalization scales as $M_b \propto e^{-4\int H(t)\,dt}\sigma^4$, where $\sigma$ is the velocity dispersion and $H(t)$ is the time varying Hubble parameter. Using this framework on a sample of 39 galaxies ranging from ultra-faint dwarfs to massive cluster ellipticals, we show that the observed offset between galaxy populations arises naturally from differences in their formation epochs. Ultrafaint dwarf galaxies yield ages of $12\pm0.8$ Gyr, in excellent agreement with independent Hubble Space Telescope stellar population ages showing synchronization within $\sim 1$ Gyr. Later-type dwarfs show systematically younger ages of $3.5-6.0$ Gyr. The dynamical age reported by the evolving FJR measures the time since a galaxy last achieved virial equilibrium. Dark matter-deficient galaxies exhibit ancient stellar populations but very young dynamical ages, consistent with a recent violent collision. Independent validation using metallicity-based stellar population ages reveals a Pearson correlation coefficient of $r=0.961$ with our dynamically derived ages for undisturbed systems, providing strong empirical support for the framework. The evolving FJR links pressure-supported systems at every mass scale, making galaxy scaling relations accurate markers of both formation and disturbance events.