Chronokinetic Thermodynamics: Thermal Velocity Variance and the Metric Proper-Time Budget

Standard pedagogical presentations of relativistic dynamics treat momentum, kinetic energy, and velocity through non-compact hyperbolic geometry, obscuring the compact symmetry of the underlying spacetime capacity. In this paper, we extend the circular velocity ledger, v² + T² = c²—where T ≡ c(dτ/dt) = c/γ represents the laboratory coordinate rate of proper-time progression—into relativistic dynamics and macroscopic electrodynamics. First, we prove the existence of an exact dynamical reciprocal invariant relating total energy and proper-time progression: E · T = m₀c³ This relation reframes kinetic energy not as an extrinsic additive quantity, but as the coordinate measure of an internal proper-time deficit: K = E - m₀c² = E(1 - T/c) As spatial velocity increases and taxes the invariant speed-of-light capacity, proper time is systematically depressed, creating an exact energetic balance between temporal deceleration and spatial momentum. Second, we bridge this microscopic kinematic capacity to macroscopic electrodynamics in dielectric media. By analyzing light propagation at phase velocity v_eff = c/n through Walter Gordon’s 1923 effective optical metric: g_μν(opt) = g_μν + (1 - 1/n²) u_μ u_ν we demonstrate that refractive deceleration in matter does not destroy speed-of-light invariance. Instead, polarization interactions redistribute the photon wave-packet budget into an effective temporal phase coupling: T_eff = c√(1 - 1/n²) From this metric coupling, the relativistic Fresnel drag coefficient k = 1 - 1/n² emerges as an elementary trigonometric consequence of the velocity circle, eliminating the need for multi-frame Lorentz transformation boosts. These results provide an intuitive, geometrically unified framework for teaching relativistic dynamics and moving media electrodynamics.