Deriving Hamiltonian Dynamics and Lorentz Covariance from the Propagation of Defects in a Linear Medium

Special Relativity is conventionally founded upon geometric postulates regarding the structure of spacetime. This work investigates whether these kinematic laws can instead be derived as theorems of a fundamental wave mechanics. Modeling a fundamental particle as a spatially localized, oscillating defect (a standing wave) within a linear elastic medium with finite propagation speed c, we employ the Hamilton-Jacobi formalism to analyze its propagation. We demonstrate that the requirement of phase stability—the condition that the defect’s internal vibration remains coherent while in motion—imposes a specific transformation law on the coordinate system. We prove that this dynamic constraint is mathematically equivalent to the Lorentz transformations. Consequently, the Minkowski metric emerges not as an a priori geometry, but as the effective acoustic metric required to describe wave propagation in the medium. Finally, we recover the relativistic action S = −mc^2τ as a direct count of the defect’s internal cycles, unifying the de Broglie hypothesis with classical Hamiltonian dynamics. This framework suggests that relativity is the natural hydrodynamic consequence of matter’s wave nature.