Dynamic Enforcement of Lorentz Invariance for Translating Wave Packets in a Linear Elastic Continuum

In a previous work, we demonstrated that gravitational lensing can be modeled as anisotropic refraction in a hyper-elastic continuum. A longstanding objection to medium-based models of the vacuum is the null result of interferometric experiments (e.g., Michelson–Morley), which are commonly interpreted as ruling out the existence of a preferred reference frame. We revisit this problem by modeling physical matter not as rigid bodies, but as self-stabilized standing-wave structures governed by the continuum’s linear wave equation. We show that, for such structures to maintain phase coherence while translating uniformly through the medium, they must undergo dynamic length contraction and time dilation. The Lorentz transformations are derived analytically from the classical wave equation, demonstrating that Lorentz invariance emerges as a stability condition of the matter–medium coupling rather than as a fundamental postulate of an otherwise structureless background. Consequently, a linear elastic vacuum is observationally indistinguishable from relativistic spacetime in the weak-field, linear regime.