Isomorphic Emergence of Lorentz and Gauge Symmetries--A Constructive Interpretation Based on Continuum Mechanics

Abstract

Lorentz symmetry and gauge symmetry constitute the mathematical cornerstones of modern physics, yet their ultimate physical origins remain elusive. From the standpoint of a constructive interpretation, this paper demonstrates that both symmetry structures can emerge isomorphically from a unified classical source: dynamical constraints on wave-packet excitations in a continuous elastic substrate medium (SM). Neither symmetry is posited as fundamental, nor does their emergence rely on quantization. Three core results are established. First, taking the transverse wave speed of a homogeneous, isotropic SM as the benchmark under a conventionalist synchronization scheme, Minkowski-type spacetime arises isomorphically, with Lorentz transformations as effective coordinate transformations between inertial frames. This reinterprets the Michelson--Morley null result: observers are composite wave-packet excitations, and medium-induced kinematic corrections are encoded in their measurement frameworks. Second, for wave packets endowed with $SU(N)$ intrinsic symmetry, the requirement of identity invariance drives the SM to spontaneously generate gauge fields. This requires invariant equivalence classes of intrinsic states under propagation, yielding gauge structures isomorphic to Yang-Mills theory. The gauge group $SU(N)$ is uniquely determined by the number of stable normal modes supported by the medium. Third, through the same conventionalist measurement protocols, inhomogeneous distributions of the SM emerge isomorphically as curved spacetime geometry, governed by Einstein-type field equations from variational extremization of the medium's deformation free energy. This paper advances a unified interpretive account of the physical origin underlying the mathematical structures of both Relativity and the Standard Model.