Simultaneous emergence of Lorentz symmetry, de Broglie quantum relationships, and gravitational behavior in a 1D classical non-relativistic, non-quantum, non-gravitational system

Abstract The non-relativistic, classical 1D continuum governed by the sine–Gordon Lagrangian density gives rise to a field equation admitting “breather” solutions which: can be Lorentz-boosted and still remain solutions; have the total mass/energy localized within a characteristic transverse diameter (proper scale ℓ₀ in Sec. 4); have an internal stationary oscillation frequency proportional to their total energy; have an amplitude-modulation spatial frequency proportional to their total linear momentum. Furthermore, if the mechanical index of refraction n(x) > 1 is allowed to vary gently while maintaining the impedance matching condition, the breather spatial diameter (same ℓ₀ notation) is re-scaled by a factor 1/√n(x); the breather internal stationary oscillation frequency is re-scaled by a factor 1/√n(x), and therefore, the internal unit of standard time is re-scaled by √n(x); the breather experiences a drift acceleration proportional to +(d/dx)n(x), which is independent of the mass/energy content of the breather itself. Property 1) is an implementation of the constructive special-relativity approach of Larmor, Lorentz, Poincaré, Selleri, Bell, Brown, etc., where the Lorentz–Minkowski symmetry is an emergent property of the solutions of the (non-relativistic) field equations and not a fundamental symmetry of the underlying mechanics. Properties 2)–4) are an implementation of the fundamental principles of the “Mécanique ondulatoire” of de Broglie, where the known relationships E = ℏω, p = ℏk emerge from understanding particles as dynamical concentrations of field energy and the diffraction/interference properties of matter are reinterpreted as classical effects of the evolution of an underlying continuous field defined in space and time instead of invoking causation from an abstract mathematical object (the wavefunction) as in Copenhagen/axiomatic quantum mechanics. Properties 5)–7) represent a mechanical derivation of fundamental principles of general relativity: gravitational potential as the space-time metric itself, gravitational length-contraction, gravitational time-dilation, free-fall of massive bodies in a gravitational field, and the principle of equivalence (here re-derived as a general opto-mechanical theorem). In essence, I show how to start from a mechanical continuum which is non-relativistic, non-quantum, and non-gravitational, and derive the principles of special relativity, quantum mechanics, and general relativity as emergent properties. Technical detail and supporting references are [8, 9]; broader programmatic context is [1, 2, 3, 4].