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Derived Dynamics in Trembling Spacetime Relativity : A Unified Geometric Basis for the Principles of Inertia, Equivalence, Action, and Energy–Mass Relations
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Classical physics rests on inertia and the equivalence principle, with gravity described geometrically, while quantum theory adds measurement postulates, nonlocal correlations, and discrete charges. This paper shows how these structures follow from a single geometric premise: Trembling Spacetime Relativity (TSRT), in which the Lorentzian metric is written as a smooth background plus a bounded, deterministic trembling component that preserves causal order. Proper time along worldlines is strictly monotone, and particles are modeled as trembling eigenmodes whose paths extremize an effective proper-time functional. The trembling sector carries a universal action scale, established in earlier TSRT work on corpuscular radiation, atomic structure, and cosmology, which underlies the geometric origin of Planck’s law and the unified emergence of the energy–frequency and energy–mass relations within a single causal framework.
Within this geometry, the core structures of classical dynamics are recovered. Inertial response appears as a Machian overlap susceptibility between a localized trembling envelope and a long-wavelength cosmological background. Gravitational response is governed by the same overlap, making the equality of inertial and gravitational mass a derived consequence, with deviations suppressed by probe size and field gradients. The least-action principle emerges as the weak-field, slow-motion limit of curvature-stationary worldlines rather than as an independent axiom. A causal Noether structure then links internal trembling symmetries to emergent gauge interactions: the Abelian reduction yields electromagnetic potentials and the Lorentz force as metric projections, while admissible non-Abelian symmetries generate Yang--Mills-type connections. Spin--statistics and charge quantization follow from closed-cycle holonomy and large gauge transformations tied to the universal action scale.
Measurement, entanglement, the arrow of time, and the cosmology–laboratory bridge integrate consistently into the same geometric framework. Measurement outcomes arise from boundary reconditioning of the trembling geometry; interference loss and Bell-type correlations follow from shared metric fluctuations with strict no-signaling. An arrow of time emerges because cosmic expansion relaxes curvature and enlarges the set of admissible trembling configurations, producing entropy-like growth without probabilistic postulates. Long-wavelength trembling modes that modulate cosmic expansion also induce small directional anisotropies in inertial response, enabling joint cosmological and laboratory tests. TSRT thus replaces a collection of axioms with a single principle: causally trembling spacetime with monotone proper time, from which inertia, equivalence, least action, energy–mass relations, gauge structure, spin–statistics, charge quantization, measurement behavior, and time’s arrows follow.
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