zenodo.txt

This paper proposes a covariant, testable reformulation of relativity in which proper time is a dynamical field and the "speed of light" is an emergent, strictly local invariant rather than a global constant. The framework is built on a single spacetime manifold endowed with two metrics: a gravitational metric g{\mu\nu} and a causal (matter) metric \tilde{g}{\mu\nu} to which all non-gravitational fields and clocks couple. The metrics are related by a controlled disformal map, \tilde{g}{\mu\nu} = A^2(\phi) g{\mu\nu} + B(\phi) \nabla\mu\phi \nabla\nu\phi, where \phi is the time field, A(phi) = \exp(\betaA \phi/M{\text{Pl}}) is a universal conformal factor, and B(\phi) encodes tiny, direction-dependent deformations of the light cone consistent with GW170817-class multi-messenger constraints (|c\gamma - cg|/c \lesssim \text{few}\times10^{-15} today). Proper time is elevated to a field by postulating that all matter, electromagnetism, and quantum phases evolve with respect to \tilde{g}-proper time \tau; in local freely falling frames, this guarantees exact local Lorentz invariance and a locally invariant c, while globally it implies that synchronization procedures and one-way light-time measurements can become path-dependent in the disformal/non-exact sector of a dynamical-time background. The covariant action, field equations, conservation laws, Parametrized Post-Newtonian (PPN) mapping, and screening mechanisms are developed to reconcile terrestrial tests with cosmological dynamics. The breakdown of global simultaneity is formalized using a synchronization-transport law, deriving a convention-independent "synchronization holonomy," an invariant measure of non-integrability of time transport around closed loops. In the purely conformal subclass this holonomy vanishes after subtraction of the full GR, kinematic, clock-scale, and reference-frame synchronization model; nonzero holonomy at leading order requires residual non-exact synchronization structure, supplied in the minimal TEP model by disformal coupling B(\phi)\neq0, and in more general extensions by non-metricity or other explicitly non-exact transport structure. Explicit small-B formulas are provided for the holonomy and the effective photon phase speed, showing how the measured one-way asymmetry is related to \phi-gradients and disformal scales under current constraints. The analysis demonstrates that Einstein's assumption of a universal c was a brilliant local theorem arising from the Temporal Equivalence Principle; transcending it demands dynamical time: c remains exactly invariant locally, but global, one-way-inferred "c" values differ by path-dependent amounts that experiments can detect or bound. A staged, falsifiable experimental program is outlined: (1) a closed-loop, multi-leg, one-way time-transfer "triangle test" designed to detect synchronization holonomy at the 10^{-19} fractional level (after averaging) and subtracting known GR effects; (2) interplanetary one-way optical time transfer targeting picosecond-level asymmetries over AU baselines; (3) distance-correlation analysis and environment-dependent screening maps with precision clock networks; (4) multi-messenger searches for distance-correlated photon–gravitational-wave arrival differentials consistent with tightly bounded disformal propagation; (5) matter-wave interferometry and torsion-balance tests sensitive to environment-dependent couplings. Cosmologically, the time field can modify late-time distance and growth observables. Early-universe closure is achieved natively without a Big Bang singularity or phenomenological thermal screening: the framework establishes a regular temporal horizon with an unbounded local proper-time history, while the associated conformal temporal geometry preserves the observed CMB acoustic structure. Known weaknesses in the variable-c literature are addressed by supplying a correct, operationally invariant observable (holonomy), clarifying when conformal couplings cannot produce a signal, and providing realistic, constraint-consistent signal forecasts with explicit error budgets and statistical plans (pre-registration, blinding, publicly released code and data). The resulting theory preserves the empirical pillars of relativity (local Lorentz invariance, gravitational-wave causality, PPN bounds) while extending its conceptual foundation: simultaneity is not only relative but can become globally non-integrable; the speed of light is not a global constant but the local echo of a deeper, dynamical temporal geometry.

Long-standing confusions about "variable c" are resolved by replacing convention-dependent statements with invariant observables tied to measurement procedures. A synchronization one-form \tilde{\sigma} is defined on spacelike slices of the matter metric; its curl d\tilde{\sigma}, after subtraction of the full GR, kinematic, clock-scale, and reference-frame synchronization model, yields a residual "temporal holonomy" H that vanishes in GR and becomes nonzero only when time is dynamical in this sense. Two key theorems are proven: (i) conformal matter coupling preserves null cones, so photons and gravitons share the same causal structure at late times; (ii) a static \phi-gradient generates no first-order one-way light-time anisotropy, placing effects in the femto-to-picosecond regime over astronomical baselines under current bounds. Disformal tilts (B \neq 0) are tightly constrained by GW170817-class multi-messenger observations but can source holonomy at levels within reach of next-generation metrology. The covariant action is presented; field equations, conservation laws, invertibility/causality conditions, and a 3+1 decomposition are derived to make the observables explicit. Screening via a continuous Temporal Topology governed by non-linear superposition of field gradients (Temporal Shear) reconciles precision local tests with cosmological evolution, with mapping to Parametrized Post-Newtonian parameters and to the EFT-of-dark-energy \alpha-functions with c_T = 1 enforced. Decisive experiments with quantitative error budgets are outlined: (1) a ground–ground–satellite triangle time-transfer experiment targeting holonomy at below 10^{-18} fractional after GR subtraction; (2) portable-clock "clock anholonomy" around closed paths at the 10^{-19} level over days; (3) multi-species clock networks seeking phase-locked annual modulations at 10^{-19}–10^{-17}; (4) interplanetary one-way optical links at picoseconds over AU; (5) altitude-dependent screening maps with optical clocks and atom interferometers; and (6) ensemble multi-messenger tests. A cosmological pipeline plan for CLASS/HyRec modifications and MCMC inference is provided, with commitment to open data and blinded analyses.

Einstein's postulate of universal c was a brilliant, operationally perfect approximation in regimes where time's flow is effectively uniform. In a universe where the rate of time is dynamical yet locally Lorentzian, "the speed of light" emerges as an invariant in every local lab but ceases to be globally universal. The new invariant content resides in path-dependent synchronization defects and holonomies of time transport, not in naive one-way "speeds." If detected, these invariants would inaugurate a post-Einsteinian era: from dynamic geometry to dynamic time.

Keywords: temporal equivalence principle; dynamic time; emergent light speed; synchronization holonomy; conformal disformal coupling; theoretical physics; quantum gravity; cosmology; general relativity

Website: https://mlsmawfield.com/tep/theory/
Repository: https://github.com/matthewsmawfield/TEP
DOI: 10.5281/zenodo.16921911

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