Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed

Author: Matthew Lukin Smawfield
Version: v0.8 (Jakarta)
Date: 18 Aug 2025
Updated: 29 Apr 2026
Status: Preprint
DOI: 10.5281/zenodo.16921911
Website: https://mlsmawfield.com/tep/theory/

Abstract

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_μν and a causal (matter) metric g̃_μν to which all non-gravitational fields and clocks couple. The metrics are related by a controlled disformal map, g̃_μν = A²(φ) g_μν + B(φ) ∇_μφ ∇_νφ, where φ is the time field, A(φ) = exp(β φ/M_Pl) is a universal conformal factor, and B(φ) encodes tiny, direction-dependent deformations of the light cone consistent with multi-messenger constraints (|c_γ − c_g|/c ≲ 10^-15 today). Proper time is elevated to a field by postulating that all matter, electromagnetism, and quantum phases evolve with respect to g̃-proper time τ; 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 become path-dependent in a dynamical-time background. The full 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 purely conformal theories this holonomy vanishes once general-relativistic Sagnac and Shapiro terms are removed; a nonzero holonomy at leading order requires non-exact structure provided either by (i) disformal photon coupling B(φ) ≠ 0 or (ii) more general non-metricity. 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 φ-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 decisive, near-term 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 shift the sound horizon and late-time expansion, potentially easing H_0 and S_8 tensions while preserving Big Bang Nucleosynthesis (BBN) and CMB acoustic physics. 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 generally 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 σ̃ is defined on spacelike slices of the matter metric; its curl dσ̃, after subtraction of general-relativistic (GR) Sagnac/gravito-magnetic terms, yields a residual "synchronization holonomy" $H_{\text{resid}}$ 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 φ-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 ≠ 0) are tightly constrained by GW170817-class multi-messenger observations but can source holonomy at levels within reach of next-generation metrology. A covariant action is presented, field equations, conservation, and invertibility/causality conditions are derived, and a 3+1 decomposition is supplied that makes observables explicit. Screening via Temporal Topology (governed by Temporal Shear $\Sigma_\mu = \nabla_\mu \ln A$) reconciles precision local tests with cosmological dynamics; a controlled disformal sector allows minute, bounded photon-cone tilts that provide clean targets for holonomy experiments. The theory is falsifiable with realistic experiments and promises to clarify persistent cosmological tensions.

Einstein moved physics from absolute time to relative simultaneity and dynamic geometry. The next step is to recognize that the flow of time itself is dynamical, and that "the speed of light" is the local echo of a deeper temporal geometry. If the predicted holonomies are observed, physics will enter a new epoch in which dynamic time joins dynamic geometry as a foundation. If not, uniquely strong bounds will have been set and the operational bedrock of c and simultaneity clarified to unprecedented precision.

Key Findings

This paper develops a covariant bi-metric framework where proper time is a dynamical scalar field φ coupled conformally to matter via A(φ) = exp(βφ/M_Pl). The theory preserves local Lorentz invariance while generating path-dependent synchronization holonomy $H_{\text{resid}}$ that vanishes in GR but becomes nonzero when time is dynamical. Key predictions include: (1) distance-structured correlations in precision timing networks with λ ~ 1,000–10,000 km, (2) screened local physics via Temporal Topology governed by Temporal Shear $\Sigma_\mu = \nabla_\mu \ln A$, and (3) falsifiable holonomy signatures in next-generation metrology. The disformal sector is constrained by GW170817 (|c_γ − c_g|/c ≲ 10⁻¹⁵), while the conformal sector remains unconstrained—providing a window for TEP phenomenology.

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The TEP Research Program

Paper	Repository	Title	DOI
Paper 0	TEP (This repo)	Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed	10.5281/zenodo.16921911
Paper 1	TEP-GNSS	Global Time Echoes: Distance-Structured Correlations in GNSS Clocks	10.5281/zenodo.17127229
Paper 2	TEP-GNSS-II	Global Time Echoes: 25-Year Analysis of CODE Precise Clock Products	10.5281/zenodo.17517141
Paper 3	TEP-GNSS-RINEX	Global Time Echoes: Raw RINEX Validation of Distance-Structured Correlations in GNSS Clocks	10.5281/zenodo.17860166
Paper 4	TEP-GL	Temporal-Spatial Coupling in Gravitational Lensing: A Reinterpretation of Dark Matter Observations	10.5281/zenodo.17982540
Paper 5	TEP-GTE	Global Time Echoes: Empirical Validation of the Temporal Equivalence Principle	10.5281/zenodo.18004832
Paper 6	TEP-UCD	Universal Critical Density: Unifying Atomic, Galactic, and Compact Object Scales	10.5281/zenodo.18064366
Paper 7	TEP-RBH	The Soliton Wake: A Runaway Black Hole as a Gravitational Soliton	10.5281/zenodo.18059251
Paper 8	TEP-SLR	Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging	10.5281/zenodo.18064582
Paper 9	TEP-EXP	What Do Precision Tests of General Relativity Actually Measure?	10.5281/zenodo.18109761
Paper 10	TEP-COS	The Temporal Equivalence Principle: Suppressed Density Scaling in Globular Cluster Pulsars	10.5281/zenodo.18165798
Paper 11	TEP-H0	The Cepheid Bias: Resolving the Hubble Tension	10.5281/zenodo.18209702
Paper 12	TEP-JWST	The Temporal Equivalence Principle: A Unified Resolution to the JWST High-Redshift Anomalies	10.5281/zenodo.19000827
Paper 13	TEP-WB	The Temporal Equivalence Principle: Temporal Shear Recovery in Gaia DR3 Wide Binaries	10.5281/zenodo.19102062

How to cite

• Main DOI (always latest version): 10.5281/zenodo.16921911

@misc{Smawfield_TEP_2025,
  author       = {Matthew Lukin Smawfield},
  title        = {The Temporal Equivalence Principle: Dynamic Time, Emergent Light Speed},
  year         = {2025},
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.16921911},
  url          = {https://doi.org/10.5281/zenodo.16921911},
  note         = {Preprint}
}

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Contact: matthew@mlsmawfield.com
Website: https://mlsmawfield.com/tep/theory/
Zenodo: https://doi.org/10.5281/zenodo.16921911
ORCID: 0009-0003-8219-3159