Temporal Equivalence Principle: Dynamic Time & Emergent Light Speed

Author: Matthew Lukin Smawfield
Version: v0.10 (Jakarta)
Date: 18 Aug 2025
Updated: 7 Aug 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 $g_{\mu\nu}$ 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, $A(\phi) = \exp(\beta_A \phi/M_{\text{Pl}})$ is a universal conformal factor, and $B(\phi)$ encodes tiny, direction-dependent deformations of the light cone consistent with GW170817-class multi-messenger constraints ($|c_\gamma - c_g|/c \lesssim \text{few}\times10^{-15}$ 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.

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(β_Aφ/M_Pl). The theory preserves local Lorentz invariance while generating path-dependent synchronization holonomy $H_{\rm resid}$ 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.

---

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.18064365
Paper 7	TEP-RBH	The Soliton Wake: A Runaway Black Hole as a Gravitational Soliton	10.5281/zenodo.18059250
Paper 8	TEP-SLR	Global Time Echoes: Optical-Domain Consistency Test via Satellite Laser Ranging	10.5281/zenodo.18064581
Paper 9	TEP-EXP	What Do Precision Tests of General Relativity Actually Measure?	10.5281/zenodo.18109760
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.19102061
Paper 14	TEP-GNSS-MGEX	Global Time Echoes: MGEX Multi-GNSS Clock Replication, 2025–2026	10.5281/zenodo.20572727
Paper 15	TEP-EFA	Temporal Equivalence Principle: Temporal Shear in the Earth Flyby Anomaly	10.5281/zenodo.19454863
Paper 16	TEP-J0437	Synchronization Holonomy in Pulsar Scintillation	10.5281/zenodo.19454620
Paper 17	TEP-LLR	Lunar Laser Ranging and the Nordtvedt Effect	10.5281/zenodo.19446029
Paper 18	TEP-HC	Temporal Equivalence Principle: Native hi_class Conformal Implementation, Linear Perturbation Closure, and CMB Acoustic Peak Preservation	10.5281/zenodo.20682752
Paper 19	TEP-LENS	Temporal Equivalence Principle: A Blind-Prediction Residual Test in Multiply-Imaged Supernovae	https://github.com/matthewsmawfield/TEP-LENS
Paper 20	TEP-LHC	Temporal Equivalence Principle: A Sidereal Modulation Audit of LHC Luminosity Data	https://github.com/matthewsmawfield/TEP-LHC
Paper 21	TEP-NIST	Temporal Equivalence Principle: Spatial Variance of Laboratory Gravitational Constants as Local Temporal Topology	https://github.com/matthewsmawfield/TEP-NIST
Paper 22	TEP-LVK	Temporal Equivalence Principle: A Standard-Siren Test of Bi-Metric Gravitational-Wave Propagation	https://github.com/matthewsmawfield/TEP-LVK
Paper 23	TEP-QF	Temporal Equivalence Principle: The Dirac Limit of Dynamical Proper Time	https://github.com/matthewsmawfield/TEP-QF
Paper 24	TEP-SPIN	Temporal Equivalence Principle: A Topological Fermion Model for Spin and the g−2 Anomaly	https://github.com/matthewsmawfield/TEP-SPIN
Paper 25	TEP-KIN	Temporal Equivalence Principle: Disformal Kinematics and the Measurement Landscape	https://github.com/matthewsmawfield/TEP-KIN
Paper 26	TEP-C0	Temporal Equivalence Principle: A Covariant Alternative to Cosmic Expansion	10.5281/zenodo.20370143
Paper 27	TEP-TH	Temporal Equivalence Principle: Temporal Horizon Cosmology and the Absence of a Physical Big Bang Singularity	10.5281/zenodo.20723059
Paper 28	TEP-BH	Temporal Equivalence Principle: Black Holes and the Temporal Horizon	10.5281/zenodo.21677827
Paper 29	TEP-BBN	Temporal Equivalence Principle: Dynamical Proper Time and the Illusion of Primordial Deuterium	10.5281/zenodo.21841148

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}
}

---

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