A scalar-tensor framework for dynamical proper time, clock-network correlations, and synchronization holonomy
The Temporal Equivalence Principle proposes that proper time is a dynamical field. Its central claim is not that local relativity fails, but that proper-time accumulation and synchronization possess global structure that standard local, reciprocity-even tests are not designed to measure.
In TEP, matter and clocks couple to a causal metric governed by a scalar time field. The spatial and covariance structure of this field is called Temporal Topology; its locally active gradients are Temporal Shear. Local Lorentz invariance and locally measured c are preserved. The new observables are global: distance-structured clock correlations, environment-dependent rate effects, and residual synchronization holonomy around closed loops.
GR already predicts that clocks tick differently in different gravitational potentials. TEP asks whether, after standard GR redshift, Sagnac, Shapiro, Doppler, station-motion, and reference-frame effects are modeled and subtracted, reproducible residual structure remains in clock-rate covariance, Temporal Shear, or closed-loop synchronization.
The primary empirical programme tests this structure using global timing networks. GNSS analyses report distance-structured correlations in multi-center clock products, a 25-year CODE baseline, and raw RINEX-derived observables; satellite laser ranging provides an optical-domain consistency test. Astrophysical papers then test whether the same temporal-field structure organizes lensing, pulsars, Cepheids, wide binaries, and high-redshift galaxies.
All manuscripts, analysis code, derived outputs, and reproducibility instructions are released for open review. The next decisive steps are independent reproduction of the timing-network correlations, technology-independent clock and ranging tests, and direct closed-loop measurements of residual synchronization holonomy.
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These three concepts define the operational core of TEP. Temporal Topology describes the clock-rate landscape, Temporal Shear describes its local gradient / slope, and Synchronization Holonomy gives the cleanest closed-loop experimental discriminator.
Operationally, these are distinct observables: GNSS-style clock networks primarily probe covariance structure CΘ, local fifth-force or rate-sector tests probe the active gradient Σμ, and closed-loop timing tests probe non-exact synchronization transport.
Using the compact notation Θ = ln A:
Matter and clocks couple to the causal matter metric:
The landscape picture is only an analogy; the formal objects are ln A, CA, and g̃μν.
Using Θ ≡ ln A:
In screened or effectively saturated regimes, the observable shear/source-charge sector is suppressed:
with 𝒮Σ(ℰ) → 0 in locally screened regimes, where ℰ denotes environmental state.
Equivalently:
Foundational framework, primary timing-network evidence, synthesis and scaling tests, candidate astrophysical applications, and direct holonomy and solar-system stress tests.
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 gravit...
Most high-precision tests of general relativity constrain reciprocity-even, largely local observables within single-metric frameworks. This leaves open a specific underdetermination between General Relativity (GR) and a class of two-metric disformal scalar-tensor modifications, exemplified here b...
Phase-coherent spectral analysis of 62.7 million station-pair measurements from 364 GNSS stations (2023–2025) reveals systematic distance-structured correlations in clock networks. These correlations follow an exponential decay with a median Temporal Topology correlation length λT = 3,...
Analysis of 25.3 years of global GNSS timing data (165.2 million station pairs) documents persistent velocity-dependent correlations in atomic clock networks. Critically, we propose that standard GNSS processing algorithms, designed to remove energetic (common-mode) errors via datum constraints, ...
This paper validates that distance-structured correlations in GNSS clocks exist in raw observations using broadcast ephemerides, not just precise products—strongly constraining precise-product processing artifacts. Broadcast ephemerides still contain control-segment information, so Satellite Lase...
An independent, optical-domain test of the Temporal Equivalence Principle (TEP) is presented using 11 years (2015–2025) of Satellite Laser Ranging (SLR) data from passive ILRS geodetic satellites (LAGEOS-1/2 and Etalon-1/2). This analysis constrains "clock-artifact" explanations by employing two-...
The Temporal Equivalence Principle (TEP) predicts correlated phase-coherent disturbances in GNSS timekeeping with a spatial correlation length of order thousands of kilometres, an east–west anisotropy exceeding the north–south counterpart, coupling to Earth orbital velocity, and a preferred axis ...
Visualize the reported distance-dependent coherence pattern and compare it with the null expectation used in the analysis. Explore the correlation decay directly in the browser.
Analysis of 25.3 years of GNSS timing data (2000–2025) reveals a persistent, distance-structured correlation in global atomic clock networks that tests an empirically untested assumption of general relativity: the global integrability of proper time. Examination of 165.2 million station pairs fro...
Dark matter observations across cosmological scales exhibit a regularity: the characteristic radius at which Newtonian dynamics fails scales as R ∝ M^(1/3), implying a universal critical density ρ_T. This scaling appears in galaxy rotation curves (SPARC database), ultra-diffuse galaxies (DF2/DF4)...
Standard gravitational lensing analysis relies on the *Isochrony Axiom*—the implicit assumption that the observed image represents a synchronous spatial snapshot of the source. For evolving sources, this approximation breaks down in the presence of conformal metric couplings, creating a "temporal...
Gravitational time dilation in General Relativity is verified to 10⁻⁵ precision in the Solar System. At intermediate astrophysical scales, however, persistent anomalies emerge—rotation curves, cluster dynamics, cosmic acceleration—that conventionally require invisible matter or exotic energy. The...
The Hubble Tension—the persistent 5σ discrepancy between local distance-ladder measurements (H₀ ≈ 73 km/s/Mpc) and early-universe CMB inference (H₀ = 67.4 ± 0.5 km/s/Mpc)—represents a significant challenge in precision cosmology. This study proposes that the tension arises from a systematic, envi...
JWST high-redshift anomalies—excessive star formation efficiencies and anomalous mass ratios—appear preferentially in deep gravitational potentials. This paper tests whether that common pattern arises from a single isochrony-axiom violation under TEP.
The Gaia DR3 wide-binary anomaly is tested as Temporal Shear recovery in weak-field environments. From 341,315 systems, a screening transition at Rs = 2,646 AU is identified, with environmental ordering consistent with TEP.
The runaway supermassive black hole RBH-1 ($z \approx 0.96$) presents a thermal paradox: JWST spectroscopy reveals a 650 km/s velocity discontinuity coexisting with cold, star-forming gas. Higher-resolution Keck/LRIS spectroscopy yields a narrow apex dispersion ($\sigma \approx 31 \pm 4$ km/s), f...
SN Refsdal provides a rare blind-prediction test of potential-dependent temporal propagation: seven GR lens-modelling teams predicted the SX reappearance delay before the image was observed, while the later Kelly et al. (2023) measurement provides an independent comparator. This single-system pro...
The May 2026 release of the LVK GWTC-5.0 catalog expands the public gravitational-wave standard-siren sample. The ongoing Hubble Tension between local distance ladder measurements (H_0 ≈ 73 km/s/Mpc) and early-universe CMB data (H_0 ≈ 67.4 km/s/Mpc) remains unresolved within standard ΛCDM cosmolo...
Twelve Earth gravity assist flybys spanning nine spacecraft are analyzed within the Temporal Equivalence Principle (TEP) framework. TEP posits that global simultaneity is inherently non-integrable, with the rate of time represented as a dynamical scalar field φ. All non-gravitational matter coupl...
Standard scintillation theory treats each scattered ray as carrying a scalar delay, so differential delays around a closed triplet cancel identically: \(\tau_{ij}+\tau_{jk}+\tau_{ki}=0\). The Temporal Equivalence Principle (TEP) instead predicts that proper-time transport is path-dependent in low...
The Temporal Equivalence Principle (TEP) is a scalar-tensor theory in which proper time is a dynamical field $\phi$ that couples universally to all matter via a conformal metric $\tilde{g}_{\mu\nu} = A^2(\phi) g_{\mu\nu}$. In deep gravitational potential wells, high ambient density suppresses the...
General Relativity is extensively validated in the deeply screened, high-proximity regime of the Solar System, but cosmological tensions—specifically the Hubble discrepancy—suggest a scale-dependent breakdown of the isochrony axiom. The Temporal Equivalence Principle (TEP) resolves these late-unive...
The cosmological extension of TEP proposes that evidence normally interpreted as cosmic expansion may instead involve large-scale Temporal Shear. Analysis of 1,701 Pantheon+ supernovae and Planck 2018 acoustic anchors shows the late universe preferring TEP geometry over ΛCDM, while CMB bounds act as the ultimate cosmological boundary condition.
This paper audits CERN LHC luminosity data for sidereal modulation predicted by the Temporal Equivalence Principle (TEP). Analyzing uncalibrated CMS Run 2 data and fill-level LHC SuperTable summaries, the pipeline tests for a sidereal residual beyond conventional mechanical tides and operational controls.
The April 2026 NIST metrology publication reported a redetermination of the gravitational constant G at Gaithersburg, Maryland, using an identical torsion-balance geometry to the prior French BIPM setup. The new measurement reveals a systematic relative drop of 2.81×10⁻⁴ compared to the Sèvres ba...
In the unscreened regime of the Temporal Equivalence Principle, this paper proposes that virtual force carriers and statistical wavefunctions are tangent-limit descriptions of a deeper disformal geometry of the temporal field, governed by the coupling coefficient B(φ). In the screened limit, wher...
Standard Quantum Field Theory and the Dirac equation are low-resolution, flat-frame tangent limits of a deeper dynamical proper-time phase transport governed by the Temporal Equivalence Principle (TEP). By treating proper time τ as a dynamical scalar field φ rather than a universal parameter, fiv...
Dismantling the point-particle paradigm: replacing intrinsic spin with fluid vorticity and eliminating renormalization through geometric boundaries.
This research programme develops the Temporal Equivalence Principle as a theory in which time is a dynamical field. More precisely, TEP proposes that proper-time accumulation is governed by a scalar field coupled to the matter/clock metric. The central question is whether synchronization and proper-time comparison remain globally integrable after standard relativistic corrections are modeled and removed.
The framework is formulated as a covariant scalar-tensor theory with a gravitational metric and a matter/clock metric. Local physics remains Lorentz invariant. The new effects appear in global observables: distributed clock correlations, synchronization holonomy, Temporal Shear, and environment-dependent Temporal Shear recovery.
The primary empirical tests use GNSS because global navigation systems form a planetary-scale network of atomic clocks. The reported signal is a distance-structured correlation pattern with characteristic length λT≈4,200 km, seen in processed clock products, a 25-year longitudinal analysis, and raw RINEX-derived observables. Satellite laser ranging provides an optical-domain consistency check.
The astrophysical papers then test whether the same temporal-field structure organizes phenomena normally treated separately: lensing mass discrepancies, globular-cluster pulsar timing, Cepheid distance-ladder biases, JWST high-redshift anomalies, and Gaia wide-binary dynamics. These applications are not presented as independent proof of TEP, but as cross-regime tests of a common Temporal Topology / Temporal Shear framework.
All papers, code, and data products are released for open review. The clearest near-term tests are external reproduction of the timing-network correlations, multi-constellation checks, optical-clock or fiber-link experiments, non-GNSS timing systems, and closed-loop one-way synchronization holonomy.
This interactive visualization displays the reported exponential decay pattern across processed clock products and raw RINEX-derived observables. The dual axes compare phase alignment in processed clock products with magnitude-squared coherence in raw RINEX-derived observables. Although these quantities are measured on different scales, both exhibit distance-structured decay consistent with the Global Time Echoes timing-network result.
The dotted trend lines show averaged exponential fits with Temporal Topology correlation lengths λT ranging from 600–4,500 km and fit qualities R²=0.87–0.99. The flagship multi-center processed-clock analyses report several-thousand-kilometre scales. Shorter apparent scales in raw or reduced demonstrations are shown as consistency diagnostics, not as replacements for the primary λT estimate.
All manuscripts, analysis code, processing pipelines, derived outputs, and reproducibility notes are publicly available under open-source licenses. The analyses are not synthetic simulations: they are reproducible outputs of documented code applied to real public geodetic and astronomical datasets. The interpretation remains model-dependent and is explicitly open to external audit.
Independent replication by other research groups is the decisive next step. Researchers interested in replication may find Paper 1 (TEP-GNSS I) the most accessible entry point, using publicly available CODE/IGS/ESA clock products (compact .clk files). Paper 2 (TEP-GNSS II) extends this with 25 years of CODE clock data. Paper 3 (TEP-GNSS III) provides the most direct raw-data test via RINEX processing but requires more substantial computational resources. Feedback on methodology, interpretation, or potential collaboration is welcomed.
Core repository containing the theoretical framework, mathematical models, and LaTeX source. Includes derivation scripts, figure generation code, and the full manuscript source.
Complete analysis pipeline for Paper 1 (TEP-GNSS I). Features multi-center cross-validation scripts, processing logs, JSON statistical outputs, and generated figures for correlation decay quantification.
Research compendium for Paper 2 (TEP-GNSS II). Contains longitudinal analysis scripts, orbital coupling logs, CMB alignment data, and comprehensive JSON result files.
End-to-end pipeline for Paper 3 (TEP-GNSS III). Includes SPP processing scripts, raw RINEX analysis logs, consistency-test datasets, and resulting anisotropy figures from 1 billion samples.
Codebase for Paper 4 (TEP-GL). Contains phantom mass modeling scripts, rotation curve data, analysis logs, and the Python notebooks used to generate manuscript figures.
Synthesis framework for Paper 5 (TEP-GTE). Includes integration scripts, cross-study correlation logs, consolidated JSON datasets, and summary figures demonstrating signal convergence.
Scaling analysis codebase for Paper 6 (TEP-UCD). Features saturation-density calculation scripts, scaling law verification logs, and figure generation for the universal critical density ρT model.
Analysis codebase for Paper 7 (TEP-RBH). Contains JWST NIRSpec data processing scripts, soliton wake modeling, line-profile decomposition tools, and visualization for the RBH-1 runaway black hole candidate analysis.
Complete analysis pipeline for Paper 8 (TEP-SLR). Includes SLR data parsers, residual computation logs, correlation analysis scripts, and final result figures.
Codebase for Paper 9 (TEP-EXP). Methodological taxonomy of precision tests of general relativity, discriminating experiments, and analysis of which observables constrain conformal, disformal, local-gradient, clock-covariance, and synchronization-holonomy sectors.
Codebase for Paper 10, Suppressed Density Scaling in Globular Cluster Pulsars. Contains pulsar catalog compilation, field-control comparisons, cluster-density scaling tests, and adversarial dynamics checks.
Codebase for Paper 11 (TEP-H0). Includes SH0ES data processing, environment stratification, TEP correction optimization, and scripts testing whether an environment-dependent Cepheid correction can reduce the Hubble tension.
Codebase for Paper 12 (TEP-JWST). JWST high-redshift galaxy analysis, isochrony axiom tests, temporal shear calculations, SUSPENSE survey kinematic comparisons, and star formation efficiency modelling.
Codebase for Paper 13 (TEP-WB). Gaia DR3 wide-binary analysis, Temporal Shear recovery tests, disk/halo environmental ordering, and field-transition radius fitting.
Codebase for Paper 15 (TEP-EFA). Analysis of twelve Earth gravity assist flybys within TEP framework, showing restricted TEP model favored over Null and Anderson empirical baseline through Temporal Topology screening and scalar force modeling.
Codebase for Paper 16 (TEP-J0437). Detection of non-zero Phase Closure in PSR J0437-4715 scintillation using 19,167 triplets, rejecting scalar-delay null hypothesis and demonstrating TEP-predicted synchronization holonomy.
Codebase for Paper 17 (TEP-LLR). Analysis of 26,207 INPOP19a LLR residuals detecting synodic Nordtvedt parameter η = -3.91 × 10⁻⁴ (6.94σ), supporting TEP's compactness-dependent Strong Equivalence Principle violation.
TEP is a scalar-tensor framework proposing that time is a dynamical field. More precisely, proper-time accumulation is governed by a scalar field coupled to the matter/clock metric, while local Lorentz invariance and locally measured c are preserved.
TEP is proposed as a generalization of GR, not a simple replacement. GR is recovered to current experimental precision where the locally observable shear/source-charge sector is screened or saturated. Differences are expected in global or weakly screened observables: distributed clock correlations, closed-loop synchronization holonomy, and environment-dependent rate or screening transitions.
No. GR already predicts gravitational time dilation: clocks tick differently in different gravitational potentials. TEP accepts that and does not claim it as new. The proposed new physics is residual structure after the standard GR timing model is removed: distance-structured clock covariance, environment-dependent Temporal Shear, and possible closed-loop synchronization holonomy.
TEP is first a theoretical framework, but the strongest empirical motivation is the timing-network programme. The astrophysical papers are transfer tests and candidate applications.
TEP does not deny the lensing, timing, dynamical, or cosmological phenomena usually attributed to dark matter. It challenges the assumption that those phenomena uniquely require a new invisible particulate substance. Temporal-field gradients and nontrivial proper-time transport can project into lensing, timing, and dynamical inference as an apparent mass-like component, termed Phantom Mass.
In the conservative interpretation, this is a testable correction to time-domain and variability-dependent lensing observables. In the stronger interpretation, tested across the series, part of the dark-sector phenomenology may be temporal in origin: an effect of analyzing a universe with nontrivial time transport under the assumption of global isochrony.
TEP predicts that Cepheid periods may acquire an environment-dependent bias in deep gravitational potentials. In the current SH0ES-host analysis, correcting for this proposed bias shifts the inferred local value toward the Planck value. This should be read as a candidate distance-ladder systematic, not as a final resolution until tested blindly on independent Cepheid, TRGB, maser, and SN host samples.
Yes, in the intended parameter regime. GW170817 tightly constrains differential photon–graviton propagation: any disformal cone tilt must be extremely small. TEP's main conformal-sector effects are different. If electromagnetic and gravitational signals travel along the same path through the same conformal temporal landscape, the effect is common-mode rather than a photon–graviton speed split. Conformal sectors remain indirectly constrained by PPN, equivalence-principle, source-screening, redshift, and clock-comparison tests.
The most direct falsifier is demonstrating that the reported GNSS covariance structure is fully reproduced by a known satellite, station-network, ephemeris, clock-product, or environmental systematic with the same distance, direction, and time dependence. Beyond this, the framework is designed to be tested through several increasingly independent pathways:
No. GNSS works extraordinarily well. TEP does not claim navigation is failing. It asks whether residual clock-network covariance, after standard modeling and differencing, contains spatial structure normally treated as noise, covariance, or processing residual. The claim is about subdominant correlation structure, not operational GNSS accuracy.
Both TEP and MOND address phenomenology attributed to dark matter, but through distinct mechanisms. MOND proposes a universal acceleration threshold (a₀ ≈ 1.2×10⁻¹⁰ m/s²) below which gravitational behavior deviates from Newtonian predictions. TEP instead tests environment-dependent Temporal Shear recovery organized around the Temporal Topology saturation scale ρT ≈ 20 g/cm3, which produces environmental ordering that differs from the MOND/EFE parameterizations tested in the wide-binary analysis. The two frameworks make qualitatively different predictions for environmental stratification.
The full manuscript series is freely available at mlsmawfield.com with Zenodo DOIs. Analysis code and data pipelines are hosted on GitHub. All manuscripts, code, and data products are released under Creative Commons CC-BY-4.0 and MIT licenses.
Note on Disambiguation: The acronym TEP (Temporal Equivalence Principle) addresses the foundations of spacetime geometry. It is distinct from the thermoelectric power coefficient (Seebeck TEP) in condensed matter, the Total Extraperitoneal surgical procedure, or Thermal Expansion and Properties in engineering.