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_μν and a causal (matter) metric g̃_μν to which all non-gravitational fields and clocks couple. In isotropic shorthand 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_A φ/M_Pl) is a universal conformal factor, and B(φ) encodes tiny, direction-dependent deformations of the light cone consistent with GW170817-class multi-messenger constraints (|c_γ - c_g|/c ≲ few×10⁻¹⁵ today); decomposed along the unit cosmic temporal flow U_μ ≡ -∇_μφ̄/√(2X̄) (g^μνU_μ U_ν = -1, orthogonal spatial projector Π_μν = g_μν + U_μ U_ν), this resolves into g̃_μν = -A_loc²(u_grav) U_μ U_ν + Aₛₚ²(u_drift) Π_μν + B(φ) Π_μ^αΠ_ν^β∇_αφ∇_βφ, separating the stationary baryon-sourced ADM lapse N_TEP(x) = A_loc(u_grav(x))N_GR(x) from the secular matter-frame spatial ruler factor Aₛₚ(u_drift(t,x)) = exp(-u_drift(t,x)) on static gravitational 3-space γ_ij(x). 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 can become path-dependent in the disformal/non-exact sector of a dynamical-time background. The covariant action, field equations, conservation laws, and PPN mapping are developed; screening is formulated via continuous Temporal Topology and nonlinear Temporal Shear (P_,X = k√(ξ) + 2ξ, k≃ 16 fixed by g_t and the empirical scale a₀). 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(φ)≠0, 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 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. Cosmologically, TEP adopts an eternal, spatially infinite, inhomogeneous background whose physical volume does not undergo cosmological expansion. Neither space nor matter changes size: the gravitational 3-metric γ_ij is static, while the drift enters through the matter-frame ruler factor Aₛₚ, which recalibrates units rather than stretching geometry. Distances and densities expressed in matter units therefore evolve without any physical stretching, allowing the observed redshift and apparent Hubble relation to emerge directly from the macroscopic historical drift of the inhomogeneous temporal network over coordinate time, governed principally by the intergalactic ruler ratio 1+z = Aₛₚ(tₒ)/Aₛₚ(tₑ) integrated in flight across unscreened space, with constrained non-exact propagation corrections through the temporal topology. An exactly solved closed-static branch establishes that the specified action admits a static, eternal gravitational configuration with conformal unit drift; that branch is an analytical benchmark, neither an attractor nor capable of near-flat matter geometry. Numerical noncompact solutions now demonstrate a constraint-consistent, on-shell inhomogeneous temporal landscape with near-zero native matter-frame spatial curvature (|Ω_k^native| = 1.40× 10⁻⁴), sustained secular temporal roll, and a positive-definite linear fluctuation spectrum across both the canonical (λ_0 = +0.413 > 0, λ_1 = +39.1) and jointly on-shell two-branch kinetic (λ_0 = +0.563 > 0, λ_1 = +555.4) sectors, bounded from below by an all-twist Bloch gap λ ≥ min m_eff² > 0 on a positive evolving constant-mean-curvature initial-slice lapse; the master-potential family V(u) = V_matter(u)e^-(u/u_s)⁴ + V₀ e^-(u_s/u)⁴ reproduces the reconstructed background profile to fractional deviation ∼ 10⁻¹⁶ across the matter domain while flattening onto the deep-well floor V₀. Early-universe and background closure—in which the Big Bang gravitational singularity is replaced by a geodesic-complete matter-frame temporal horizon without phenomenological thermal screening—is developed quantitatively through the companion transport and Boltzmann analyses (Papers 18, 26, 27, 29, 30), reproducing the Pantheon+/DESI DR2/Planck θ_* background history, the baryon-sourced (ω_cdm=0) CMB acoustic and lensing spectra, and the four-observable light-element reaction-flow equilibrium. 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 benchmark sensitivity windows 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 the full GR, kinematic, clock-scale, and reference-frame synchronization model, yields a residual "temporal holonomy" H_resid that vanishes in GR and becomes nonzero only when time is dynamical in this sense. Embedding the preferred-foliation geometry in a spatially covariant ADM/khronon effective action establishes four key results: (i) conformal matter coupling preserves null cones, and electromagnetic waves and transverse-traceless gravitons share the identical principal wave operator □_TEP ≡ -(N_TEPc)⁻²∂_t² + Aₛₚ⁻²(t,x)γ^kl∂_k∂_l, enforcing c_GW(t,x) = c_γ(t,x) at every event (Δ t_γ–GW ≡ 0 at B=0, satisfying GW170817); (ii) the static-space WKB tensor measure √Aₛₚ⁺¹· Aₛₚ⁻¹ ≡ 1 eliminates FLRW Hubble friction, preserving 1/r graviton decay, the standard-siren discriminator Ξ(z) = d_L^GW/d_L^EM = 1/(1+z), and the inverse detector chirp-mass map ℳ_det = ℳ̃/(1+z) at pinned interior response; (iii) a static φ-gradient produces no direct propagation asymmetry in the static purely conformal limit—the cancellation is exact, not merely first-order; and (iv) the selected two-branch kinetic sector (P_,X = k√(ξ) + 2ξ, with k≃ 16 fixed by the empirical scale a₀ ∼ cH₀ and zero per-system fit parameters) screens Solar-System post-Newtonian and preferred-frame observables (|γ_PPN-1|≃ 4.32× 10⁻⁸ and |α_2|≃ 2.16× 10⁻⁸ < 2.4× 10⁻⁷ at 1.6 R_⊙, |α_1|≃ 5.86× 10⁻⁵ < 10⁻⁴ at 1 AU), lunar laser ranging (|Ġ_loc/G_loc|_LLR≲ 1.7× 10⁻¹⁸ yr⁻¹), and binary-pulsar dipole radiation well inside observational bounds while passing the wide-binary joint test at 0.4–1.7σ with zero per-system free parameters and generating flat galactic rotation curves and unstretched-space gravitational lensing via A_loc(u_grav). Disformal tilts (B ≠ 0) can source holonomy subject to multi-messenger constraints. While conformal and kinetic environmental screening responses are fixed with zero per-system free parameters by the selected operator P_,X = k√(ξ) + 2ξ, non-exact loop holonomy scales with the dimensionless disformal normalization B₀: a corrected Earth–Sun–satellite triangle calculation gives a converged benchmark |H_resid|≃4.33×10⁻¹³ |B₀| s per loop (with zero circulation in the exact-connection control), providing a direct experimental observable for bounding or detecting B₀. The effective covariant architecture 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 α-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. Cosmological inference is implemented through native hi_class and Cobaya analyses (Papers 18, 26), 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

Open Science Statement: This work is a preprint and is open to community review, ideas, and collaboration. All analysis code, configuration files, and manuscripts are open source. Feedback and contributions to further test these results are welcome.