1. Introduction: From a Universal Speed to a Universal Principle of Time

Relativity and quantum theory disagree about time. General relativity (GR) makes proper time geometric—$d\tau^2 = -g_{\mu\nu} dx^\mu dx^\nu / c^2$—dynamical, and observer-dependent. Quantum mechanics (QM) treats time as an external parameter t in the Schrödinger equation, $i\hbar \partial_t|\psi\rangle = \hat{H}|\psi\rangle$. The clash has haunted quantization programs for a century and manifested operationally as subtle ambiguities in defining one-way light speeds and simultaneity across extended regions. Precision clocks and time-transfer links now measure gravitational redshifts and velocity-dependent dilations at $10^{-18}$ fractional levels, while cosmology exhibits persistent $H_0$ and $S_8$ tensions. In this context, Einstein's postulate of a universal speed c must be sharpened: it is exact for any local freely falling laboratory, but it cannot be a global property if the flow of time itself is dynamical.

We propose the Temporal Equivalence Principle (TEP): all non-gravitational dynamics, signals, and quantum phases evolve according to the proper time defined by a single causal metric $\tilde{g}_{\mu\nu}$ that couples universally to matter. The rate at which proper time accrues is a field. This elevates "when" to the status "where" acquired in 1915: as space was geometrized, now time's rate is dynamized. The unavoidable consequence is that synchronization conventions are no longer globally integrable: even after removing known GR effects (Sagnac, Shapiro), closed-loop time transport does not necessarily return a clock to its initial epoch, and one-way times differ in opposite directions. The measured "speed of light" between distant clocks is revealed as an emergent ratio of distance to accrued proper time, not a fundamental constant comparable across regions without a theory of time's flow. Locally, c remains exactly invariant; globally, dynamical time imprints tiny, path-dependent asymmetries that we can detect or bound.

2. Axioms and the Temporal Equivalence Principle

We adopt four axioms:

These axioms encode Einstein's local invariance as a theorem and extend it: c is exactly invariant for every tangent space of $\tilde{g}_{\mu\nu}$, but global synchronization and one-way timing depend on the dynamical field $\phi$.

2.1 Relation to Other Modified Gravity Frameworks

The TEP framework can be situated within the broader landscape of modified gravity and scalar-tensor theories. To clarify its unique features, we provide a comparative analysis:

The multi-messenger constraint from GW170817, requiring the speed of gravitational waves $c_g$ to be equal to the speed of light $c_γ$ to high precision, has severely constrained this landscape. Our model, with a universal conformal coupling $A(\phi)$ and a disformal term $B(\phi)$, automatically satisfies $c_g = c_γ$ in the conformal limit ($B=0$). The non-zero $B(\phi)$ introduces a small, calculable deviation $c_γ ≠ c_g$, which will be the target of future multi-messenger constraints.

Conceptually, TEP differs from many scalar-tensor theories by its foundational principle: the universal coupling of matter to a single metric $\tilde{g}_{\mu\nu}$. This is philosophically closer to Bekenstein's TeVeS theory, which also introduced a separate matter metric, but our framework is more minimal, using only a scalar, and makes a unique prediction in the form of the synchronization holonomy.

3. Operational Foundations: Measurement, Simultaneity, and One-Way Light

3.1 What we actually measure

No measurement of c uses null proper time along the photon's worldline; that would be $c = dx/d\tau$ with $d\tau = 0$. Instead, we compare a distance with an elapsed proper time on an observer's clock. For two spatially separated clocks A and B with worldlines $\gamma_A$, $\gamma_B$ and a light path $\lambda$ from A to B, the measured one-way time $t_{AB}$ is the difference in B's proper time between emission and reception, after a synchronization convention has assigned simultaneity between A and B. Two-way measurements are convention-independent; one-way measurements are not, unless we provide an invariant observable.

3.2 The synchronization problem in a dynamical-time background

Einstein synchronization assumes (i) reciprocity of propagation and (ii) homogeneity of time standards along the path. If the rate $d\tau/dt$ varies spatially or directionally, synchronization by light exchange over extended loops becomes path-dependent. The proper formalism invokes the congruence $\tilde{u}_\mu$ of observers (clocks) and the null geodesics $k^\mu$ of $\tilde{g}_{\mu\nu}$. Define simultaneity as orthogonality to $\tilde{u}_\mu$ (where Frobenius integrability allows it), and define time transport by mapping a proper-time interval at A to B using $\tilde{g}$-null signals. In GR, the non-closure of such transports arises from rotation of the congruence (vorticity) and spacetime curvature (Sagnac/Shapiro); both can be modeled and removed. In a dynamical-time geometry with disformal couplings, there is an additional, tiny non-exact contribution to time transport that cannot be removed by coordinate choices: a synchronization holonomy.

3.3 A convention-independent observable: synchronization holonomy

Define the holonomy H as the total non-closure in proper time accumulated by transporting synchronization once around a closed, oriented loop C embedded in spacetime:

where $d\tau_{\text{prop}}$ is the infinitesimal proper-time transport inferred along each leg by $\tilde{g}$-null exchange. In SR and pure GR (after subtracting modeled Sagnac/Shapiro), H = 0 for loops small enough that curvature terms cancel to the relevant order. In a disformal time geometry H generically does not vanish. Crucially:

• H is an operational scalar: it depends only on measured proper times and the loop.
• H vanishes in the conformal-only limit ($B = 0$) after GR effects are subtracted; any claim to the contrary confuses convention with physics.
• H reflects the curvature of an effective time-transport connection that includes a disformal, non-exact component.

This reframes "variable c" claims: the invariant diagnostic is not a raw one-way c, but H. A nonzero H means simultaneity is not integrable beyond GR; this is what dynamic time does to measurement.

4. Disformal Invertibility, Causality, and Hyperbolicity

Invertibility and signature

The inverse of $\tilde{g}_{\mu\nu}$ exists for $A>0$ and $1 + (B/A^2)(\partial\phi)^2 \neq 0$:

For Lorentzian signature, require $A>0$ and $B(\partial\phi)^2 > -A^2$. We assume $B$ small and gradients bounded in all regimes of interest.

Causality

The matter cone is inside or equal to the gravitational cone when $B \geq 0$ and gradients are modest; no closed causal curves arise for small $B$. With $B\to0$ at late times, gravitational and matter null cones coincide ($c_T = c_{\text{EM}}$). With $B$ small, any phase differences in propagation are minute and bounded by multi-messenger results.

Hyperbolicity

The scalar's canonical kinetic ensures hyperbolic evolution with well-posed Cauchy problem where $V''(\phi) > 0$ near minima. Maxwell's equations in a Lorentzian matter metric remain hyperbolic. Linearized gravity propagates on the Einstein-frame cone. No ghosts or gradient instabilities arise for canonical $\phi$ and small $B$. The EFT is valid below the disformal scale $M$, with higher-dimensional operators suppressed.

5. Local Lorentz Invariance, Proper Time, and the Emergence of c

Proper time

Clocks measure $d\tau^2 = -\tilde{g}_{\mu\nu} dx^\mu dx^\nu/c^2$. In a local lab with small velocities and $\partial_0\phi \ll |\nabla\phi|$, $-\tilde{g}_{00} \approx A(\phi)^2 (-g_{00}) - B(\partial_0\phi)^2$, so

up to $O(B)$ corrections. This "dynamical time law" rescales all frequency standards locally.

Local c

All small labs measure an invariant $c$; the conformal factor rescales both clocks and rulers uniformly, preserving null cones. Thus the empirical fact "$c$ is constant" remains a theorem of the theory at the local level.

Global measures

Global "speeds" involve synchronized endpoints. Because $d\tau$ depends on $\phi$'s history, the global assignment of simultaneity acquires non-integrability when $\phi$ varies in space or time. That non-integrability—not a local violation of Lorentz invariance—is the source of observable departures.

6. Synchronization in a Dynamical Time: One-Way Light and Holonomy

Operational definitions

Two-way light speed is synchronization-independent and has established $c$'s local invariance. One-way measures require synchronized clocks at A and B; Einstein synchronization assumes time-orthogonal slices and propagation symmetry. In a dynamical $\phi$ background, slow clock transport and one-way synchronization are path and history dependent.

Key theorems

Synchronization one-form and holonomy

Decompose $\tilde{g}_{\mu\nu}$ in 3+1 form:

Define the synchronization one-form $\tilde{\sigma} = -\tilde{N}^{-1} \tilde{N}_i dx^i$. For a closed spatial loop $C$, the synchronization holonomy is

In GR with stationary spacetimes, $H$ reproduces the Sagnac/gravito-magnetic effect; we subtract these known contributions using geodesy and ephemerides. The residual holonomy beyond GR is

with $\Sigma$ spanning $C$. This residual vanishes in SR/GR with $B=0$ and stationary $A$; it is generically nonzero with disformal corrections ($B \neq 0$) and/or temporal variation of $A$ combined with motion through $\nabla\phi$.

Gauge and protocol invariance

$\tilde{\sigma}$ is a one-form on the spatial slice induced by the matter metric and a chosen time function $t$. Changing the synchronization protocol shifts $\tilde{\sigma} \to \tilde{\sigma} + d\chi$ for some scalar $\chi$ when the change corresponds to a redefinition of time labels without physical alterations; the loop integral of an exact form vanishes: $\oint d\chi = 0$. Therefore $H_{\text{resid}}$, constructed from measured proper times and two-way calibrations and expressed via $d\tilde{\sigma}$ after GR subtraction, is invariant under admissible synchronization changes. This addresses the critique that holonomy is conventional: the nonzero part is the curvature (curl) of $\tilde{\sigma}$, not its exact part.

Disformal corrections to σ̃

For small $B$ and modest $\partial\phi$,

with $n^\mu$ the unit normal to slices. Then

and

with $\Delta$ indicating the residual beyond GR. Time dependence of $\phi$ and motion through $\nabla\phi$ generate non-vanishing curl.

7. Screening, PPN, Equivalence Principle, and Disformal Bounds

Screening

A density-dependent effective potential $V_{\text{eff}}(\phi; \rho) = V(\phi) + [A(\phi) - 1] \rho$ makes $\phi$ acquire a large effective mass in dense environments (short-range interactions, "thin shells") and remain light cosmologically. This reconciles local null tests with cosmological dynamics.

PPN mapping

In unscreened regimes, the PPN parameter $\gamma - 1 \approx -2 \alpha_0^2/(1 + \alpha_0^2) \approx -2 \alpha_0^2$ with $\alpha_0 = \alpha(\phi_\infty)$; Cassini's $|\gamma - 1| < 2.3\times10^{-5}$ implies $\alpha_0 \lesssim 3\times10^{-3}$. Screening further suppresses effective couplings near massive bodies. Other PPN parameters remain GR-like for small $\alpha_0$.

Equivalence principle

TEP ensures universality in the matter frame. In the Einstein frame, the apparent fifth force is bounded by Eötvös experiments; MICROSCOPE gives $\eta \lesssim 10^{-14}$. Sectoral dilaton-like couplings (to $\alpha$, $\mu$, quark masses) are constrained to $|d| \lesssim 10^{-5}$–$10^{-6}$ by composition tests and clock ratios.

Disformal constraints

GW170817/GRB170817A constrain $|c_\gamma - c_g|/c \lesssim \text{few}\times10^{-15}$ at $z \approx 0.01$. This requires $B(\phi_0) (\partial\phi)^2 \ll 1$ today along astrophysical paths. We adopt $B$ effectively negligible at late times on cosmological paths; small residual $B$ remains a controlled source for holonomy without violating bounds.

8. Cosmology: Background Evolution, Growth, and EFT Mapping

Background

In spatially flat FLRW (Einstein frame),

In the matter frame, the continuity equation for $\tilde{\rho}_m$ is standard; the Einstein-frame exchange encodes energy flow between $\phi$ and matter. During radiation domination, $T \approx 0$ and $\phi$ is overdamped; it thaws during matter domination and can act as dark energy.

EFT of dark energy mapping

The theory maps to EFT-of-DE with $\alpha_T = 0$ enforced to satisfy $c_T = 1$. The braiding $\alpha_B$ and Planck-mass run $\alpha_M$ are both small and controlled by $\alpha(\phi)$ and $\dot{\phi}$. Bounds from large-scale structure and ISW require $|\alpha_M|$, $|\alpha_B| \ll 1$ at late times. Our parameter choices respect these.

Recombination and H₀

Early dark energy fraction $\Omega_\phi(z \approx 1100) \lesssim \text{few percent}$ and small $\alpha(\phi_{\text{rec}})$ keep recombination microphysics intact (HyRec constraints) and shift the sound horizon $r_s$ by $\lesssim1$–2%, enough to ease $H_0$ tension. Sectoral couplings $d_e$ at recombination must be tiny to preserve $\alpha$, $m_e$; we adopt negligible $d_e$ at early times.

Growth and S₈

In unscreened low-density regions, the effective Newton constant is $G_{\text{eff}}(k,a) = G [1 + 2 \alpha_0^2/(1 + m_\phi^2 a^2/k^2)]$, giving mild scale-dependent growth, raising $f\sigma_8$ on large scales while screening suppresses changes in high-density regions. A net reduction of $S_8$ inferred from weak lensing by ~0.01–0.03 is possible while preserving CMB lensing. Detailed MCMC fits with CLASS/HyRec modifications are planned.

Standard sirens

In the late-time conformal limit, EM and GW share null cones; standard siren distances are unaffected kinematically. Subtle, detector time-standard effects may introduce an integrated $\phi$-history contribution at the sub-second level over ~100 Mpc; ensemble analyses can bound or detect this.

9. Quantum Clocks, Interferometry, and Composition Dependence

Quantum evolution

Proper-time Schrödinger evolution $i\hbar d|\psi\rangle/d\tau = \hat{H}|\psi\rangle$ becomes $i\hbar d|\psi\rangle/d\tilde{t} = A(\phi) \hat{H} |\psi\rangle$ in the matter frame. Transition frequencies scale as $\nu \propto A(\phi)$ with tiny sectoral corrections from dilaton-like couplings $d_e$, $d_\mu$, $d_q$:

Species sensitivity

Different clock transitions carry different $K$-coefficients; by comparing ratios, one can disentangle the universal $A(\phi)$ scaling from composition dependence and constrain $d_e$, $d_\mu$, $d_q$.

Interferometry

Phases acquire contributions proportional to the integral of $A(\phi)$ along arms. Atom interferometers with vertical baselines can sense $\partial_h\phi$ at $10^{-21}$–$10^{-22}$ m$^{-1}$; photon interferometers are sensitive at different bands.

Decoherence

Rapid $\phi$ variations would induce dephasing at rates $O(\partial_t\phi)$, strongly bounded by clock stabilities. We assume adiabatic evolution in the lab, consistent with null drift bounds.

10. Experimental Proposals and Falsifiability

We propose a suite of decisive, cross-checking experiments that collectively falsify or confirm the theory. All observables are dimensionless ratios or calibrated residuals; all designs include nulls, blinding, and open data.

11. Statistical and Open Science Principles

Pre-registration. Publish analysis plans (models, priors, nulls, thresholds) for triangle and portable-clock experiments; record any deviations.

Blinding. Blind event-time stamps and calibration offsets; employ independent teams for calibration and analysis.

Open data/code. Release raw timestamps, environmental monitors, calibration logs, and pipelines with DOIs; release CLASS/HyRec modification, atom-sensitivity database, and meta-analytic code.

Hierarchical inference. Use heavy-tailed likelihoods (Student-t) to mitigate outliers; multi-level random effects by domain; selection models for publication bias; explicit covariance modeling for shared systematics.

12. Addressing Critiques and Clarifying Claims

No EM–GW kinematic delay in conformal subclass. With $B=0$ today, EM and GW share null cones; any observed delays are astrophysical/source or detector-time-standard effects. We correct earlier overstatements and align with GW170817 constraints.

Static $\phi$ gradients do not generate first-order one-way anisotropy. We provide an explicit derivation (Appendix A) showing cancellation, correcting prior misinterpretations.

Holonomy invariant and not a synchronization artifact. Our observable $H_{\text{resid}}$ is constructed from physical proper-time measurements and is the integral of a curvature two-form $\Delta(d\tilde{\sigma})$ beyond GR; protocol changes shift $\tilde{\sigma}$ by exact differentials, which integrate to zero over closed loops.

Causality and hyperbolicity. We give explicit invertibility and signature conditions and restrict to canonical kinetics and small disformal couplings; matter-frame causality is preserved. A full hyperbolicity proof for generic $B(\phi)$ is delegated to a companion technical paper; our small-$B$ regime is safe.

Cosmology claims are hypotheses to be tested. We outline how modest $\phi$ dynamics can ease $H_0/S_8$ while respecting early-universe bounds; quantitative MCMC fits are required and planned.

13. Philosophical and Conceptual Implications

Simultaneity beyond Einstein. Special relativity makes simultaneity observer-dependent; dynamic time makes synchronization non-integrable at the global scale. The invariant content is a holonomy of time transport: moving clocks around closed loops in a dynamical time background returns path-dependent offsets after subtracting GR effects.

Constants clarified. The speed of light is an invariant in local tangent spaces; globally, "$c$" is not a number but a family of operational ratios dependent on clock histories in a time field. Variation of constants becomes a question about dimensionless ratios across environments and epochs.

Machian undertone. The rate of time responds weakly to the stress-energy of matter through $\alpha T$, giving a principled, covariant flavor to the idea that the "rest of the universe" influences local rates, without violating local physics.

14. Conclusions

We have articulated a covariant framework in which the rate of time is a dynamical field with universal matter coupling. The architecture preserves local Lorentz invariance and null-cone structure (in the conformal limit), conforms to multi-messenger bounds, and yields new invariant observables—synchronization holonomy and clock anholonomy—that vanish in GR and are measurable with modern clocks. Screening reconciles local nulls 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 us 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, we will have set uniquely strong bounds and clarified the operational bedrock of $c$ and simultaneity to unprecedented precision.

@misc{Smawfield_TEP_2025,
  author       = {Matthew Lukin Smawfield},
  title        = {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}
}