The Angular Dual-Time Spacetime Paradigm: Evidence from Zero-Parameter Turbulence and a Binary Crucial Test

1. Introduction

1.1. Research Background

Time t has been central to physics since Newton's absolute time. Classical mechanics treats t as a universal, linear parameter, together with spatial coordinates forming inertial frames. Special relativity (SR) showed that time dilates with motion [4], and general relativity (GR) further geometrized spacetime, linking time flow to gravity [5]. Nevertheless, current relativistic frameworks still treat t as a single, monotonic parameter rooted in translation symmetry. All precision time measurements—from cesium to strontium lattice clocks—depend on electromagnetic transition frequencies; they measure linear time t, the parameter that tracks translation along spatial paths. This translation-centric perspective struggles with rotational motion: for instance, the causality of rotating frames, frame-dragging in Kerr black holes, and the behavior of quantum entanglement in rotating systems [6,7,8,9]. Historically, the formulation of rotational dynamics has lagged behind that of translational motion. The rotational symmetry of spacetime—which implies angular momentum conservation—was recognized decades after linear momentum [10]. Similarly, the discovery of intrinsic spin [11] and the nearly 50-year gap between the Schwarzschild and Kerr solutions [12] reflect this persistent asymmetry. These difficulties hint at the possible existence of a separate time dimension for rotational motion.

1.2. The φ(τ) Framework: A Complementary Geometric Perspective

To address these issues, we introduced in earlier work the angular displacement spacetime framework φ(τ), formulated via the complex-time construction Time = t + iτ, in which angular time τ stands as an orthogonal dimension alongside linear time t [1].

• t: Describes translational motion (denoted: s_lin, conventional seconds);

• τ: Describes rotational motion (denoted: s_ang, seconds of angular displacement)

The theory postulates a universal maximum angular velocity Ωₘₐₓ = c/(2π) ≈ 4.77×10⁷ rad/s, analogous to the speed of light c for translation.

The present work is a continuation of our previous φ(τ) study [1]. Its purpose is not to challenge or replace established relativistic theory, but to provide a supplementary geometric framework for rotational dynamics and, more importantly, to propose a concrete falsifiable experimental scheme that allows the hypothesis to be tested empirically.

1.3. Outline of the Paper

Section 2 briefly recapitulates the core postulates of φ(τ) and the local co-boundary coupling strategy. Section 3 presents the zero-parameter turbulent scaling law as indirect macroscopic evidence. Section 4 develops the operational definition of τ via three independent angular-clock schemes. Section 5 describes the binary crucial experiment under the zero-area limit. Section 6 discusses further implications and limitations. Section 7 concludes.

2. Theoretical Background

2.1. Three Postulates of the φ(τ) Theory

The φ(τ) framework rests on three postulates [1]:

Postulate I (Orthogonal Duality of Bitemporal Dimensions): Spacetime comprises linear time t and angular time τ, unified in the complex time Time=t+iτ. Here t governs translation (conventional seconds, denoted s_lin), while τ governs rotation (seconds of angular displacement, denoted s_ang). The two-time dimensions are independent.

Postulate II (Maximum angular velocity): Nature imposes a universal upper bound on angular velocity: Ωₘₐₓ = c/(2πR₀) ≈ 4.77×10⁷ rad/s, where R₀=1 m is a normalization constant (analogous to the unit radius in radian measure). This limit is an intrinsic property of spacetime, not a material constraint. Even if future experiments find Ωₘₐₓ=k⋅c/(2π) with finite k, the φ(τ) framework remains valid as long as a finite maximum angular velocity exists in nature.

Postulate III (Alternative Relativity Principle): The rotational physical laws governing structured objects (e.g., rotating disks) exhibit form invariance in uniformly rotating reference frames (Alternative Lorentz Covariance).

2.2. Impossibility of Global Proportionality ξ=dt/dτ

A global proportionality constant ξ = dt/dτ cannot be consistently maintained, for three reasons:

(1) Conflation of Orthogonal Evolution Manifolds: While linear time t (denoted s_lin) and angular time τ (denoted s_ang) share the identical SI base dimension [T]—a mathematical necessity for constructing the complex time Time=t+iτ—they parameterize fundamentally orthogonal physical manifolds: t tracks translational progression along spatial paths, whereas τ tracks the topological accumulation of rotational phase. Forcing a global constant ξ=dt/dτ is conceptually analogous to imposing a rigid, universal proportionality between translational kinetic energy (½mv²) and rotational kinetic energy (½Iω²). While both strictly share the exact same SI dimension [M⋅L²⋅T⁻²], they describe completely distinct physical degrees of freedom. Such a global constant would erroneously conflate two independent evolutionary axes into a single rigid scalar, violating the core principle that τ must be defined operationally through pure rotational geometry.

(2) Violation of Lorentz Invariance: In curved or non-inertial frames, the Lorentz factor ($\gamma =1/\sqrt{(1-\mathrm{v}²/\mathrm{c}²)}$) depends on position. A global ξ would demand a fixed ratio dt/dτ, contradicting the fact that at different radii of a rotating disk the local dt differ. No locally covariant equation can accommodate such a global constant.

Note: In φ(τ) framework, the rotating disk is not required to be a rigid body. When a uniformly rotating elastic disk reaches a steady state such that its shape no longer changes, it can be equivalently treated as "rigid"—the "rigidification principle." Here "rigidification" only signifies steady-state shape invariance of an elastic disk, and does not imply superluminal motion of a classical rigid body, thereby preserving local Lorentz covariance.

(3) Confusion Between Boundary Conditions and Definitions: Assumptions like ξ=1 at the equator of an extreme Kerr black hole (a*=1) are boundary conditions, not universal definitions. Away from such limits, gravitational potentials and frame-dragging drive t and τ to evolve independently.

2.3. Local Co-boundary Coupling

Instead of a global constant, t and τ are coupled dynamically at local co-boundary events:

(1) Translational dynamics follow standard relativity parameterized by t; rotational dynamics are described by φ(τ) equations parameterized by τ.

(2) At a chosen co-boundary point (e.g., the center of a rotating platform), one measures Δt with an atomic clock and Δτ with an angular clock, yielding a local factor ξ_local= (Δt/Δτ)_local that encodes all local physical conditions.

(3) Once calibrated at one point, the theory predicts how ξ_local should vary with the system’s rotational state, which can be tested experimentally.

This local coupling strategy keeps φ(τ) compatible with established physics while endowing τ with its own dynamical behavior.

3. Indirect Experimental Evidence: Zero-Parameter Turbulent Scaling

The φ(τ) framework postulates a universal maximum angular velocity, Ωₘₐₓ​=c/(2π)≈4.77×10⁷ rad/s (which manifests as Ω_FluidMAX ≡ Ωₘₐₓ​ in the turbulence context), inducing an angular time dilation factor ${\gamma }_{\tau }=\sqrt{1-{\displaystyle \frac{{\omega }^2}{{\Omega }_{\mathrm{F}\mathrm{l}\mathrm{u}\mathrm{i}\mathrm{d}\mathrm{M}\mathrm{A}\mathrm{X}}^2}}}$​​, When applied to turbulent flows, this rotational constraint generates a unified, zero-parameter scaling law for the turbulent exponents, derived entirely from first principles in our previous work [1, 2-Pu, Phys. Rev. Fluids, under review (FT10319),3-Preprint 202606.0653]:

$${\zeta }_p={\displaystyle \frac{p}{3}}+{\displaystyle \frac{p\left(3-p\right)}{8{\mathsf{\pi }}^2}}+{\displaystyle \frac{\mathit{m}\mathit{a}\mathit{x}(0,p-6)}{4{\mathsf{\pi }}^2}}$$

This closed-form expression exhibits a clear two-regime structure dictated by the rotational saturation at Ω_FluidMAX ​:

• For p≤6 (perturbative regime where the continuum hypothesis holds), the formula reduces to strict quadratic corrections to Kolmogorov scaling: ζ_p=p/3+p(3−p)/8π² ​;
• For p≥7 (hard-truncation regime where the continuum breaks down), the max function activates a linear additional term (p−6)/4π²​, reflecting the phase transition from smooth leakage to angular condensation.

Remarkably, this zero-parameter formula reproduces all high-precision DNS/experimental data for p=1~9 with a relative error of less than 0.3%. At p=9 (the highest currently measurable order), the absolute error is only 0.0020, outperforming the standard She-Leveque model by a factor of 8.7. Crucially, this achievement represents a rare instance in modern physics where a purely theoretical prediction preceded experimental verification. Unlike phenomenological models that are retrospectively fitted to existing data, the unified ζ_p ​ formula was derived from the geometric constraints of φ(τ) spacetime before the high-precision DNS data for p=7∼9 became widely available. This predictive power distinguishes φ(τ) from speculative multi-time theories that lack experimental contact.

The fact that a single topological constant—1/(4π²), arising directly from the intrinsic 2π periodicity of angular time τ—can explain the entire anomalous scaling behavior of turbulence provides strong prima facie indirect evidence for the physical reality of the τ dimension.

4. Operational Definition Framework for Angular Time τ

To render τ empirically accessible, we outline three independent schemes for defining and measuring τ without relying on conventional t-clock frequency standards. All three measure τ through pure geometric angular displacement or quantum phase accumulation, with τ = θ / Ωₘₐₓ.

4.1. Optical Torque Angular Clock (OTAC)

OTAC exploits angular momentum conservation of circularly polarized photons. Each photon carries spin ±ℏ. When n photons are absorbed, the total transferred angular momentum is Jₚᵤₗₛₑ = nℏ. If the absorber is a torsion balance with moment of inertia I and the pulse duration is Δt, the induced rotation angle is θ=J_pulseΔt/I=nℏΔt/I. Calibrating the relation between θ and J_pulse then gives a direct mapping to τ:

$$\tau \equiv {\displaystyle \frac{\mathsf{\theta }}{{\Omega }_{max}}}(\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathsf{\Omega }\mathrm{ₘ}\mathrm{ₐ}\mathrm{ₓ}=\mathrm{c}/(2\mathsf{\pi })\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s})$$

(1)

This approach avoids traditional frequency-to-time conversions, instead defining τ geometrically through angular displacement.

4.2. Gravitational Torsion Geometric Clock (GTGC)

GTGC defines τ from the rotation of a macroscopic free rotor in vacuum. A torque free rigid body conserves angular momentum and rotates without dissipation about its center of mass. Angular time τ is the ratio of angular displacement ϕ to Ωₘₐₓ. One complete 2π turn defines the basic unit τ₀, with period

$${\tau }_0\equiv {\displaystyle \frac{2\pi }{{\Omega }_{max}}}$$

(2)

Laser interferometry directly measures the angular displacement of the dumbbell arms relative to a fixed reference, with interference fringe shifts proportional to Δϕ. No time integration is required.

4.3. Superconducting Flux Quantum Clock (SFQC)

SFQC uses the quantized magnetic flux in a superconducting quantum interference device (SQUID) to define τ. The flux quantum Φ₀ = h/(2e) directly connects Planck’s constant and the electron charge to the quantum phase. In a superconducting loop, the magnetic flux is an integer multiple of Φ₀, corresponding to a phase difference that is an integer multiple of 2π. Each 2π phase cycle—i.e., one Φ₀ crossing the loop—defines a fundamental angular time unit τ₀. This amounts to counting phase cycles in quantum phase space, independent of conventional time standards.

4.4. Cross-Validation Network

The three clocks rest on distinct physical principles—photon angular momentum, classical geometric rotation, and superconducting flux quantization. If all three yield mutually consistent τ readings, they form a triple cross-validation network that strengthens the case for τ as a physically meaningful observable. Detailed engineering specifications of the three clocks are not repeated here; the present paper focuses on their conceptual role in making τ experimentally testable.

5. Binary Crucial Experiment under the Zero-Area Limit

To distinguish φ(τ) predictions from standard relativistic predictions, one must isolate angular-velocity effects from radius-dependent effects. Conventional rotating-frame experiments operate at finite radius r ≠ 0, where transverse Doppler shift, centrifugal potential, and the Sagnac effect are all present. These effects can mask or mimic any τ-dependent signal.

5.1. Coordinate vs. Intrinsic Angular Time in φ(τ)

The above sections defined three coordinate angular times (geometric descriptions):

$${\tau }_{coord}\equiv {\displaystyle \frac{\varphi }{{\Omega }_{max}}}={\displaystyle \frac{\omega t}{{\Omega }_{max}}}$$

(3)

From literature [13], the intrinsic angular time (measurement result) in φ(τ) is derived as:

$${\tau }_{proper}={\tau }_{coord}\sqrt{1-{\displaystyle \frac{{\omega }^2}{{\Omega }_{max}^2}}}$$

(4)

At low angular velocities (ω≪ Ωₘₐₓ),$\sqrt{1-{\displaystyle \frac{{\omega }^2}{{\Omega }_{max}^2}}}\approx 1{,\tau }_{coord}\approx {\tau }_{proper}$, reducing nonlinearity to linearity.

5.2. Physical Foundation of the Zero-Area Limit

We adopt the zero-area limit: place the test clock exactly at the rotation center (r = 0). In this configuration:

(1) Geometric Configuration

Place the atomic clock at the rotation center (r=0); an identical clock on the ground serves as the reference.

(2) Sagnac Elimination:

The effective enclosed area A = πr² → 0, so the Sagnac effect vanishes (Δt_Sagnac∝Aω→0).

(3) No Relativistic Effects

At r=0, v=ωr=0 and acceleration a=ω²r=0; both SR and GR predict zero frequency shift$(\mathsf{\Delta }\mathrm{f}/\mathrm{f}=0$).

5.3. Competing Predictions

The core difference between φ(τ) and relativity lies in whether "time dilation depends on angular velocity ω" rather than radius r.

(1) Joint Prediction of SR & GR

In a rotating reference frame, the frequency offset of atom clocks is determined by transverse Doppler effect (SR) and gravitational redshift (GR):

$${\displaystyle \frac{\Delta f}{f}}=-{\displaystyle \frac{v^2}{{2c}^2}}-{\displaystyle \frac{\mathsf{\varphi }}{c^2}}$$

(5)

In equation (5), the linear velocity v=ωr and the centrifugal potential $\mathsf{\varphi }=-{\displaystyle \frac{{\mathsf{\omega }}^2{\mathrm{r}}^2}{2}}$.

At the center point, v=0 and ϕ=0. Therefore:

$${\left({\displaystyle \frac{\Delta f}{f}}\right)}_{SR/GR}=0$$

(6)

Traditional theories thus predict synchronization between the central rotating clock and the ground stationary clock.

(2) Prediction of φ(τ)

In the φ(τ) framework, τ dilation is directly governed by angular velocity ω (decoupled from radius r) [1,13]:

$${\displaystyle \frac{\Delta f}{f}}=-{\displaystyle \frac{{\omega }^2}{2{\Omega }_{max}^2}}$$

(7)

This offset scales with ω² and is observable even at r=0 (as long as ω≠0). Measuring the linear relationship between Δf/f and ω² allows indirect determination of Ωₘₐₓ without prior assumptions.

5.4. Secondary Verification Logic and Decision Criteria

To address uncertainties in "which quantity the atom clock responds to," this experiment adopts a "primary atom clock verification + secondary angular clock cross-testing" two-tiered decision logic to ensure rigor.

(1) Primary Verification (Atom Clock Comparison)

• Decision Criterion 1: If Δf/f ≠ 0 and scales as ω² (Eq. 7), φ(τ) is supported; τ is physically real. A fit of Δf/f vs. ω² yields Ωₘₐₓ without prior assumption.

•Decision Criterion 2: If Δf/f=0, we proceed to secondary verification to exclude the possibility that atomic clocks are insensitive to τ.

(2) Secondary Verification (Angular Clock Cross-Testing)

Replace the atom clock with OTAC/GTGC/SFQC angular clocks and repeat the angular velocity gradient experiment:

•Decision Criterion 3: If the angular clocks (OTAC/GTGC/SFQC) show τ dilation (τ_proper following Eq. 4), φ(τ) is supported, and atomic clocks are simply blind to τ.

•Decision Criterion 4: If no angular clock detects τ dilation, φ(τ) is falsified; τ would be a mathematical artifact.

5.5. Sensitivity Analysis and Feasibility Assessment

Under current technological limits, the experiment exhibits high feasibility.

(1) Effect Magnitude at Current Technology Limits

The predicted offset is Δf/f = ω²/ (2${\Omega }_{max}^2$). For ω = 300 rad/s, Δf/f ≈ 2.0×10⁻¹¹, which is many orders of magnitude above the sensitivity of current optical lattice clocks (≤1×10⁻¹⁸ [14]). At ω=50 rad/s, Δf/f≈5.5×10⁻¹³, still well within detectable limits.

(2) Systematic Error Control Strategies

•Common-Mode Suppression: The r = 0 design eliminates centrifugal forces (a = 0) and gravitational potential effects.

•Cross-Validation Redundancy: The three angular clocks (OTAC/GTGC/SFQC) rely on different physical principles, offering independent checks against systematic biases

This approach combines theoretical rigor with experimental feasibility, ensuring robust validation of the τ-dimensional hypothesis.

5.6. Why the Effect Was Previously Overlooked

The τ-dilation effect is uniform across the entire rotating frame (common-mode). Traditional Mössbauer rotor experiments perform differential measurements between the edge and center [15,16,17], which subtract out any uniform background shift. In such designs, the τ signal would likely be discarded as a constant baseline drift or instrumental offset. The zero-area central measurement proposed here specifically targets this common-mode effect, filling a long-standing blind spot in rotational time experiments.

A full review of the extensive Mössbauer literature is beyond the scope of this paper; the key point is that existing differential-edge experiments are not designed to detect a uniform ω² shift and therefore cannot be used to rule it out. Furthermore, Linear time clocks measure electronic transition frequencies (1/tₚᵣₒₚₑᵣ), may inherently lacking sensitivity to τ-dimensional evolution. This limitation resembles measuring temperature with a ruler—only specialized angular clocks (OTAC, GTGC, SFQC) designed to geometrically or quantum-mechanically capture τ can reveal its existence.

6. Discussion

6.1. Dual Interpretation of τ: Unity of Dimensional Time and Topological Discreteness

The angular time defined in this work exhibits dual physical properties—geometric continuity (dimensional time) and topological discreteness (dimensionless quantity)—which may offer new insights into the quantumization of time.

(1) Geometric Continuity (Dimensional Time)

At macroscopic scales, τ appears as a continuous manifold parallel to linear time t. Through the maximum angular velocity limit, τ acquires temporal dimension (unit: s_ang), enabling dynamic coupling with t via the local factor ξ_local and integration into existing differential geometric frameworks.

(2) Topological Discreteness (Dimensionless)

Reinterpreting τ’s definition: when rotating at maximal angular velocity, the time required to complete one full cycle (2π radians) is t₀=2π/Ωₘₐₓ. This defines a normalized time τ′=t/t₀=${\Omega }_{max}\bullet t/\left(2\pi \right)$. For uniform rotation (θ=Ωt), substituting Ω=Ω_max yields τ′=θ/2π. Here, τ′=1 strictly corresponds to one complete geometric rotation—each accumulation of 2π radians increases τ by one unit, embodying the discrete nature of angular time. However, this is not mere cyclic repetition; each rotation is accompanied by system state changes (energy, entropy, phase), consistent with the second law of thermodynamics (dS/dτ>0).

It is emphasized that these two interpretations are complementary facets of the same physical nature, serving distinct research objectives. The dimensional definition (τ ≡ θ/Ωₘₐₓ) underpins φ(τ)’s dual-time framework, granting τ independent physical status through coupling with linear time t via ξ_local. The dimensionless definition (τ′=θ/2π) aligns with rotational motion’s discrete counting feature, as evidenced by material fatigue life analysis—where mechanical design relies on stress cycle counts (N, i.e., τ) rather than duration [18]. Together, these definitions form a unified physical picture of τ’s "continuity and discreteness," enriching its theoretical foundation without contradiction.

6.2. Compatibility with Existing Theories: Expansion Rather Than Revolution

The φ(τ) framework is proposed as a complementary description for rotational dynamics, not as a refutation of special or general relativity. At low angular velocities and in purely translational settings, it reduces smoothly to the standard single-time description. Integration with general relativity can be pursued via the local co-boundary factor, which encodes gravitational potential and frame-dragging effects.

(1) Relationship with Special Relativity (SR)

SR describes translational motion relativity and has been extensively validated. φ(τ) treats SR’s linear time t as the translational evolution benchmark, introducing angular time τ for rotational scenarios. At low angular velocities (ω≪ Ωₘₐₓ), τ’s dilation effect reduces to linearity. The center-ground comparison experiment avoids traditional relativistic effects, focusing on τ’s rotational dilation to distinguish between theories.

(2) Integration with General Relativity (GR)

GR couples spacetime geometry with gravitational fields. The Kerr black hole’s frame-dragging and rotating singularities challenge single-time descriptions. φ(τ)’s dual-time framework provides a novel perspective: treating black hole rotation as τ’s evolution baseline can describe rotating spacetime curvature near horizons, while local co-boundary factors ξ_local incorporate gravitational potential effects to achieve dynamic coupling between t and τ. Weak gravitational field experiments (e.g., different altitudes on Earth) could probe ξ_local modulation by gravitational potentials, providing experimental evidence for φ(τ)–GR integration.

(3) Synergy with Quantum Mechanics

SFQC combines superconducting magnetic flux quantization (Φ₀=h/2e) with phase cycles, while OTAC leverages photon angular momentum (J=nℏ), both achieving precise quantum-level τ counting. This synergy may bridge quantum mechanics’ discreteness with general relativity’s continuity—a central issue in quantum gravity.

6.3. Limitations and Open Questions

Several important limitations should be acknowledged:

(1) The turbulent scaling law provides only indirect statistical evidence; direct proof of τ requires the zero-area experiment.

(2) The three angular-clock schemes remain conceptual designs; their practical realization and t-independence purity require experimental demonstration.

(3) The full mathematical structure of φ(τ) at the interface with general relativity is still under development.

For these reasons, the present work should be regarded as a testable theoretical proposal with encouraging indirect support, rather than an established result.

6.4. Outlook

If the zero-area experiment detects a nonvanishing ω² shift consistent with φ(τ), the implications would extend beyond fluid turbulence into relativistic physics, quantum rotation, and spacetime foundations. Even a null result would be valuable, as it would place a tight experimental bound on any possible angular-time effect and reinforce the standard single-time description under extreme rotation.

7. Conclusion

We have advanced the φ(τ) angular displacement spacetime framework by providing an operational definition of angular time τ, presenting zero-parameter turbulent scaling laws as the first macroscopic indirect evidence, and designing a binary crucial experiment under the zero-area limit.

The turbulent scaling result shows that a single topological constant derived from the 2π periodicity of τ can account for the full anomalous scaling of turbulence. The zero-area experiment offers an unambiguous falsification path: either a nonvanishing ω²-dependent frequency shift is observed at the rotation center, supporting the physical reality of τ, or the null result stands, reinforcing the standard single-time description even under extreme rotation.

Either outcome will advance our understanding of rotational spacetime. The experiment is feasible with existing technology and requires only a reconfiguration of existing clock-comparison setups to place the probe at the rotation center.