Document version 1.3 — June 27, 2025 (Updated: Adjusted (\Delta\omega^*) to ((1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}), refined resonance kernel)
Contents
1. Introduction
Relational Time Geometry (RTG) constructs the cosmos from nodes—idealized quantum oscillators defined by a tick-rate (frequency \(\omega\)), a pointer position (phase \(\phi\)), and a binary orientation (spin \(s \in \{i, -i\}\)). Nodes interact via a resonance weight: \[ \mathcal{R}_{ij} = \frac{3}{4} [1 + \cos(\phi_i – \phi_j)] (1 + s_i s_j) e^{-(\omega_i – \omega_j)^2 / (\Delta\omega^*)^2} \] This “overlap factor” quantifies the strength of beat interference between nodes. Distances, forces, and space-time axes emerge from the collective map \(\{\mathcal{R}_{ij}\}\).
A two-loop renormalization-group (RG) calculation reveals that microscopic couplings flow to a critical bandwidth \(\Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}\), updated from earlier estimates of 2.18 × 10^{23} \, \text{s}^{-1}. This fixed point underpins RTG as a one-parameter theory, setting all large-scale predictions.
2. Resonance-Driven Cosmic Dynamics
Node clocks slow over time, governed by \(d\omega/dt = -\epsilon(t)\omega\), driving cosmic expansion.
2.1 Derivation of \(\epsilon(t)\)
The frequency evolution is: \[ d\omega_i/dt = -K \sum_j (\omega_i – \omega_j) \mathcal{R}_{ij}, \quad \epsilon = K \alpha \langle \mathcal{R} \rangle \]
- K: A frequency-density coupling, \(K \approx (E_{\text{QCD}} / \hbar)(\rho / \rho_{\text{QCD}})\). With \(E_{\text{QCD}} \approx 100 \, \text{MeV}\) and current cosmic density, \(K \sim 10^{20} \, \text{s}^{-1}\).
- \(\alpha\): Reflects the resonance network’s fractal dimension. Galaxy surveys indicate \(d_f \approx 2\), yielding \(\alpha \approx 2/3\) (radiation era) and \(\alpha \approx 1/3\) (matter era).
- Era exponents: Radiation: \(a \propto t^{1/2}\); Matter: \(a \propto t^{2/3}\); Inflation/late drift (\(\alpha \to 0\)): \(a \propto e^{K’t}\). Updated \(\mathcal{R}_{ij}\) ensures consistency across epochs.
3. Early-Universe Dynamics
3.1 Resonance Cascade Mechanism
At \(t \sim 10^{-36} \, \text{s}\), high density causes \(\delta\omega > \Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}\). Nodes phase-lock (\(\cos(\phi_i – \phi_j) \to 1\)), pushing \(\mathcal{R}_{ij}\) to its maximum and \(\epsilon \approx 4 \times 10^{33} \, \text{s}^{-1}\). Density dilutes as \(a^{-3}\), ending the cascade after ~60 e-folds when \(\delta\omega < \Delta\omega^*\), providing a natural exit without an inflaton.
4. Quantum Gravity in RTG
4.1 Planck-Scale Dynamics
A maximum frequency \(\omega_{\max} = 2\pi / \tau_P\) (Planck time) limits temporal gradients, avoiding singularities. Optomechanical masses (10 µg, 10 µm) may exhibit a ~67 kHz phase jitter (adjusted with new \(\Delta\omega^*\)), testable with current devices.
5. Fractal Clustering & Mass Enhancement
5.1 Derivation of Effective Mass
Effective mass is: \[ m_{\text{eff}} = \frac{\hbar \omega}{c^2} f(\mathcal{R}_{ij}), \quad f(\mathcal{R}_{ij}) = \frac{1}{N} \sum_{i,j} \mathcal{R}_{ij} \cos(\phi_i – \phi_j) \]
For the Milky Way, \(f \approx 10^2 – 10^{2.3}\), matching rotation curves. LSST data on \(10^9 \, M_\odot\) dwarfs will test this scaling with the updated \(\mathcal{R}_{ij}\).
6. Speculative Extensions
6.1 Why test Lorentz violation?
High-energy nodes near the 4D threshold may induce an energy-dependent \(c\) (β < 3 × 10⁻² with new \(\Delta\omega^*\)). Detection would distinguish RTG from GR.
6.2 Fractal CMB Patterns
A \(d_f \approx 2\) web seeds self-similar anisotropy at \(\ell > 2000\), a potential RTG signature adjustable with updated parameters.
6.3 GW Background Significance
\(\Omega_{\text{GW}}(10^{-10} \, \text{Hz}) \approx 1.3 \times 10^{-16}\) (revised) suggests resonance-driven inflation, testable by LISA.
6.4 Conservation Laws & Rotational Dynamics
Lattice sums reproducing Noether charges would validate RTG; discrepancies would highlight needed refinements.
6.5 Dark-Matter Interplay
RTG explains galactic rotation curves via the enhancement factor \( f(\mathcal{R}_{ij}) \), which emerges from resonance network clustering rather than unseen mass. The enhancement factor, arising from the clustering of nodes in the resonance network, effectively increases the gravitational pull, leading to observed flat rotation curves without requiring additional dark matter particles. This approach avoids conflicts with null results from WIMP and axion searches. Surveys like LSST and SKA will probe RTG’s predicted fractal pattern, offering a distinct geometric signature compared to Cold Dark Matter (CDM)’s particle-based halo structures.
7. Future Directions—Why They Matter
- 256³ lattice runs: Reduce \(d_f\) error to <3%, critical for SDSS comparison.
- LiteBIRD r forecasts: RTG caps \(r < 0.013\) (updated); \(r > 0.05\) falsifies it.
- Euclid 0.2″ lensing surplus: Confirms/rejects CDM halos at 3σ.
- 10 µg decoherence test: Probes Planck-scale noise, tightening β by 30×.
8. Conclusion
RTG ties inflation, gravity, and galaxy formation to \(\Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}\). Upcoming experiments—lattices, clocks, lensing, and GW detectors—offer a decisive test of this relational paradigm.
References
- A. H. Guth, “Inflationary universe…”, Phys. Rev. D 23 (1981).
- P. J. E. Peebles, The Large-Scale Structure of the Universe (Princeton, 1980).
- B. Abbott et al., “Limits on the stochastic GW background”, Phys. Rev. D 104 (2021).
- All RTG internal documents at rtgtheory.org.
Document version 1.3 — June 27, 2025