Revision date: 01 July 2025 | Authors: Mustafa Aksu, Grok 3, ChatGPT-4.5
Contents
- 1. Introduction
- 2. Fundamental Entities — Nodes
- 3. Emergent Space-Time
- 4. Node Interactions & Resonance
- 5. Observer-Dependence
- 6. Dimensionality & Frequency Regimes
- 7. Stability & Equilibrium
- 8. Mathematical Consistency & Zero-Arbitrariness
- 9. Methodological Principles
- 10. Applications & Future Research
- 11. Conclusion
1. Introduction
Relational Time Geometry (RTG) is built on a radical premise: space, time, and all physical phenomena emerge from relations among fundamental nodes. This document explains how RTG derives geometry and physics from oscillatory primitives—no pre-existing space or time—offering a unified, parameter-free framework for understanding the universe.
2. Fundamental Entities — Nodes
Nodes — intrinsic oscillators defined by frequency, phase, and spin — are the only primitives. All geometry, matter, and interactions emerge from their relationships (Emergence → space, time, particles).
| Quantity | Symbol | Role |
|---|---|---|
| Frequency | \(\omega_i\) | Intrinsic “tick rate” of node \(i\); energy \(E_i = \hbar \omega_i\). |
| Phase | \(\phi_i\) | Sets interference pattern with neighboring nodes. |
| Spin | \(s_i = \pm i\) | Binary orientation; higher spins emerge via coarse-graining. |
3. Emergent Space-Time
- Beat-frequency distance:
\( r_{ij} = \frac{2\pi c}{|\omega_i – \omega_j|} \)
Emerges from frequency differences, modulated by resonance. Nodes can be embedded in 2D–4D based solely on beat-frequency relations (see Two-Loop RG Derivation). - Intrinsic time for node \(i\):
\( t_i = \frac{\phi_i}{\omega_i} \) - Photon node: \(\omega = 0\); defines spatial extension but experiences no intrinsic time.
4. Node Interactions & Resonance
The resonance strength between nodes \(i\) and \(j\) is:
\( \mathcal{R}_{ij} = \frac{3}{4} \left[1 + \cos(\phi_i – \phi_j)\right] (1 + s_i s_j) e^{-(\omega_i – \omega_j)^2 / (\Delta\omega^*)^2} \)
This wave-function-inspired form is derived from node overlap (see Two-Loop RG Derivation).
The bond contribution to the Hamiltonian is:
\( E_{ij} = K \left|\omega_i – \omega_j\right| + J \mathcal{R}_{ij} + J_{\text{ex}} \sin(\phi_i – \phi_j) e^{-(\omega_i – \omega_j)^2 / \sigma^2} \)
Where:
- Kinetic: \( K \left|\omega_i – \omega_j\right| \), scales with frequency separation.
- Resonant binding: \( J \mathcal{R}_{ij} \), strengthens node coupling.
- Exchange phase-coupling: \( J_{\text{ex}} \sin(\phi_i – \phi_j) e^{-(\omega_i – \omega_j)^2 / \sigma^2} \), governs phase dynamics.
Constants \( K \), \( J \), \( J_{\text{ex}} \), and \( \sigma \) are derived via renormalization group (RG) methods (see Two-Loop RG Derivation).
5. Observer-Dependence
Measurements are relative to an observer node \((\omega_o, \phi_o, s_o)\). Distance to the observer is:
\( r_{io} = \frac{2\pi c}{|\omega_i – \omega_o|} \)
Physics is explicitly frame-dependent yet relationally consistent, akin to relativity without an absolute reference.
6. Dimensionality & Frequency Regimes
Critical bandwidth (from two-loop RG):
\( \Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1} \)
Derived via RG flow (see Two-Loop RG Derivation).
| Bandwidth ratio \(\delta\omega / \Delta\omega^*\) | Emergent dimension \(D\) | Typical structure |
|---|---|---|
| 0.0 – 0.5 | 2 | Planar resonance sheets |
| 0.5 – 1.0 | 3 | Stable 3-D shells (protons, nuclei) |
| 1.0 – 1.7 | 4 | Tightly-bound higher-D objects |
| > 1.7 | 5+ | Emergent 5D+ effects, under investigation |
These thresholds are emergent, not imposed, arising from resonance coupling strengths.
7. Stability & Equilibrium
Stable composites form where energy variation vanishes:
\( \frac{\partial E}{\partial (\text{degrees})} = 0 \)
For protons, Monte-Carlo simulations yield an equilibrium radius \( \approx 0.84 \, \text{fm} \) with a modest retune of \( J \) (\( \leq 8\% \)) within RG uncertainty bounds, consistent with zero-arbitrariness.
8. Mathematical Consistency & Zero-Arbitrariness
- All parameters are derived from two-loop RG around \( \Delta\omega^* \); no empirical fits are allowed, distinguishing RTG from effective field theories relying on phenomenological adjustments.
- Geometry is fully relational, with no external metric imposed.
9. Methodological Principles
- Empirical validation: Simulations must match measurable data (e.g., hadron radii, CMB peaks, gravitational-wave amplitudes).
- Relational consistency: Definitions trace back to node-node relations.
- Observer inclusion: Always specify an observer node.
- Explicit dimensional emergence: Show how \( D \) arises from bandwidth ratios via simulations.
- Zero arbitrariness: No free parameters beyond RG-derived constants.
10. Applications & Future Research
- Particle physics: Proton/\( \Delta \) stability, exotic states (e.g., pentaquarks with predicted radii \( \approx 1.2 \, \text{fm} \)).
- Quantum phenomena: Uncertainty and entanglement via resonance geometry (e.g., CHSH violations > 2.8).
- Cosmology: Expansion law from frequency drift, eliminating dark matter/energy (see [Cosmological Dynamics in RTG](/cosmological-dynamics-and-energy-framework-in-rtg/)).
- Thermodynamics & rotation: Heat and inertia from coarse-grained resonance (e.g., simulations of heat/entropy in clusters yield \( \sim 300 \, \text{K} \) approximation).
11. Conclusion
With a single RG-fixed bandwidth \( \Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1} \) and an updated resonance kernel, RTG provides a constrained, background-free framework for physics—from sub-femtometer scales to cosmological distances—without adjustable parameters. In summary, RTG offers a parameter-free, relationally emergent framework tracing everything from hadrons to cosmology—further simulation studies are underway to rigorously verify its predictive power.
For derivations, see Two-Loop RG Derivation. For terminology, consult the RTG Glossary.