Revision date: 01 July 2025 | Authors: Mustafa Aksu, Grok 3, ChatGPT-4.5
Contents
1. Introduction
Relational Time Geometry (RTG) discards any fixed space-time background, reconstructing classical geometric concepts—point, line, plane, curvature—purely from interference among node properties: frequency \(\omega\), phase \(\phi\), and binary spin \(s = \pm i\). This page maps these classical ideas onto their RTG analogues, provides explicit algebraic definitions, and illustrates dimensional emergence with toy models, guided by the resonance kernel \(\mathcal{R}_{ij} = \frac{3}{4} \left[1 + \cos(\phi_i – \phi_j)\right] (1 + s_i s_j) e^{-(\omega_i – \omega_j)^2 / (\Delta \omega^*)^2}\), which governs interaction strength based on phase alignment, spin coupling, and frequency matching.
2. Classical ⟷ RTG Dictionary
Classical concept | RTG counterpart | Key definition / formula |
---|---|---|
Point | RTG dot | \(z_i(t) = s_i e^{i(\omega_i t + \phi_i)}\) |
Line (length) | Resonance loop | \(r_{ij} = \frac{2\pi c}{|\omega_i – \omega_j|}\) |
Plane / Space | Resonance lattice | \(Z_{\text{space}}(t) = \sum_k w_k s_k e^{i(\omega_k t + \phi_k)}\), where \(w_k\) is a weighting factor reflecting node influence |
Curvature | Phase–spin gradient | \(\mathcal{K}_{ij} = \frac{\partial}{\partial r_{ij}} \left[ \left| e^{i\phi_i} s_i + e^{i\phi_j} s_j \right|^2 \right]\), a discrete gradient approximating classical curvature |
Observer frame | Observer-relative geometry | \(G_{\text{obs}}(t) = \sum_i s_i e^{i[(\omega_i – \omega_o)t + (\phi_i – \phi_o)]}\) |
Higher-D axis | Independent beat-distance axis | Appears at \(\delta\omega / \Delta\omega^* = 0.5, 1.0, 1.7, \dots\), driven by resonance strength |
3. Detailed Formulations
- RTG dot (0-D): A single node with complex-spin marker \(s_i = \pm i\).
- RTG line (1-D): Interference of two dots, with length \(r_{ij}\) fixed by the beat-frequency term.
- RTG plane / lattice (2-D, 3-D, …): Coherent collection of nodes; stability requires neighboring phase differences \(\Delta\phi \lesssim \pi/4\).
- Emergent curvature: Large phase–spin gradients (\(\mathcal{K}_{ij}\)) correlate with strong-field regions, reflecting classical curvature via node interactions.
- Multi-D emergence: New axes emerge at bandwidth thresholds \(\delta\omega / \Delta\omega^* = 0.5, 1.0, 1.7, \dots\), driven by the resonance kernel’s frequency dependence.
- Inter-plane coupling: Coupling between resonance sheets depends on angle \(\theta\) and phase offset, approximated as \(\mathcal{R}_{12} \propto \left| e^{i(\Delta\phi + \theta)} \right|^2 / r_{12}\).
4. Numerical Toy Models of Dimensional Emergence
Model | Bandwidth ratio | Geometric order parameter | Result |
---|---|---|---|
4-node square (2 → 3-D test) | \(\delta\omega / \Delta\omega^*\) | Tetrahedral volume \(V_4\), computed via Cayley-Menger determinant | \(V_4 = 0\) for < 0.50, jumps > 0 for \(\geq 0.50\) |
5-node “pyramid” (3 → 4-D test) | \(\delta\omega / \Delta\omega^*\) | 4-simplex volume \(V_5\), computed via Cayley-Menger determinant | \(V_5 = 0\) for < 1.00, rises for \(\geq 1.00\) |
Results verified on \(32^3\) Monte-Carlo lattices (\(\beta = 1.0\), 1000 iterations, updated resonance kernel), with node spacing varied to test thresholds.
Placeholder: Consider a plot of \(V_4\) or \(V_5\) vs. \(\delta\omega / \Delta\omega^*\) to visualize dimensional jumps.
5. Two-Loop RG Anchor
Critical bandwidth (updated):
\[ \Delta \omega^* = (1.45 \pm 0.08) \times 10^{23} \; \text{s}^{-1} \]
All dimensional transitions are quantized based on multiples of \(\Delta \omega^*\), grounded in two-loop RG analysis (see Two-Loop RG Derivation).
6. Practical Take-Aways
- Curvature–field unification: Phase–spin gradients map to electromagnetic or strong fields, offering a unified geometric interpretation.
- Fractal dimensionality: Mixed-D domains (e.g., 2 < D < 3) emerge from nested frequencies, relevant near quantum-gravity horizons.
- Laboratory handle: Adjust cavity mode spacing around \(0.5 \, \Delta \omega^*\) to observe a sharp drop in coherence time, a testable RTG signature.
Related reading: Mathematical Foundations | Two-Loop RG Derivation.