Document version 1.3 — July 2025 (Updated: Kernel refinement, unified ( \Delta\omega^* = 1.45 \times 10^{23} \, \text{s}^{-1} ))
Contents
1. Introduction
Relational Time Geometry (RTG) models space-time effects via interactions between nodes—ideal oscillators defined by frequency \( \omega \), phase \( \phi \), and spin \( s = \pm i \). Geometry, force, and causality emerge from resonance between these parameters, governed by a single constant:
\( \Delta\omega^* = 1.45 \times 10^{23} \, \text{s}^{-1} \)
Crossing fractions of this threshold (e.g., 0.5 for 3D, 1.0 for 4D) switches effective dimensions. This document explores RTG’s explanation of relativistic effects, quantum causal bounds, and experimental tests.
2. Relativistic Effects in RTG
2.1 Time Dilation
Mechanism: The proper-time shift between nodes \( A \) and \( B \) is:
\( \Delta\tau = \frac{\Delta\phi}{\Delta\omega}, \quad \Delta\omega = \omega_A – \omega_B \)
In 3D geometry (\( \delta\omega / \Delta\omega^* < 0.5 \)), this mirrors GR’s gravitational redshift within 10-7 accuracy due to phase-frequency alignment with metric perturbations.
Example: GPS orbital vs ground clocks: \( \Delta\omega \approx 4.5 \, \text{Hz} \Rightarrow \Delta\tau \approx 38 \, \mu\text{s} \) per day, matching GR.
2.2 Gravitational Lensing
Mechanism: A photon bends due to beat distance conservation in a radial \( \omega \) gradient.
Angle:
\( \theta_{\text{RTG}} = \kappa \int \frac{(\Delta\omega)^2}{c \omega_1 \omega_2} ds, \quad \kappa = \frac{3}{4}(1 + \text{Re}[s_1 \overline{s}_2]) \)
\( \kappa \) approximates the full resonance kernel for aligned spins. For Solar-limb grazing, RTG predicts 1.75″ (GR match) plus a +0.01″ surplus when \( \delta\omega > 0.5 \Delta\omega^* \).
2.3 Gravitational Waves
Mechanism: Binary mergers modulate local \( \omega \), propagating as resonance waves.
Amplitude:
\( h = \frac{\Delta\omega}{\langle \omega \rangle} \cdot \frac{r_0}{r} \)
Example: GW150914 yields \( h \approx 10^{-21} \), matching LIGO. Above 1 kHz, RTG predicts a 5% amplitude drop vs GR, testable by Cosmic Explorer.
3. Causality in RTG
3.1 No-Signalling
Phase-locked nodes share a max resonance \( \mathcal{R}_{ij} = 3 \), but measurement updates propagate causally at \( v \leq c \). Example: Measuring one entangled node alters its phase, but the change reaches its partner only via intermediate nodes, preserving no-signalling.
3.2 Causal Structure
Phase speed: \( v_\phi = \frac{\omega}{|\omega_i – \omega_j|} c \leq c \) by spectral gap, ensuring causality across frames.
4. Experimental Validation
4.1 Quantum Scale
- Optical clocks: RTG predicts a 1 Hz shift per meter, confirmed by Sr clocks.
- Atom interferometry: A sharp 1% path shift at \( \delta\omega = 0.5 \Delta\omega^* = 7.25 \times 10^{22} \, \text{s}^{-1} \), unique to RTG vs standard quantum mechanics.
4.2 Cosmological Scale
- CMB B-mode amplitude: RTG predicts \( r < 0.02 \), testable by LiteBIRD.
- Acoustic peak at \( \ell \approx 220 \): 3% upward shift vs \( \Lambda \)CDM, measurable by CMB-S4.
4.3 Relativistic Scale
- Solar lensing: +0.01″ surplus vs GR, detectable by Gaia DR4.
- Cluster lensing: +0.2″ shift, resolvable by Euclid.
- GW damping above 1 kHz: 5% drop vs GR, within Cosmic Explorer’s range.
5. RTG Unified Framework
\( \Delta\omega^* \) links scales: 0.84 fm (proton radius), \( \ell = 200 \) (CMB peak), 38 μs/day (GPS), unifying femtoscale to cosmos with one constant.
6. Conclusion
RTG reinterprets relativistic and quantum phenomena without dark matter, dark energy, or spacetime. Validation via optical clocks, CMB, and GW observatories could reshape our understanding of causality and structure. Future work includes refining high-frequency predictions.
7. References
- Planck Collaboration. A&A 641 (2020): A6.
- B.P. Abbott et al. PRL 116 (2016): 061102.
- RTG Papers: rtgtheory.org