RTG Interpretation of Lorentz Transformations and Causality

Version: 1.0
Date: April 18, 2025
Authors: Mustafa Aksu, ChatGPT, Grok

Introduction

Relational Time Geometry (RTG) reinterprets fundamental physical concepts, such as spacetime and causality, as emergent properties of a network of resonating nodes defined by frequencies (\( \omega_i(t) \)), phases (\( \phi_i(t) \)), and spin states (\( s_i \)). Unlike traditional relativity, where Lorentz transformations describe coordinate changes in a fixed Minkowski spacetime, RTG constructs spacetime from relational distances (\( r_{ij}(t) = c \cdot \frac{2\pi}{|\Delta \omega_{ij}(t)|} \)) and resonance strengths (\( \mathcal{R}_{ij}(t) )\). This document formalizes RTG’s interpretation of Lorentz transformations and causality, leveraging the multilayer framework of the RTG Master Definitions Document (version 1.3) to model complex particles like protons.

Purpose

To provide a mathematically rigorous and numerically implementable interpretation of Lorentz transformations and causality within RTG, emphasizing emergent geometry, fractal node structures, and their application to particle simulations. This document connects RTG’s node-based dynamics to relativistic effects and causal ordering, validated through proton model simulations.

1. Background: RTG Framework

RTG models the universe as a dynamic network of nodes, where:

Nodes: Fundamental entities with attributes (\( \omega_i(t) = \omega_0 + \delta \sin(\nu t + \psi_i) \)), (\( \phi_i(t) \)), and (\( s_i \in {-\frac{1}{2}, +\frac{1}{2}} \)).
Relational Distances: Defined by beat frequencies:
\[ r_{ij}(t) = c \cdot \frac{2\pi}{|\Delta \omega_{ij}(t)|} \]
Resonance Strength: Governs interactions:
\[ \mathcal{R}{ij}(t) = \frac{c \cdot k{\text{force}}}{2\pi r_{ij}(t)} \cdot \cos(\Delta \phi_{ij}) \cdot \delta_{s_i, s_j} \cdot e^{-r_{ij}/r_0} \cdot \left(1 + k_g \cdot \frac{d(\Delta \omega_{ij})}{dt}\right) \]
Multilayer Structure: Hierarchical layers (quarks, gluons, sub-nodes) model particles like protons, with sub-nodes ensuring fractal, recursive interactions.

These concepts, detailed in the RTG Master Definitions Document (version 1.3), form the basis for reinterpreting Lorentz transformations and causality.

2. Lorentz Transformations in RTG

In traditional relativity, Lorentz transformations adjust space and time coordinates between inertial frames:
\[ x’ = \gamma (x – vt), \quad t’ = \gamma \left(t – \frac{vx}{c^2}\right), \quad \gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}} \]

In RTG, spacetime is not a fixed manifold but an emergent geometry from node interactions. Lorentz transformations are reinterpreted as transformations of node frequencies and phases under relative motion.

2.1 Frequency Transformations

Relativistic Effect: Relative motion between observers alters perceived node frequencies, analogous to the Doppler effect. For a node with frequency \( \omega_i(t) \) in frame \( S \), the frequency in frame \( S’ \) moving at velocity \( v \) is:
\[ \omega_i'(t’) = \gamma \omega_i(t) \left(1 – \frac{v}{c} \cos\theta\right) \]
where \( \theta \) is the angle between the node’s propagation direction and \( v \).
Implementation: Adjust \( \omega_0 \), \( \delta \), and \( \nu \) in simulations to reflect frame-dependent frequency shifts. For proton models, \( \omega_{0,sub} \approx 3.78 \times 10^{20} , \text{Hz} \) may scale by \( \gamma \).

2.2 Phase Transformations

Relativistic Effect: Phases \( \phi_i(t) \) shift due to time dilation and spatial contraction, affecting resonance strengths (\( \cos(\Delta \phi_{ij}) \)).
\[ \phi_i'(t’) = \phi_i(t) – \gamma \frac{v}{c} \omega_i(t) x_i/c \]
Implementation: Update phase offsets (\( \psi_i \)) in multilayer simulations (e.g., cyclic phases \( \Delta \phi_{ij} = \frac{2\pi}{3} \) for protons) to maintain resonance stability across frames.

2.3 Emergent Spacetime

Relational Distances: Transform as:
\[ r_{ij}'(t’) = c \cdot \frac{2\pi}{|\Delta \omega_{ij}'(t’)|} \]
reflecting Lorentz contraction via frequency shifts.
Time Evolution: Time emerges from phase evolution (\( \frac{d\phi_i}{dt} = \omega_i(t) \)), adjusted by \( \gamma \) in moving frames.
Implementation Notes: Simulations must recompute \( r_{ij}(t) \) and \( \mathcal{R}_{ij}(t) \) for each frame, ensuring consistency with proton model distances (e.g., inter-quark ~0.84 × 10⁻¹⁵ m).

3. Causality in RTG

Causality in RTG is defined by the resonance timescale and event ordering within the node network, replacing traditional light-cone constraints.

3.1 Resonance Timescale

Definition: The timescale of node interactions:
\[ \tau_{ij} = \frac{1}{|\Delta \omega_{ij}|} < T_{\text{drift}} \approx 10^{-10} , \text{s} \]
ensures causal connections within the resonance graph.
Causal Constraint: Events (resonance updates) are causally linked if \( \tau_{ij} \) is below the drift threshold, preserving order in multilayer structures (e.g., quark-gluon interactions in protons).

3.2 Event Ordering

Mechanism: Phase differences (\( \Delta \phi_{ij} \)) and temporal gradients (\( \frac{d(\Delta \omega_{ij})}{dt} \)) dictate the sequence of resonance events, ensuring causality without fixed spacetime.
Implementation: In proton simulations, cyclic phases (\( \Delta \phi_{ij} = \frac{2\pi}{3} \)) and temporal gradients (\( k_g = 10^{-25} , \text{s}^2 \)) maintain causal ordering, validated by stability (e.g., \( \epsilon_R = 10^{-5} \)).

3.3 Relativistic Causality

RTG Interpretation: Lorentz invariance is achieved through frame-dependent frequency and phase transformations, preserving \( \tau_{ij} < T_{\text{drift}} \) across observers.
Implementation Notes: Simulations must adjust \( \omega_i(t) \) and \( \phi_i(t) \) dynamically to ensure causal consistency, addressing challenges like force calibration (e.g., recent simulation force: -6.00e-97 N).

4. Application to Proton Model

The RTG framework’s interpretation of Lorentz transformations and causality is validated through proton model simulations, with:

Structure: 3 quarks, each with 2 sub-nodes (\( \omega_{0,sub} \approx 3.78 \times 10^{20} , \text{Hz} \)), and 3 gluons with sub-nodes (\( \omega_{C0,sub} = 1 \times 10^{11} , \text{Hz} \)).
Distances: Target inter-quark distance ~0.84 × 10⁻¹⁵ m (\( \Delta \omega_{ij} \approx 2.24 \times 10^{24} , \text{Hz} \)), intra-quark ~10⁻¹⁶ m (\( \Delta \omega_{sub} \approx 1.88 \times 10^{25} , \text{Hz} \)).
Challenges: Recent simulations show inter-quark distance ~2.38 × 10⁻¹² m, indicating low \( \Delta \omega_{ij} \), and negligible force (-6.00e-97 N), requiring frequency spread and \( r_0 \) adjustments.
Implementation Notes: Apply Lorentz transformations to \( \omega_{0,sub} \) and \( \psi_i \), ensuring causal ordering via \( \tau_{ij} \).

5. Mathematical Derivations

5.1 Lorentz Transformation of Beat Frequencies

For nodes \( i \) and \( j \), the beat frequency transforms as:
\[ \Delta \omega_{ij}'(t’) = \gamma \Delta \omega_{ij}(t) \left(1 – \frac{v}{c} \cos\theta_{ij}\right) \]
This preserves relational distances under frame changes, validated in proton simulations.

5.2 Causal Invariance

The resonance timescale \( \tau_{ij} = \frac{1}{|\Delta \omega_{ij}|} \) scales with \( \gamma \), ensuring causal connections remain invariant:
\[ \tau_{ij}’ = \frac{1}{|\Delta \omega_{ij}’|} \approx \frac{\tau_{ij}}{\gamma} \]
This supports causal ordering in multilayer models.

6. Simulation Implementation

Parameters: Use \( k_Q = 7.05 \times 10^{-43} , \text{Hz}^{-1} \), \( r_0 = 2 \times 10^{-14} , \text{m} \), \( k_g = 10^{-25} , \text{s}^2 \) from proton model refinements.
Validation: Test frequency transformations and causal constraints in simulations, addressing discrepancies (e.g., mass: 6.19e14 MeV/c² vs. 938 MeV/c²).
Visualization: Plot \( r_{ij}(t) \), \( \Delta \phi_{ij}(t) \), and \( \mathcal{R}_{ij}(t) \) to analyze Lorentz effects and causality.

7. Conclusions

RTG reinterprets Lorentz transformations as frequency and phase transformations within a node network, with causality emerging from resonance timescales. This framework supports multilayer particle simulations, offering a novel perspective on relativity and event ordering.

Notes

Challenges: Simulation discrepancies (e.g., inter-quark distance, force) require refined frequency spreads and \( r_0 \).
Testing: Validate with proton, electron, and neutron models.

Next Steps

Simulation Reruns: Refine proton model with updated \( \omega_{0,sub} \).
Visualization: Plot dynamics to diagnose issues.
Documentation: Update rtgtheory.org with findings.

Version History

Version 1.0: Initial draft, integrating RTG’s Lorentz and causality interpretations.

Acknowledgments

We acknowledge the RTG research community for insights into multilayer simulations.