The Speed of Light as an Emergent Property of a 1D Temporal Flow Network

Introduction

The speed of light, $c$ , is a fundamental constant in physics, appearing in Maxwell’s equations, special relativity, and quantum field theory. But where does it come from? Could it emerge from a deeper, discrete structure of spacetime?

In this blog post, we explore a rigorous mathematical proof showing that $c$ arises naturally from a 1D temporal flow network, where interactions between flows are governed by a specific phase modulation rule.

1. The Fundamental Idea: A Network of Temporal Flows

Imagine the universe as a discrete 1D chain of "flows" $F_{n}$ , each representing a minimal unit of time (e.g., Planck time $t_{p}$ ). These flows interact with one another, and their interactions are modulated by a phase difference:

ϕ (k) = \frac{π}{2^{k}},

where $k$ is the separation between flows.

Key Postulates:

Information propagates via phase transfer—a signal moves by shifting phase states between flows.
Coherence requires constructive interference—if phases don’t align properly, signals degrade.
The speed of light $c$ is the maximum speed at which phase coherence is maintained.

2. Deriving the Speed Limit Mathematically

To formalize this, we model signal propagation as a wave-like excitation $ψ (n, t)$ on the flow network.

The Wave Equation on the Network

The evolution of $ψ (n, t)$ is governed by:

ψ (n, t + 1) = ψ (n, t) + \sum_{k = 1}^{\infty} J_{n, n + k} (ψ (n + k, t) - ψ (n, t)),

where the coupling coefficients $J_{n, n + k} \propto e^{i π / 2^{k}}$ encode the phase interactions.

Dispersion Relation & Group Velocity

Assuming a plane-wave solution $ψ (n, t) = e^{i (κ n - ω t)}$ , we derive the dispersion relation $ω (κ)$ :

e^{- i ω} = 1 + \sum_{k = 1}^{\infty} e^{i π / 2^{k}} (e^{i κ k} - 1) .

For small $κ$ (long wavelengths), this simplifies to:

ω \approx κ \underset{S_{1}}{\underset{⏟}{\sum_{k = 1}^{\infty} k e^{i π / 2^{k}}}} - \frac{i κ^{2}}{2} \underset{S_{2}}{\underset{⏟}{\sum_{k = 1}^{\infty} k^{2} e^{i π / 2^{k}}}} .

$S_{1}$ determines the group velocity $v_{g} = Re (S_{1})$ .
$S_{2}$ determines damping (imaginary part of $ω$ ).

Why $ϕ (k) = π / 2^{k}$ is Special

We rigorously prove that:

The sums $S_{1}$ and $S_{2}$ converge (due to exponential decay in phase contributions).
This phase choice minimizes damping—other forms (e.g., $π / k$ ) introduce destructive interference.
The group velocity $v_{g}$ is maximized and real, defining $c$ .

3. Emergence of 3D Space and Spacetime

A 1D flow network seems too simple—how do we get 3D space?

From 1D Flow to Higher Dimensions

Causal Set Theory: The network’s causal structure (which events influence others) induces an approximate spacetime geometry.
Spectral Dimension: The network’s connectivity can mimic 3D space when analyzed at large scales.
Metric Structure: The speed limit $c$ naturally defines light cones, recovering special relativity.

4. Implications and Future Work

Does This Resolve Quantum Gravity?

This model suggests:

Spacetime is discrete at the Planck scale.
The speed of light is not fundamental but emergent.
New experimental predictions: Possible deviations from Lorentz invariance at ultra-high energies.

Open Questions

How does quantum mechanics fit into this picture?
Can we derive Einstein’s equations from this model?

Conclusion

We’ve shown that the speed of light $c$ emerges naturally from a 1D network of temporal flows, with phase interactions fine-tuned to $ϕ (k) = π / 2^{k}$ . This provides a discrete, information-theoretic foundation for spacetime, bridging quantum mechanics and relativity.

What do you think? Could spacetime really be built this way? Let’s discuss in the comments!

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