Emergent Physics from Temporal Flow: Newton, Maxwell, Einstein, and Quantum Theory from First Principles

John Gavel

Introduction

Temporal Flow Physics (TFP) is a theoretical framework proposing that time is fundamental, and all familiar physical structures—space, particles, energy, and forces—emerge from quantized causal interactions between discrete temporal flows. Instead of assuming pre-existing spatial dimensions, TFP derives space as an emergent property of comparisons between flows.

This post explores how fundamental physical laws naturally emerge from the first principles of TFP. We begin by deriving Newton’s laws from flow accumulation principles, then show how Maxwell’s equations arise from entropy-regulated causal connectivity. We next examine how Einstein’s field equations emerge from flow distortions. Finally, we delve into Quantum Mechanics (QM) and Quantum Field Theory (QFT), exploring how wave-like behavior, uncertainty, and gauge interactions arise through entropy-mediated causal structures.

Entropy: The Core Regulatory Field in TFP

Before deriving physics laws, we must clarify entropy’s role in TFP. Unlike thermodynamic entropy (which relates to heat and disorder) or Shannon information entropy (which quantifies uncertainty), TFP entropy functions as a causal regulator that dictates which flows interact and segment into localized structures.

Mathematically, entropy at node $ii$ is defined as:

$$S_i=\exp (-\frac{(i-i_0{)}^2}{2{\sigma }^2})S\_i\; =\; \backslash exp\; \backslash left(\; -\backslash frac\{(i\; -\; i\_0)^2\}\{2\backslash sigma^2\}\; \backslash right)$$

where $\sigma \backslash sigma$ controls the width of entropy localization. The contrast between neighboring entropy states modulates interaction weights:

$$W_{ij}=\exp (-\frac{\mathrm{\mid }\mathrm{\Delta }{F}_{ij}{\mathrm{\mid }}^2}{ϵ+(\mathrm{\Delta }{S}_{ij}{)}^2})W\_\{ij\}\; =\; \backslash exp\; \backslash left(\; -\backslash frac\{|\backslash Delta\; F\_\{ij\}|^2\}\{\backslash epsilon\; +\; (\backslash Delta\; S\_\{ij\})^2\}\; \backslash right)$$

• High entropy contrast suppresses causal connectivity → leading to discrete, coherent particle-like segments.

• Low entropy contrast preserves continuous flows → enabling long-range coherence.

This entropy mechanism ensures that physical structures naturally emerge as modular, localized units, rather than being arbitrarily imposed.

1. Newton’s Laws from Flow Accumulation

Newton’s laws describe motion and force interactions. In TFP, motion corresponds to the accumulation of flow misalignments, regulated by entropy barriers.

First Law (Inertia)

Newton’s first law states that objects remain in motion unless acted upon by a force. In TFP, a flow state $F_{i}F\_i$ remains unchanged unless influenced by surrounding entropy gradients:

$$\frac{d{F}_i}{dt}=0,\text{if }\sum _j{W}_{ij}\mathrm{\Delta }{F}_{ij}=0\backslash frac\{dF\_i\}\{dt\}\; =\; 0,\; \backslash quad\; \backslash text\{if\; \}\; \backslash sum\_j\; W\_\{ij\}\; \backslash Delta\; F\_\{ij\}\; =\; 0$$

This means a system with uniform entropy distribution evolves without resistance, mirroring inertia.

Second Law (Force = Mass × Acceleration)

The acceleration of a node in TFP corresponds to the rate at which flow misalignment accumulates, defining a force-like quantity:

$$\sum _j{W}_{ij}\mathrm{\Delta }{F}_{ij}=M_i\frac{d^2{x}_i}{d{t}^2}\backslash sum\_j\; W\_\{ij\}\; \backslash Delta\; F\_\{ij\}\; =\; M\_i\; \backslash frac\{d^2\; x\_i\}\{dt^2\}$$

where:

• $W_{ij}W\_\{ij\}$ represents entropy-modulated interaction strength.

• $M_{i}M\_i$ arises from flow resistance effects, acting as an emergent mass.

Third Law (Action-Reaction)

Since interactions are relational in TFP, force propagation must conserve momentum:

$$\mathrm{\Delta }{F}_{ij}=-\mathrm{\Delta }{F}_{ji}\backslash Delta\; F\_\{ij\}\; =\; -\backslash Delta\; F\_\{ji\}$$

This ensures that flow distortions always affect neighboring nodes in equal and opposite ways, mirroring Newton’s third law.

2. Maxwell’s Equations from Entropy-Driven Gauge Fields

Maxwell’s equations govern electromagnetic interactions. In TFP, gauge fields emerge naturally from entropy-regulated causal connectivity.

Gauss’s Law (Charge Sources Fields)

Classically:

$$\mathrm{\nabla }\cdot \mathbf{E}=\rho \backslash nabla\; \backslash cdot\; \backslash mathbf\{E\}\; =\; \backslash rho$$

In TFP, entropy gradients generate local flow accumulation:

$$\mathrm{\nabla }\cdot \mathrm{\Delta }{S}_i={\sigma }_i\backslash nabla\; \backslash cdot\; \backslash Delta\; S\_i\; =\; \backslash sigma\_i$$

where ${\sigma }_i\backslash sigma\_i$ acts as an emergent charge density arising from entropy flow misalignment.

Faraday’s Law (Changing Fields Induce Circulation)

Classically:

$$\mathrm{\nabla }\times \mathbf{E}=-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t}\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\{\backslash partial\; t\}$$

In TFP:

$$\mathrm{\nabla }\times \mathrm{\nabla }S=-\frac{\mathrm{\partial }A}{\mathrm{\partial }t}\backslash nabla\; \backslash times\; \backslash nabla\; S\; =\; -\backslash frac\{\backslash partial\; A\}\{\backslash partial\; t\}$$

showing how coherent entropy flows induce phase curvature, similar to electromagnetic induction.

3. Einstein’s Field Equations from Flow Metric Emergence

In General Relativity (GR), spacetime curvature is driven by mass-energy density:

$$G_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }G\_\{\backslash mu\backslash nu\}\; =\; \backslash frac\{8\backslash pi\; G\}\{c^4\}\; T\_\{\backslash mu\backslash nu\}$$

In TFP, curvature is caused by entropy misalignment driving causal flow distortions:

$$g_{\mu \nu }=\sum _j{W}_{ij}\mathrm{\Delta }{F}_{ij}g\_\{\backslash mu\backslash nu\}\; =\; \backslash sum\_j\; W\_\{ij\}\; \backslash Delta\; F\_\{ij\}$$

where:

• $g_{\mu \nu }g\_\{\backslash mu\backslash nu\}$ acts as the emergent metric.

• $W_{ij}W\_\{ij\}$ governs interaction strength, defining curvature effects.

This suggests that spacetime is not fundamental, but an emergent geometric structure shaped by entropy-modulated causal flows.

4. Quantum Mechanics & Quantum Field Theory in TFP

Quantum States as Flow Configurations

Instead of presupposing probability waves, quantum behavior emerges from coherent flow interactions:

$$F_i(t)=A_i{e}^{i{\theta }_i(t)}F\_i(t)\; =\; A\_i\; e^\{i\; \backslash theta\_i(t)\}$$

where:

• $A_{i}A\_i$ represents flow amplitude.

• ${\theta }_i(t)\backslash theta\_i(t)$ evolves dynamically via entropy gradients:

$$\frac{d{\theta }_i}{dt}=\sum _j{W}_{ij}\mathrm{\Delta }{S}_{ij}\backslash frac\{d\; \backslash theta\_i\}\{dt\}\; =\; \backslash sum\_j\; W\_\{ij\}\; \backslash Delta\; S\_\{ij\}$$

Uncertainty Principle from Flow Misalignment

Instead of assuming Heisenberg’s uncertainty principle:

$$\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\mathrm{\hslash }}{2}\backslash Delta\; x\; \backslash Delta\; p\; \backslash geq\; \backslash frac\{\backslash hbar\}\{2\}$$

TFP derives uncertainty from entropy-driven causal constraints:

$$\mathrm{\Delta }{F}_i\mathrm{\Delta }{S}_i\ge ϵ\backslash Delta\; F\_i\; \backslash Delta\; S\_i\; \backslash geq\; \backslash epsilon$$

where $ϵ\backslash epsilon$ regularizes low-scale entropy effects, setting a minimal limit for precise flow alignment.

Quantum Fields as Continuous Flow Networks

Standard quantum field theory treats fields as continuous:

$$\mathrm{\square }\varphi +m^2\varphi =0\backslash Box\; \backslash phi\; +\; m^2\; \backslash phi\; =\; 0$$

TFP’s entropy-weighted formulation gives:

$$\sum _j{W}_{ij}\mathrm{\Delta }{F}_{ij}+m_{\text{eff}}^2{F}_i=0\backslash sum\_j\; W\_\{ij\}\; \backslash Delta\; F\_\{ij\}\; +\; m\_\{\backslash text\{eff\}\}^2\; F\_i\; =\; 0$$

where $m_{\text{eff}}m\_\{\backslash text\{eff\}\}$ arises from entropy gradients controlling interaction strength, making mass an entropy-coherent emergent property.

Gauge Interactions as Flow Adjustments

TFP maps QED’s interaction Lagrangian:

$${\mathcal{L}}_{\text{int}}=-e{A}_{\mu }\bar{\psi }{\gamma }^{\mu }\psi \backslash mathcal\{L\}\_\{\backslash text\{int\}\}\; =\; -e\; A\_\{\backslash mu\}\; \backslash bar\{\backslash psi\}\; \backslash gamma^\backslash mu\; \backslash psi$$

to entropy-weighted causal effects:

$${\mathcal{L}}_{\text{TFP}}=-\xi ({\mathrm{\partial }}_{\mu }S)F_i^2\backslash mathcal\{L\}\_\{\backslash text\{TFP\}\}\; =\; -\backslash xi\; (\backslash partial\_\{\backslash mu\}\; S)\; F\_i^2$$

suggesting that gauge interactions arise naturally from entropy-driven flow realignments.

Alright, so what did we actually do here? I started with a basic idea: time is the real foundation, and everything else—space, forces, particles—comes from the way flows interact. From there, I asked a bunch of questions: How do Newton’s laws show up? What about electromagnetism? Can I get gravity out of this? And what happens to quantum mechanics in a world like this?

I've been working on these questions and this is how I answer them. Forces come from flow misalignments, electricity and magnetism show up because of entropy changes, and gravity works when flow interactions start looking like curved space. Even quantum uncertainty comes naturally when you realize that high entropy makes things blurry, while low entropy makes them sharp. Simple concepts, for the most part.

Basically, instead of just assuming these laws exist, I showed they come out of something deeper—causal connections between flows that depend on entropy. And if that’s true, physics might not be built on space and particles like we always thought. It might be built on something even simpler.

I also have to say I'm not exactly comfortable with how entropy is normally portrayed or even thought about. However I'm building a framework that happens to align with entropy-driven principles, but not because I set out to prove entropy’s role in physics—it just naturally fits with what I'm doing. 

So, that’s the big idea, explained as plainly as I can say it. Hope it makes sense.