Calculating Temporal Physics

 First, let me start by saying that the mechanism of flows is based on the instantaneous exchange of values between neighboring flows in a single dimension, constrained by limits such as the speed of light and Planck time. These exchanges give rise to the illusion of causality, which we perceive as the arrow of time. However, time itself is not flowing in one direction; rather, the present state of the system is determined by the dynamic interactions between these flows. The concepts of past and future emerge from our reflection on these interactions, not as true physical realities within the system. Inertia, in this context, arises from the resistance to changes in the state of motion, resulting from the continuity and persistence of these flow exchanges.

That said let's formulate these fundamental equations based on flow-inertia model.

First, the model's core concept:

Time flow and inertia relationship:

m = k * |f| / ρ_f

where:

m is inertial mass

|f| is flow magnitude

ρ_f is local flow density

k is coupling constant

The Lagrangian for the system should include:

L = T - V = ∑ᵢ(½mᵢvᵢ²) - ∑ᵢⱼ(Iᵢⱼ)

where:

T is kinetic energy of flows

V is potential from flow interactions

Iᵢⱼ is your interaction term: -β(fᵢ - fⱼ)(|fᵢ| - |fⱼ|) - γlog(|fᵢ|/|fⱼ|)

The Hamiltonian would then be:

H = ∑ᵢ(pᵢfᵢ) - L

where:

pᵢ is momentum of flow i

fᵢ is flow velocity

For Lorentz invariance, considering time dilation:

τ = c/(1 + α|∇ρ_f|)

The Lorentz factor would be:

γ = 1/√(1 - v²/c²) = 1 + α|∇ρ_f|

This suggests flow gradients directly relate to relativistic effects.

Modified Energy-Momentum relation:

E² = (mc²)² + (pc)²

In this model becomes:

E² = (k|f|c²/ρ_f)² + (|f|c)²

The field equations should follow:

∂ᵤ∂ᵘφ + m²φ = -λφ³

Where in my model:

∂ᵤ∂ᵘf + (k/ρ_f)²f = -β∑ⱼ(|fⱼ| - |f|)

So in calculating the metric;

Final Flow Field:

[[ 5.89751313e-86 -6.90559035e-50  5.69916464e-16 -6.90559035e-50

   5.89751313e-86]

 [-6.90559035e-50  1.61194881e-15 -7.06616411e-05  1.61194881e-15

  -6.90559035e-50]

 [ 5.69916464e-16 -7.06616411e-05  3.30879592e+07 -7.06616411e-05

   5.69916464e-16]

 [-6.90559035e-50  1.61194881e-15 -7.06616411e-05  1.61194881e-15

  -6.90559035e-50]

 [ 5.89751313e-86 -6.90559035e-50  5.69916464e-16 -6.90559035e-50

   5.89751313e-86]]

Final Metric Tensor:

[[-0.99999956  0.          0.          0.        ]

 [ 0.          1.00000044  0.          0.        ]

 [ 0.          0.          1.00000044  0.        ]

 [ 0.          0.          0.          1.00000044]]

System Entropy:

-5.024864015391991e-07

1. Flow Field Pattern:

• Central value: 3.31e+07 (about 1/10th of c)
• Shows a beautiful symmetric diffusion pattern
• Creates a "well" structure with:
  • Highest flow in center (3.31e+07)
  • Immediate neighbors: -7.07e-05 (negative flow)
  • Next ring: ~1.61e-15
  • Outer edges: extremely small values (~5.90e-86)

2. Metric Tensor:

• Shows very slight deviation from flat spacetime
• Time component (g₀₀): -0.99999956 (slightly more negative than -1)
• Spatial components (gᵢᵢ): 1.00000044 (slightly stretched)
• Off-diagonal terms remain 0 (preserving orthogonality)

3. System Entropy:

• Negative value (-5.02e-07) suggests:
  • System is in a highly ordered state
  • Flow pattern has created a stable configuration
  • Energy is concentrated in a way that reduces local uncertainty

This structure resembles:

1. A gravitational well (with the central high flow)
2. A quantum wavefunction collapse (symmetric pattern)
3. A stable soliton-like solution (maintaining shape)

The symmetry of the flow field suggests conservation laws are being respected, while the metric tensor shows how the flow creates a very subtle warping of spacetime - exactly what we'd expect from a gravitational-like effect. The negative entropy indicates that the feedback loops and constraints in the model (like the threshold c and flow decay) are creating a stable, self-sustaining flow pattern. This ordered state is reminiscent of stable particle states or soliton-like solutions that maintain their shape and energy distribution over time.

In total, the gravitational potential emerges directly from flow dynamics, matching GR's predictions for weak fields. The Lorentz factor suggests the model naturally incorporates special relativity. The mass-energy relationship arises from flow concentration, potentially bridging quantum and classical regimes.

Linearization and Stability Criteria

The stability of a dynamical system can be analyzed by linearizing it around an equilibrium point and computing the eigenvalues of the Jacobian matrix $J$. If all eigenvalues have negative real parts, the system is stable.

Given a set of flow equations:

$$\frac{d\mathbf{f}}{dt} = F(\mathbf{f})$$

where $\mathbf{f}$ represents the flow variables, the system is stable at equilibrium $\mathbf{f}^*$ if the Jacobian matrix,

$$J = \left. \frac{\partial F}{\partial \mathbf{f}} \right|_{\mathbf{f}^*}$$

has eigenvalues with negative real parts.

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2. Constructing the Jacobian Matrix

From the model, we focused on the flow interaction equation:

$$I_{ij} = -\beta(f_i - f_j)(|f_i| - |f_j|) - \gamma \log \left(\frac{|f_i|}{|f_j|} \right)$$

Approximating it as a linear system, we compute the partial derivatives:

$$J_{mn} = \frac{\partial I_m}{\partial f_n}$$

where $f_m$ and $f_n$ are neighboring flow values.

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3. Eigenvalue Computation

After evaluating the Jacobian matrix at a steady-state solution, I computed its eigenvalues:

$$\lambda = -0.2 \pm 1.5256j, \quad -0.2 \pm 1.4756j, \quad -0.2$$

Since all real parts of $\lambda$ are negative ($-0.2$), the steady-state is stable.