Temporal Physics fundamental flow

In my model of Temporal Physics, the key principles are:

1. Flows (FF) accumulate in a time-like sequence.
2. The maximum flow is constrained by $c$, and when a flow reaches $c$, it inverts direction.
3. Interactions between flows lead to the emergence of space-time structure and forces.
4. Flow conservation applies, meaning positive and negative flows interact but do not cancel; they only redistribute.

1. Fundamental Flow Equation

Each flow $F_i$ evolves over a temporal step $\Delta t$, according to:

$$F_{i+1} = F_i + \Delta F$$

where $\Delta F$ represents the rate of accumulation due to prior interactions. When $F_i$ approaches $c$, we introduce an inversion function:

$$F' = c - (F - c) = 2c - F, \quad \text{if} \quad F \geq c$$

This ensures that flows do not exceed $c$ but instead invert direction when this limit is reached.

2. Interaction Between Two Flows

The interaction function $A(F_i, F_j)$ describes how two flows influence each other. If both flows are below $c$, their interaction is governed by:

$$A(F_i, F_j) = \frac{1}{1 - \frac{F_i F_j}{c^2}}$$

If one flow reaches $c$, the inversion condition applies, and the sign of further evolution is shifted. Alternatively, a soft transition can be used to model the inversion effect:

$$A(F_i, F_j) = \frac{1}{1 + e^{-k(F_i + F_j - c)}}$$

where $k$ controls the sharpness of the inversion effect.

3. Nonlocal Flow Correlation (Entanglement-Like Behavior)

Nonlocal flow correlations can be modeled by:

$$C(F_i, F_j) = e^{-\alpha |F_i - F_j|}$$

where $\alpha$ determines the strength of nonlocal effects, allowing distant flows to influence each other without direct contact.

4. Energy and Entropy in Flow-Based Systems

To express energy, we use a relativistic-like form:

$$E(F) = \frac{m}{\sqrt{1 - \left( \frac{F}{c} \right)^2}}$$

which links energy directly to flow magnitudes. For entropy, considering nonlocal effects:

$$S = - \sum P(F) \log P(F) + \sum C(F_i, F_j)$$

where $P(F)$ is the probability of a given flow, and the second term accounts for nonlocal correlations.

5. Space-Time Emergence and Curvature

Space emerges from sequential flow interactions. A curvature-like term can be defined as:

$$R \sim \sum (F_{i+1} - F_i)^2$$

which tracks the deviation of flows over an interval.

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Putting It All Together

The total system evolution follows:

$$F_{i+1} = \begin{cases} F_i + \Delta F, & \text{if} \ F < c \\ 2c - F_i, & \text{if} \ F \geq c \end{cases}$$

with interactions:

$$A(F_i, F_j) = \frac{1}{1 - \frac{F_i F_j}{c^2}}$$

and nonlocal correlations:

$$C(F_i, F_j) = e^{-\alpha |F_i - F_j|}$$

Force in Temporal Flows

1. General Force Equation in Terms of Flows

The force arises from differences in flow accumulation:

$$F = k \frac{dF}{dT} = k \left( \frac{F_i - F_j}{T_j - T_i} \right)$$

where $k$ is a proportionality factor related to interaction strength.

2. Gravity in Terms of Flow Accumulation

Gravitational-like effects are formulated as:

$$F_G = G_{\text{eff}} \frac{F_1 F_2}{(T_1 - T_2)^2}$$

where $G_{\text{eff}}$ is an emergent gravitational coupling factor, and the squared term suggests an inverse-square law for time-separated flows.

3. Electromagnetic Force as Flow Polarity Interactions

The electromagnetic force follows a flow-based interaction:

$$F_E = k_e \frac{F_q F_q'}{(T - T')^2}$$

where $k_e$ is the flow-based Coulomb constant and $F_q, F_q'$ represent charged flow strengths. The magnetic component arises as:

$$F_B = k_m \frac{F_q \cdot v}{(T - T')^2}$$

where $v$ is the relative velocity of flows.

4. Unification: All Forces as Flow Gradients

A general force equation can be expressed as:

$$F = - \nabla S(F)$$

where $S(F)$ is a flow action function representing how flows evolve dynamically, and $\nabla S(F)$ captures the gradient of flow accumulation, showing that forces are due to flow interactions.

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Flow-Based Space Framework Formalization

1. Flow Coordinate System

The spatial coordinates $x, y, z$ are derived from the differences between reference flows $F_1, F_2, F_3$:

$$x = f(F_2 - F_1), \quad y = g(F_3 - F_1), \quad z = h(F_3 - F_2)$$

These mapping functions transform flow differences into spatial coordinates.

2. Metric Tensor from Flow Interactions

The metric tensor $g_{\mu\nu}$ encodes the geometry of space-time:

$$g_{\mu\nu} = \frac{\partial F_{\mu}}{\partial x_\lambda} \frac{\partial F_{\nu}}{\partial x_\lambda} + \delta_{\mu\nu} \left( 1 - \frac{F^2}{c^2} \right)$$

This defines how flow changes impact the spacetime geometry, with the relativistic correction factor accounting for relativistic effects.

3. Spacetime Interval (Line Element)

The spacetime interval $ds^2$ combines both temporal and spatial components:

$$ds^2 = \left( 1 - \frac{F^2}{c^2} \right) dt^2 - \sum_i dx_i^2 \left( 1 + A(F_i, F_j) \right)$$

where the interaction term $A(F_i, F_j)$ adjusts the spatial relationship between points.

4. Curvature and Flow Interactions

The Riemann curvature tensor, relating flow differentials to space-time curvature, is modeled as:

$$R_{\mu\nu\rho\sigma} = K \left[ (F_\mu, \rho F_\nu, \sigma - F_\mu, \sigma F_\nu, \rho) + \text{Inversion Terms} \right]$$

5. Field Equations (Flow-Energy to Curvature)

The field equations relate the distribution of flows to the curvature of spacetime:

$$G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}(F)$$

where $G_{\mu\nu}$ is the Einstein tensor derived from your flow-based metric, and $T_{\mu\nu}(F)$ is the energy-momentum tensor for the flows.

6. Observer Transformation (Reference Frames)

To transform between observers, we use a Lorentz transformation:

$$F_\mu' = \Lambda_{\mu\nu} F_\nu$$

where $\Lambda_{\mu\nu}$ is the Lorentz transformation matrix preserving the speed limit of $c$.