From Temporal Flow to Spacetime Geometry

 From Temporal Flow to Spacetime Geometry: A Rigorous Framework for Dimensional Emergence

Temporal Flow Physics (TFP) is a radically minimalistic framework where time is fundamental, and space, gauge fields, entropy, and gravity emerge from quantized one-dimensional temporal flows $F_i(t)$. Unlike conventional physics, TFP does not assume a preexisting manifold, coordinates, or field content. Everything arises from the dynamics and relationships between these temporal flows.

In this blog, I develop a precise and mathematically rigorous construction of how spatial dimensions, gauge structures, and geometric curvature naturally emerge from nested comparisons of temporal flows. I also derive explicit equations of motion, construct an emergent metric, and show how entropy corresponds to curvature energy — paving the way toward gravity from flow dynamics.

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 Foundations: The Action Principle

We begin with the discrete action governing a network of temporal flows $F_i(t):$

$$S[F_i] = \sum_i \int dt \left[ \frac{1}{2} \left(\frac{dF_i}{dt}\right)^2 - \frac{\lambda}{2} \sum_{j \in \mathcal{N}(i)} \left(\frac{dF_i}{dt} - \frac{dF_j}{dt}\right)^2 + V(F_i) \right]$$

Terms:

• $u_i = \frac{dF_i}{dt}$: flow rate

• $\lambda$: coupling constant

• $\mathcal{N}(i)$: neighborhood of node $i$

• $V(F_i)$: potential energy term (e.g. double-well form for vacuum stability)

This Lagrangian includes:

• Kinetic energy: local evolution of each flow

• Alignment energy: penalizes differences in flow rates — source of spatial coherence

• Potential energy: flow self-interaction, symmetry breaking

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Equations of Motion: Euler-Lagrange Derivation

Applying the Euler-Lagrange equation to this action yields:

$$\frac{d^2F_i}{dt^2} = \lambda \sum_{j \in \mathcal{N}(i)} \left( \frac{d^2F_j}{dt^2} - \frac{d^2F_i}{dt^2} \right) - V'(F_i)$$

This simplifies to:

$$(1 - \lambda d_i) \ddot{F}_i + \lambda \sum_{j \in \mathcal{N}(i)} \ddot{F}_j = - V'(F_i)$$

This is a coupled system of ODEs, describing how acceleration at node $i$ is determined by local flow interactions and potential.

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 Dimensional Emergence from Flow Comparisons

1. Zero-Order: Fundamental flow

$F(t)$: pure temporal evolution

2. First-Order: Temporal change

$\Delta F_i = F(t_i) - F(t)$: introduces directed comparison

3. Second-Order: Spatial interval

$\delta_{ij} = F(t_i) - F(t_j)$: enables planar relations

4. Third-Order: Spatial geometry

$\delta_{ijk} = F(t_i) - 2F(t_j) + F(t_k)$: angular/volume structure

5. Higher-Order: Saturation

$\delta_{ijkl} = \delta_{ijk} - \delta_{jkl}$: redundant refinement, no new dimensions

Result:

Dimensionality emerges from nested flow comparisons. 3D space arises when non-coplanar comparisons are saturated — no more independent axes are possible.

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Emergent Metric from Flow Fluctuations

Define:

• Background flow: $\bar{F}(x)$
  

• Fluctuation: $\delta F(x) = F(x) - \bar{F}(x)$
  

Then define the emergent metric:

$$g_{\mu\nu}(x) = \langle \partial_\mu \delta F(x) \, \partial_\nu \delta F(x) \rangle$$

This is the 2-point correlation function of local flow fluctuations.

Why this works:

• Symmetric by construction

• Tensorial under coarse-grained coordinate transformations

• Captures local variance in directional flow changes

• Emergent Lorentzian signature arises from coherent propagation constraints

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Gauge Field as Flow Misalignment

Define:

• Local misalignment: $\delta F_{ij} = F_i - F_j$

• Local potential: $A_\mu^{(ij)} = \partial_\mu \delta F_{ij}$

Then the gauge curvature tensor is:

$$F_{\mu\nu}^{(ij)} = \partial_\mu A_\nu^{(ij)} - \partial_\nu A_\mu^{(ij)} = [\partial_\mu, \partial_\nu] \delta F_{ij}$$

Explicit proof:
In standard gauge theory, the field strength arises from noncommutativity of covariant derivatives. Here, non-smoothness or discrete topology ensures that second-order differences are non-zero, giving rise to:

$$F_{\mu\nu}^{(ij)} \neq 0 \quad \Rightarrow \quad \text{Intrinsic curvature from flow misalignment}$$

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Entropy as Curvature Energy

Define local variance of flow rates:

$$\sigma_i^2 = \frac{1}{|\mathcal{N}(i)|} \sum_{j \in \mathcal{N}(i)} \left( u_i - u_j \right)^2$$

Then total entropy is:

$$\sigma^2_{\text{total}} = \sum_i \sigma_i^2 = \sum_{\langle ij \rangle} (u_i - u_j)^2$$

This is exactly the same as the curvature energy:

$$\mathcal{F} = \frac{1}{4} \sum_{\mu,\nu} \left( F_{\mu\nu}^{(ij)} \right)^2$$

Conclusion:
Entropy = misalignment = curvature = disorder in temporal alignment
This unifies thermodynamic entropy, curvature energy, and field strength under one principle.
As well as gravity, consider  σu2​(x)∝Fμν(ij)​(x)F(ij)μν(x), with the proportionality constant α~ would then include factors like α and possibly the gauge coupling constant.