From Temporal Flow to Emergent Gravity: Why Dimensional Consistency Matters in Temporal Flow Physics

By John Gavel

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Introduction

Temporal Flow Physics (TFP) is a new framework where time itself is fundamental, and space emerges as a relational construct from quantized 1D temporal flows $F_i(t)$. This is a paradigm shift requires that all derived physics—gravity, matter, fields—fit together dimensionally and conceptually.

Today I want to share how the emergent gravitational dynamics, encapsulated by an Einstein-like equation, arise naturally and dimensionally consistently from the dynamics of the flow field $F$. This not only validates the theory's internal consistency but links it directly to known physics constants and equations.

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The Temporal Flow Field and Dimensional Basis

The fundamental object in TFP is the scalar flow field:

$$F(x^\mu)$$

where $x^\mu$ are emergent spacetime coordinates constructed relationally from flow comparisons.

Dimensionally, the flow field has units:

$$[F] = T^{-1}$$

since it represents a fundamental flow rate or frequency. The emergent coordinates $x^\mu = (t, \vec{x})$ combine time and space dimensions:

$$[x^0] = T, \quad [x^i] = L$$

Derivatives and Their Dimensions

Partial derivatives act on $F$ as:

$$\partial_\mu F = \frac{\partial F}{\partial x^\mu}$$

which have dimensions:

• For time component:

$$[\partial_0 F] = \frac{[F]}{[x^0]} = T^{-1} / T = T^{-2}$$

• For space components:

$$[\partial_i F] = \frac{[F]}{[x^i]} = T^{-1} / L = L^{-1} T^{-1}$$

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Constructing the Effective Lagrangian

The effective Lagrangian density in curved emergent spacetime $g_{\mu\nu}$ is:

$$\mathcal{L} = \frac{1}{2} g^{\mu\nu} \partial_\mu F \partial_\nu F - V(F)$$

The kinetic term dimension:

$$\left[ \frac{1}{2} g^{\mu\nu} \partial_\mu F \partial_\nu F \right] = [g^{\mu\nu}][\partial_\mu F]^2$$

Since $g^{\mu\nu}$ is dimensionless (it acts like a metric tensor raising indices), the kinetic term inherits the dimension of $[\partial_\mu F]^2$.

For the time-time component contribution:

$$[\partial_0 F]^2 = (T^{-2})^2 = T^{-4}$$

For the space-space component contribution:

$$[\partial_i F]^2 = (L^{-1} T^{-1})^2 = L^{-2} T^{-2}$$

Because the emergent spacetime metric has a Lorentzian signature, both terms combine to yield a consistent energy density dimension, which is:

$$[\text{Energy density}] = M L^{-1} T^{-2}$$

in SI units, or simply $L^{-4}$ in natural units where $c = \hbar = 1$.

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Potential Term and Dimensional Matching

The potential $V(F)$ represents flow self-interactions and must have units of energy density to appear consistently in the Lagrangian:

$$[V(F)] = [\mathcal{L}] = L^{-4} \quad (\text{natural units})$$

Given $[F] = T^{-1} = L^{-1}$ (since in natural units $c=1$ makes time and length interchangeable), the potential $V(F)$ can be constructed as powers or polynomials of $F$ scaled by fundamental constants to achieve the correct dimension:

$$V(F) \sim \Lambda F^n$$

where $\Lambda$ is a coupling constant with appropriate dimension to ensure $V(F)$ has dimension $L^{-4}$.

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Gravitational Term and Newton’s Constant

The emergent gravitational part of the action is:

$$S_g = \frac{1}{16 \pi G} \int d^4x \sqrt{-g} R$$

Here:

• $R$, the Ricci scalar curvature, has dimension:

$$[R] = L^{-2}$$

• The integration measure $d^4x$ has dimension:

$$[d^4x] = L^4$$

• $\sqrt{-g}$ is dimensionless.

So the gravitational action term has dimension:

$$\left[\frac{1}{G} R \right] L^4 = 1$$

implying Newton’s constant has dimension:

$$[G] = L^2$$

in natural units. This matches perfectly the emergent gravitational coupling in TFP.

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Emergent Einstein-Like Equations

Varying the effective action $\Gamma$ w.r.t. the metric gives:

$$\delta \Gamma = \frac{1}{16\pi G} \int d^4 x \sqrt{-g} \left( R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} \right) \delta g^{\mu\nu} + \int d^4 x \sqrt{-g} \frac{\delta \mathcal{L}_{\text{eff}}}{\delta g^{\mu\nu}} \delta g^{\mu\nu}$$

Defining the effective stress-energy tensor from the flow field:

$$T_{\mu\nu}^{\text{eff}} := - \frac{2}{\sqrt{-g}} \frac{\delta (\sqrt{-g} \mathcal{L}_{\text{eff}})}{\delta g^{\mu\nu}}$$

we arrive at the emergent Einstein equations:

$$G_{\mu\nu} := R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8\pi G T_{\mu\nu}^{\text{eff}}$$

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Flow Field Equation of Motion in Emergent Spacetime

Varying the action w.r.t. $F$ gives the curved spacetime wave equation:

$$\nabla_\mu \nabla^\mu F - \frac{\partial V}{\partial F} = 0$$

where the covariant derivatives $\nabla_\mu$ depend on the emergent metric $g_{\mu\nu}$.

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Summary: Dimensions Align, Theory Coheres

• The flow field $F$, as a fundamental 1D temporal flow, has dimensions of inverse time.

• Its kinetic and potential terms combine consistently to give an energy density dimension appropriate for a Lagrangian density.

• The gravitational coupling $G$ and curvature $R$ fit naturally into the emergent spacetime picture with correct dimensions.

• The Einstein-like equations derived from varying the effective action $\Gamma$ relate flow fluctuations' energy-momentum to curvature, exactly as in General Relativity, but emerging here from quantized temporal flows.

This dimensional consistency is a check that TFP is not just a conceptual framework but mathematically and physically sound.