Emergent Gravity: Why Dimensional Consistency and Gauge Dynamics Matter in Temporal Flow Physics

By John Gavel

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 Fi​(t). This is a paradigm shift that demands 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 underlying dynamics of the flow field F. We'll explore not only how the fabric of spacetime emerges but also how its curvature arises from fundamental "misalignments" or "twists" in these flows, quantified by a gauge field strength, and how this links directly to known physics constants and equations.

---

The Temporal Flow Field and Dimensional Basis

The fundamental object in TFP is the scalar flow field:

F(xμ)

where xμ 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 $x^{\mu }=(t,x)$ combine time and space dimensions:

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

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: $[{\partial }_{0}F]=\frac{[F]}{[x^0]}=\frac{T^{-1}}{T}=T^{-2}$

For space components: $[{\partial }_{i}F]=\frac{[F]}{[x^i]}=\frac{T^{-1}}{L}=L^{-1}{T}^{-1}$

---

The Emergent Fabric of Spacetime: The Metric

In TFP, the very spacetime metric tensor $g_{\mu \nu }(x)$ emerges from the statistical correlations of the gradients of these fundamental flow fluctuations. We define it as:

$g_{\mu \nu }(x)=⟨{\partial }_{\mu }\delta F(x)\cdot {\partial }_{\nu }\delta F(x)⟩$

Where $\delta F(x)=F(x)-\bar{F}(r)$ represents the fluctuations around a background flow Fˉ(r). This is a crucial point: the geometry of spacetime isn't fundamental; it's a coarse-grained average of the underlying flow dynamics.

Even though F is a scalar field, the metric tensor arises from bilinear combinations of its derivatives. Each component of gμν​ involves products of directional derivatives, generating a rank-2 tensor. This is similar to how, in classical field theory, the energy-momentum tensor is built from derivatives of scalar fields.

The independent components of the metric tensor correspond to correlations between flow gradients along different directions. Dimensionally, gμν​ is dimensionless, correctly acting as a metric tensor.

---

The Emergent Curvature: Gauge Field Strength from Flow Misalignments

While the metric defines the fabric of spacetime, curvature arises from how flows "twist" or "misalign" locally. This is where an emergent gauge field strength comes into play.

Consider the difference between neighboring temporal flows: $\delta F_{ij}(x):=F_i(x)-F_j(x)$

We can then define an emergent gauge potential Aμ(ij)​(x) from the derivatives of these flow differences: $A_{\mu }^{(ij)}(x):={\partial }_{\mu }\delta F_{ij}(x)$

Now, the emergent gauge field strength tensor Fμν(ij)​(x) is derived from the commutator of these potentials: $F_{\mu \nu }^{(ij)}(x):={\partial }_{\mu }{A}_{\nu }^{(ij)}-{\partial }_{\nu }{A}_{\mu }^{(ij)}=[{\partial }_{\mu },{\partial }_{\nu }]\delta F_{ij}(x)$

In a perfectly smooth, flat space with commuting partial derivatives, this would typically be zero. However, in TFP, the "derivatives" ∂μ​ are effective and approximate, acting on a discrete network of flows. This is analogous to lattice gauge theory, where the curvature (field strength) is non-zero because the underlying discrete structure allows for local misalignments or "holonomies" that measure the failure of paths to commute.

This $F_{\mu \nu }^{(ij)}$ tensor is mathematically identical to the standard gauge field strength tensor. It quantifies the "misalignment" or "twist" of flows along closed loops in the emergent spacetime, which is precisely how curvature is understood in both gauge theories and General Relativity (via the Riemann tensor, which measures the non-commutativity of covariant derivatives).

---

Connecting Disorder (Entropy) to the Energy of Curvature

A profound connection arises when we consider the "disorder" or "entropy" of these flow configurations. Let σu2​(x) be a scalar measure of the local disorder or variance of the flow differences δFij​ around point x.

We can hypothesize that this local disorder ${\sigma }_u^2(x)$ is directly related to the energy density of the emergent gauge field that quantifies the flow misalignments. In gauge theories, the energy density of the field is proportional to Fμν​Fμν. Thus, we make the crucial identification:

${\sigma }_u^2(x)\propto F_{\mu \nu }^{(ij)}(x)F^{(ij)\mu \nu }(x)$

This suggests that the "cost of misalignment" in the temporal flows, interpreted as local entropy or disorder, is precisely the energy content of the emergent gravitational field. This aligns beautifully with concepts from entropic gravity, where spacetime and gravity are seen as emergent thermodynamic phenomena.

---

Constructing the Effective Lagrangian and Emergent Einstein Equations

Now, let's build an effective Lagrangian density L in this curved emergent spacetime, incorporating the insights from our flow dynamics:

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

The kinetic term's dimension: $[\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }F{\partial }_{\nu }F]=[g^{\mu \nu }][{\partial }_{\mu }F{]}^2$

For the time-time component contribution: $[{\partial }_{0}F{]}^2=(T^{-2}{)}^2=T^{-4}$

For the space-space component contribution: $[{\partial }_{i}F{]}^2=(L^{-1}{T}^{-1}{)}^2=L^{-2}{T}^{-2}$

When combined with the Lorentzian signature of the emergent metric, these terms 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 $c=\hslash =1$).

The potential term $V(F)$ in the Lagrangian must also have units of energy density. Crucially, this potential can encapsulate the energy cost of flow misalignments, meaning V(F) could be directly related to σu2​(x) and, therefore, the gauge field strength squared. For example, $V(F)\sim \Lambda F^n$ (where Λ scales dimensions appropriately), with Fn representing the complex interactions of flows that lead to disorder.

Gravitational Term and Newton’s Constant

The emergent gravitational part of the action is:

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

Here: R, the Ricci scalar curvature, has dimension: $[R]=L^{-2}$. The integration measure d4x has dimension: $[d^{4}x]=L^4$. −g​ is dimensionless.

So the gravitational action term has dimension: $[\frac{1}{G}R]L^4=1$

implying Newton’s constant has dimension: $[G]=L^2$ (in natural units). This matches perfectly the emergent gravitational coupling. Furthermore, the Ricci scalar R itself can be seen as the macroscopic manifestation of the local flow misalignments described by $F_{\mu \nu }^{(ij)}$, integrated and averaged over the emergent spacetime.

Emergent Einstein-Like Equations

Varying the effective action Γ with respect to the metric gμν​ gives:

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

Defining the effective stress-energy tensor Tμνeff​ from the flow field (which includes contributions from the kinetic energy and the "misalignment" potential):

$T_{\mu \nu }^{eff}:=-2\frac{1}{\sqrt{-g}}\frac{\delta (\sqrt{-g}{L}_{eff})}{\delta g^{\mu \nu }}$

we arrive at the emergent Einstein equations:

$G_{\mu \nu }:=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=8\pi G{T}_{\mu \nu }^{eff}$

These equations relate the curvature of the emergent spacetime (Gμν​) to the energy and momentum of the underlying flow dynamics (Tμνeff​), just as in General Relativity.

Flow Field Equation of Motion in Emergent Spacetime

Varying the action with respect to F gives the curved spacetime wave equation:

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

where the covariant derivatives ∇μ​ depend on the emergent metric gμν​.

---

Summary: Dimensions Align, Theory Coheres, and Gravity Emerges

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

The spacetime metric $g_{\mu \nu }$ emerges from the statistical correlations of flow gradients.

The local curvature (gravity) arises from misalignments or twists in these flows, quantified by an emergent gauge field strength $F_{\mu \nu }^{(ij)}$. This "cost of misalignment" or local disorder (${\sigma }_u^2(x)$) is identified with the energy density of this emergent gauge field.

The kinetic and potential terms (which now inherently include the energy from flow misalignments) 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, with R being a macroscopic reflection of the fundamental flow misalignments.

The Einstein-like equations derived from varying the effective action Γ relate flow fluctuations' energy-momentum to curvature, exactly as in General Relativity, but emerging here from quantized temporal flows. This dimensional and conceptual consistency is a strong indicator that TFP is not just a conceptual framework but mathematically and physically sound, offering a pathway to truly emergent gravity.