TFP Section 7 (v9)

SECTION 7 — EMERGENT GRAVITY FROM MASS–CURVATURE COUPLING (v9.4)

Geometry as Response to Mass Density
By John Gavel

7.0 Overview

Gravity emerges as the geometric response of the substrate to mass density \( M(x) \). No gravitational field is postulated — curvature arises from:

• Local tension \( T(x) \) (Section 2.3.4)
• Mass density \( M(x) \propto T(x) \) (Section 2.2.2)
• Correlation-induced metric \( g_{\mu\nu} \) (Section 3)

The Einstein equation is not assumed — it is the continuum limit of substrate dynamics.

7.1 Mass as the Source of Curvature

From Section 2.3.4, mesoscopic tension is:

\[ T(x) \approx k_{\text{avg}} \big[1 - A(x)^2\big] + \frac{k_{\text{avg}} a^2}{2} |\nabla A(x)|^2 \tag{7.1} \]

The first term is small for aligned domains (\( |A| \approx 1 \)), so the dominant contribution is:

\[ T(x) \approx \frac{k_{\text{avg}} a^2}{2} |\nabla A(x)|^2 \tag{7.2} \]

From Section 2.2.2, mass density is proportional to tension:

\[ M(x) = \gamma_M T(x) \tag{7.3} \]

where \( \gamma_M > 0 \) is a dimensionless proportionality constant. Thus mass sources field gradients.

7.2 Curvature from Tension Laplacian

From Section 3.4.3, scalar curvature is defined as:

\[ R(x) = \frac{\nabla^2 T(x)}{T(x)} \tag{7.4} \]

Substituting (7.3):

\[ R(x) = \frac{\nabla^2 M(x)}{M(x)} \tag{7.5} \]

This is the substrate analogue of Poisson’s equation. In Newtonian gravity, the potential \( \Phi \) satisfies:

\[ \nabla^2 \Phi = 4\pi G \rho \tag{7.6} \]

Identifying \( \rho \leftrightarrow M \) and \( \Phi \leftrightarrow \ln M \) (since \( \nabla^2 \ln M = (\nabla^2 M)/M - |\nabla \ln M|^2 \approx (\nabla^2 M)/M \) for slowly varying \( M \)), we recover:

\[ R(x) \propto \nabla^2 M(x) \tag{7.7} \]

7.3 Emergent Metric

From Section 3.2, operational distance is:

\[ d_{ij} = -\ln |C_{ij}(0)| \tag{7.8} \]

In the continuum limit (\( d_{ij} \to 0 \)), this defines a metric tensor \( g_{\mu\nu} \) such that the squared distance is:

\[ ds^2 = g_{\mu\nu} dx^\mu dx^\nu = d_{ij}^2 \tag{7.9} \]

The Ricci tensor is derived from loop holonomy (Section 3.4.2). For a loop \( p \) containing site \( x \), holonomy is \( H_p = \sum_{(u\to v)\in p} \tau_{u\to v} \) (Eq 3.13). Averaging over loops with area \( \langle A_p \rangle \):

\[ R_{\mu\nu}(x) = \frac{1}{\langle A_p \rangle} \sum_{p \ni x} H_p \hat{e}_\mu \hat{e}_\nu \tag{7.10} \]

where \( \hat{e}_\mu \) are unit vectors along coordinate directions.

7.4 Einstein-Like Relation

From (7.5) and (7.10), curvature is proportional to mass density. The Einstein tensor is:

\[ G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \tag{7.11} \]

Define the stress-energy proxy using the motif 4-velocity \( u_\mu \) (Section 4.10):

\[ T_{\mu\nu} = M(x) u_\mu u_\nu \tag{7.12} \]

Then:

\[ G_{\mu\nu} = \kappa_G T_{\mu\nu} \tag{7.13} \]

The coupling constant \( \kappa_G \) is fixed by dimensional calibration (Section 5). From Section 5:

• Mass unit: \( M_c \)
• Length unit: \( L_c \)
• Time unit: \( T_c \)

Since \( [G_{\mu\nu}] = \text{length}^{-2} \) and \( [T_{\mu\nu}] = \text{mass} \cdot \text{length}^{-3} \), we require:

\[ \kappa_G = 8\pi G_{\text{obs}} = 8\pi \left( \frac{L_c^3}{M_c T_c^2} \right) \tag{7.14} \]

This ensures dimensional consistency and matches Newtonian gravity in the weak-field limit.

7.5 Newtonian Limit

For weak fields and slow motion (\( |\mathbf{v}| \ll c \)), the metric is:

\[ g_{00} \approx -\left(1 + \frac{2\Phi_g}{c^2}\right), \quad g_{ij} \approx \delta_{ij} \tag{7.15} \]

From (7.5), \( R \approx \nabla^2 M / M \). In the Newtonian limit, \( R \approx -2 \nabla^2 \Phi_g / c^2 \) (from linearized GR). Equating:

\[ -\frac{2 \nabla^2 \Phi_g}{c^2} \approx \frac{\nabla^2 M}{M} \quad \Rightarrow \quad \nabla^2 \Phi_g \approx -\frac{c^2}{2M} \nabla^2 M \tag{7.16} \]

Integrating and identifying \( M \leftrightarrow \rho \):

\[ \Phi_g(x) \propto -\int \frac{M(x')}{|\mathbf{x} - \mathbf{x}'|} d^3x' \tag{7.17} \]

Thus Newtonian gravity emerges as the long-range limit of mass–curvature coupling.

7.6 Breakdown of Geometry

From Section 3.2, distance is defined only when \( |C_{ij}| > 0 \). If coherence decays completely (\( |C_{ij}| \to 0 \)), then:

\[ d_{ij} = -\ln |C_{ij}| \to \infty \tag{7.18} \]

The metric becomes singular, and curvature vanishes:

\[ g_{\mu\nu} \to 0, \quad R \to 0 \tag{7.19} \]

Geometry ceases to exist below the coherence scale \( L_c \). This is the substrate resolution limit (Section 5.5).

7.7 Axiomatic Closure

Gravitational Concept	Substrate Origin	Axiom
Mass (\( M \))	Flip frequency	A2, A6, A9
Curvature (\( R \))	\( \nabla^2 T / T \)	A2, A3, A6
Metric (\( g_{\mu\nu} \))	Correlation distance \( d_{ij} \)	A2, A3
Einstein Equation	Continuum limit of \( R \propto \nabla^2 M \)	Sections 2–3
Newtonian Limit	Weak-field expansion	Section 5

7.8 Bridge to Section 8 — Fields

Section 7 provides:

• Gravity as geometric response to \( M(x) \)
• Curvature sourced by \( \nabla^2 M \)
• Breakdown scale at \( L_c \approx 10^{-17} \text{m} \)

Section 8 unifies this with electromagnetic fields in a common geometric framework, using the metric \( g_{\mu\nu} \) and flow field \( A(x,t) \) as fundamental observables.