5. GRAVITY: STRESS-INDUCED DILATANCY

5.1 The Dilatancy Mechanism

Definition: Dilatancy is the volumetric expansion of a granular medium under shear stress.

Historical Note: Osborne Reynolds (1885) observed that wet sand becomes dry when walked upon - water is sucked into the expanded void space.

Application to Vacuum:

A mass M creates stress in the surrounding vacuum grains. This stress causes the lattice to dilate (expand), reducing the local packing fraction:

$$\phi(r) = \phi_0 [1 - \beta \sigma(r)]$$

where:

• φ₀ ≈ 0.64 (baseline RCP density)

• β = dilatancy coefficient

• σ(r) = stress at distance r from mass

Refractive Index:

The vacuum's effective "stiffness" (speed of light) depends on density: $$c_{eff}(r) = c_0 \sqrt{\frac{\phi(r)}{\phi_0}}$$

Light Bending:

Photons (which are phonons) travel slower in dilated (less dense) regions. This creates a refractive index gradient: $$\nabla n = \nabla \left(\frac{c_0}{c_{eff}}\right)$$

Result: Light bends toward the mass (toward higher density = lower refractive index).

This IS gravitational lensing, but the mechanism is refraction, not spacetime curvature.

5.2 Critical Point Amplification

Problem: Laboratory granular media have weak dilatancy (β ≈ 10⁻⁹). To match observed gravitational lensing (Einstein rings, galaxy clusters), we need β

≈ 10⁻⁵.

Discrepancy: Factor of 10⁴.

Solution: The vacuum is at the jamming transition (φ ≈ φ_c). At critical points, response functions diverge:

$$\beta_{crit} = \beta_0 |\phi - \phi_c|^{-\gamma}$$

where γ ≈ 1.0 (mean-field exponent).

Numerical Estimate:

For the vacuum to match GR: $$|\phi - \phi_c| \approx 10^{-4}$$

Interpretation: The vacuum is within 0.01% of the jamming transition. This extreme proximity to criticality explains why it responds so sensitively to stress despite being microscopically rigid.

This is not fine-tuning - it's Self-Organized Criticality. The vacuum maintains itself at this critical point via quantum fluctuations (Section 2.3).

5.3 Galactic Rotation Curves (Dimensional Confinement)

Observation: Spiral galaxies show flat rotation curves - orbital velocity v(r) ≈ constant, not v(r) ∝ 1/√r (Keplerian).

Standard Interpretation: Requires dark matter halo (ρ_DM ∝ 1/r²).

SBF Interpretation: Shear-induced dimensional confinement.

5.3.1 Isotropic vs Anisotropic Vacuum

In spherical symmetry (elliptical galaxies):

Stress is isotropic (pressure). Dilatancy spreads in 3D: $$\text{Area} \propto r^2 \implies g(r) \propto \frac{1}{r^2}$$

Result: Newtonian gravity (v ∝ 1/√r).

In disk geometry (spiral galaxies):

Rotation induces shear stress parallel to the disk plane. Force chains align with the shear direction, confining gravitational flux to a cylinder:

$$\text{Area} \propto r \times h \quad \text{(cylindrical, not spherical)}$$ $$g(r) \propto \frac{1}{r}$$

Result: Flat rotation curve (v = constant).

Quantitative Prediction:

The transition from spherical (3D) to cylindrical (2D) gravity occurs when the rotational shear stress exceeds radial compression ($\tau_{\text{shear}} / \sigma_{\text{radial}} > 1$).

• Transition Radius: For typical spiral galaxies ($M \sim 10^{11} M_\odot$), this occurs at $r_{\text{trans}} \approx 3-5$ kpc.

• Falsification: If detailed velocity mapping shows no correlation between the plateau velocity $v_{\text{flat}}$ and the disk scale height $h$ (where $v_{\text{flat}} = \sqrt{2GM/h}$), the dimensional confinement model fails.

5.3.2 Mathematical Derivation

Gauss's Law for Gravity: $$\oint \mathbf{g} \cdot d\mathbf{A} = -4\pi G M$$

Spherical (3D): $$g(r) \cdot 4\pi r^2 = 4\pi GM$$ $$g(r) = \frac{GM}{r^2} \implies v = \sqrt{\frac{GM}{r}}$$

Cylindrical (2D confinement): $$g(r) \cdot 2\pi r h = 4\pi GM$$ $$g(r) = \frac{2GM}{rh} \implies v = \sqrt{\frac{2GM}{h}} = \text{constant}$$

Prediction: Rotation curve velocity depends on disk thickness h, not total mass M alone.

Observational Test: Galaxies with thicker disks should have higher v_plateau at the same mass. This IS observed (Tully-Fisher relation).

5.4 The Bullet Cluster (Plasticity & Hysteresis)

Observation: Two galaxy clusters collided at v ≈ 4500 km/s. Gravitational lensing shows mass distribution offset from gas (baryons).

Standard Interpretation: Dark matter particles pass through without interacting (collisionless), while gas collides and slows (collisional).

SBF Interpretation: Elasto-plastic deformation of the vacuum.

5.4.1 The Yield Point

Every material has a yield stress σ_y beyond which it deforms plastically (permanently).

For the granular vacuum: $$\sigma_y \approx G \times 10^{-4} \quad \text{(near jamming transition)}$$

During collision:

• Gas (baryons) creates localized, intense stress → exceeds σ_y locally

• Vacuum undergoes plastic flow (grains rearrange irreversibly)

• After collision, vacuum "remembers" the deformation (hysteresis)

Result: The lensing signal (which traces vacuum dilatancy) is offset from the current baryon distribution because it reflects the history of stress, not instantaneous position.

Analogy: A dent in sheet metal persists after the hammer is removed. The Bullet Cluster's "dark matter halo" is a ghost dent in the vacuum structure.

5.4.2 Falsification Test

Prediction: The lensing offset should correlate with:

• Collision velocity (higher v → larger offset)

• Time since collision (offset relaxes over gigayear timescales)

If observed: "Dark matter" halos show breathing modes (oscillation), SBF is falsified (plastic deformations don't oscillate).

Status: No breathing modes observed to date (consistent with SBF).