6. BLACK HOLES: CRYSTALLINE CORES

6.1 The Phase Transition

Standard GR: Gravitational collapse continues to r = 0 (singularity).

SBF: The granular vacuum has a maximum density (φ_max = 0.74, FCC packing).

At critical compression: $$\phi \to \phi_{FCC} \approx 0.74$$

The vacuum undergoes phase transition: $$\text{Liquid (RCP, } Z \approx 14.4\text{)} \to \text{Crystal (FCC, } Z = 12\text{)}$$

The core solidifies. Collapse stops.

6.2.1 Connection to Analog Gravity

Unruh (1981) showed that sound waves in flowing fluids obey equations identical to scalar fields in curved spacetime.

Sonic Horizon: Point where flow velocity v equals sound speed c_s.

SBF Mapping:

• Fluid → Granular vacuum

• Sound speed → c = √(G/ρ)

• Event horizon → Phase boundary (liquid/crystal interface)

Critical Difference:

Standard analog gravity assumes continuous fluid. SBF adds jamming constraint:

$$\phi_{max} = 0.74 \quad \text{(FCC limit)}$$

At this density, flow stops. The "horizon" becomes a material boundary, not a geometric surface.

6.2.2 Relativistic Support: The Frozen Star Convergence

The concept of a vacuum phase transition—central to SBF's definition of forces—finds independent support in recent relativistic theory. Work by Janzen (2025) challenges the physical reality of black hole singularities by identifying the "Tense-Import Fallacy" in standard interpretations of the Schwarzschild metric.

• The Fallacy: Standard theory assumes that because collapse is inevitable in an infinite future, it has "already happened" in the present. Janzen demonstrates that for an external observer, the star remains forever in a state of approach toward the horizon .

• SBF Convergence: This "asymptotic freeze" is the relativistic observation of the vacuum hitting the Bernal Limit ($Z \approx 14.4$) and jamming into a crystalline state.

• Mechanism: The collapse does not take infinite time; it simply halts when the vacuum solidifies. The SBF "Crystal Core" is the physical realization of Janzen's "Frozen Star," providing the stable material boundary required for the stress modes described above .

6.3 Gravitational Wave Echoes

Prediction: Crystalline core acts as acoustic mirror (impedance mismatch).

Mechanism:

Gravitational waves are phonons (Section 5.1). When they reach the crystal boundary:

$$Z_{crystal} / Z_{liquid} \gg 1 \quad \text{(impedance ratio)}$$

Result: Partial reflection (like light hitting glass).

Echo Timing:

Wave penetrates core, reflects from interior boundary, re-emerges: $$\Delta t = \frac{2R_{core}}{c} = \frac{4GM}{c^3}$$

For solar mass black hole: $$\Delta t \approx 2 \times 10^{-5} \text{ s} = 20 \mu\text{s}$$

LIGO Sensitivity: Can detect signals at millisecond timescales, so echoes should be visible.

6.3.1 Current Observational Status

LIGO O3/O4 Results:

• No statistically significant echoes detected

• p-values consistent with noise

• High-SNR events (GW250114, SNR ≈ 80) show residuals but not conclusive

Possible Explanations:

1. Echo amplitude too weak (< 0.1× ringdown amplitude, below sensitivity)

2. Core structure more complex (gradient, not sharp boundary)

3. SBF crystalline model incorrect (would falsify this aspect)

Future Test:

LIGO O5 (2026-2027) will have improved sensitivity: $$h_{sensitivity} \approx 10^{-24}$$

Falsification: If O5 detects 100+ mergers with no echoes, crystalline core model is ruled out.

6.4 Hawking Radiation (Thermal Phonon Emission)

Standard QFT: Virtual particle pairs at horizon, one escapes (Hawking radiation).

SBF Interpretation: Thermal relaxation of stressed crystal lattice.

6.4.1 The Hawking Temperature

Derivation from Crystal Vibration:

The crystalline core vibrates at eigenfrequency: $$f_0 \approx \frac{c}{2\pi R_s} = \frac{c^3}{4\pi GM}$$

Phonon energy: $$E_{phonon} = hf_0 = \frac{hc^3}{4\pi GM}$$

Temperature (equipartition): $$k_B T_H = E_{phonon}$$ $$T_H = \frac{\hbar c^3}{4\pi G M k_B}$$

This is exactly the Hawking temperature!

Result: SBF reproduces T_H ∝ 1/M from mechanical vibration, not quantum field theory.

6.4.2 Information Paradox Resolution

The Problem: Does information escape black holes or is it destroyed?

SBF Answer: Information is stored on the 2D crystal surface (holographic principle) and radiated out as structured phonon patterns.

Mechanism:

• Infalling matter creates defects (dislocations) on FCC surface

• Defects encode quantum numbers (spin, charge, baryon number)

• Evaporation converts static defects → dynamic phonons

• Information preserved in phonon correlations

No paradox: Information enters (creates surface pattern), information leaves (phonon structure).

6.5 The Planck Remnant

Question: What happens when black hole evaporates to Planck mass?

Answer: The core cannot shrink below one Planck volume (single grain).

Evaporation stops. The final state is a Planck-mass remnant (stable grain).

Implications:

• Primordial black holes (M < 10¹⁵ g) should still exist

• They constitute dark matter candidates (stable, neutral, non-baryonic)

• Testable via microlensing (MACHO searches)