CRR Black Hole Physics

1. Core CRR Framework (Theoretical)

Coherence Accumulation

dC/dt = L(x,t)
C(t) = ∫₀ᵗ L(x,τ) dτ

The coherence functional C accumulates over time through the Lagrangian L, which measures system organization and information integration. For black holes, this represents the integrated matter-energy density in stable orbital configurations.

Rupture Operator

δ(t - t_i) : C(t_i) ≥ C_crit
C_crit = Ω log(Λ/λ₀)

When accumulated coherence exceeds critical threshold, the system undergoes discontinuous transition (rupture). For black holes, this maps to the event horizon where spacetime geometry forces discrete state change.

Rebirth/Regeneration

Φ(t) = ∫₀ᵗ φ(x,τ) exp(C(τ)/Ω) dτ

Post-rupture dynamics weighted exponentially by accumulated coherence. For black holes, this represents Hawking radiation emerging from quantum fluctuations at the horizon, weighted by the black hole's history.

2. Translation to Black Hole Physics

Coherence → Matter Distribution

C(M,r) ≈ ∫ ρ(r) dV
where ρ = matter density in accretion disk

Coherence maps to integrated mass distribution. Stable orbital matter in the accretion disk represents coherent state. The Lagrangian L(x,t) becomes the local matter-energy density.

Rupture → Event Horizon

δ(r - R_s) where R_s = 2GM/c²
Schwarzschild radius = rupture boundary

The event horizon is the rupture surface. Once matter crosses R_s, it undergoes irreversible transition (no return trajectory exists). This is a geometric rupture in spacetime itself.

Regeneration → Hawking Radiation

R[T_H] = dS/dt
T_H = ℏc³/(8πGMk_B)
L_H = ℏc⁶/(15360πG²M²)

Hawking temperature and luminosity represent regeneration flux. Quantum fluctuations at the horizon create particle-antiparticle pairs, with outgoing radiation weighted by the black hole's mass history (accumulated C).

3. Browser-Compatible Implementation

Important: Full CRR dynamics require O(n²) integration over all past states and cannot run in real-time in browsers. The code below uses visually accurate approximations that preserve CRR structure while remaining computationally feasible.

Coherence Accumulation (Simplified)

Full equation: C(t) = ∫₀ᵗ ρ(r,τ) dV dτ
Approximation: C(t) ≈ C(t-dt) + ΔC·dt

↓

// Coherence evolution using exponential relaxation
updateCRRStates(deltaTime) {
    // Simplified coherence: exponential approach to equilibrium
    this.crr.coherenceLevel += 
        (this.crr.coherenceStrength - this.crr.coherenceLevel) 
        * deltaTime * 0.5;
    
    // This approximates: dC/dt = k(C_target - C_current)
    // instead of full integral over matter distribution
}
                    

Rupture Events (Stochastic)

Full: δ(t - t_i) when C ≥ C_crit
Approximation: P(rupture) ∝ ruptureRate·dt

↓

// Rupture as stochastic process (Poisson-like)
if (Math.random() < this.crr.ruptureRate * deltaTime) {
    this.crr.ruptureEvents++;
    // Trigger visual disruption of particle states
    this.triggerRuptureEvent();
}

// Real physics: deterministic at C = C_crit
// Visualization: probabilistic for smoother animation
                    

Hawking Radiation (Particle Emission)

Full: Φ(t) = ∫ φ(τ) exp(C(τ)/Ω) dτ
Approximation: emission rate ∝ T_H · intensity

↓

// Hawking radiation as particle escape velocity
vec3 escape = normalize(pos) 
              * time * 0.01 * hawking;

// Particles generated near horizon (r ≈ 1.05 R_s)
// Velocity weighted by Hawking temperature
// Visual intensity represents exp(C/Ω) weighting

pos += escape + sin(time + pos * 5.0) * energy * 0.02;
                    

4. Physical Constants (Used in Calculations)

const G = 6.67430e-11;       // Gravitational constant (m³/kg·s²)
const c = 299792458;         // Speed of light (m/s)
const M_sun = 1.989e30;      // Solar mass (kg)
const k_B = 1.380649e-23;    // Boltzmann constant (J/K)
const hbar = 1.054571817e-34; // Reduced Planck constant (J·s)

// Schwarzschild radius (exact)
R_s = (2 * G * mass) / (c * c);

// Hawking temperature (exact)
T_H = (hbar * c³) / (8 * π * G * mass * k_B);

// Hawking luminosity (Stefan-Boltzmann law)
L_H = (hbar * c⁶) / (15360 * π * G² * mass²);
                    

5. Accuracy Assessment

Physically Accurate (Exact Calculations)

✓ Schwarzschild radius R_s = 2GM/c²
✓ Hawking temperature T_H = ℏc³/(8πGMk_B)
✓ ISCO radius for Kerr metric (spin-dependent)
✓ Surface gravity κ = GM/R_s²
✓ Hawking luminosity (Stefan-Boltzmann)
✓ Information rate (entropy flow)

Pedagogical Approximations (For Visualization)

⚠ Coherence accumulation: uses exponential relaxation instead of full spatial integration
⚠ Rupture events: stochastic trigger instead of deterministic threshold
⚠ Particle dynamics: simplified N-body without full GR geodesics
⚠ Accretion disk: temperature profile approximated, no magnetohydrodynamics
⚠ Gravitational lensing: artistic representation, not ray-traced
⚠ Frame dragging: simplified rotation, not full Kerr metric

Bottom Line: This visualization accurately computes black hole thermodynamics but uses simplified dynamics for real-time rendering. The CRR principles (coherence accumulation → rupture → regeneration) are preserved in structure, though not with full mathematical rigor. For research-grade simulations, see numerical relativity codes (Einstein Toolkit, SpEC).

📐 Mathematical Framework: CRR → Physics → Code

1. Core CRR Framework (Theoretical)

2. Translation to Black Hole Physics

3. Browser-Compatible Implementation

4. Physical Constants (Used in Calculations)

5. Accuracy Assessment

Black Hole System Components

Physical Properties

CRR Framework

Visualisation

CRR Framework States

Matter State Dynamics

Physical Properties

Thermodynamic Properties