Black Hole Simulation
Tier-2 · Spacetime MemoryA black hole in GCD is the Γ(ω→1) pole — the structural boundary where drift cost diverges and return becomes impossible (τ_R = ∞_rec). The event horizon is not imported from general relativity; it is derived independently from Axiom-0 as the point where no signal can return.
Collapsus generativus est; solum quod redit, reale est. — At the horizon, nothing returns.
GR ↔ GCD Translation
Translatio inter camposEvery gravitational concept maps to a GCD kernel quantity. Click to expand the full table.
| GR Concept | GCD Analog | Formula |
|---|---|---|
| Event Horizon | ω → 1 pole | Γ(ω) = ω³/(1−ω+ε) → ∞ |
| Gravitational Field | dΓ/dω | Gradient of drift cost |
| Mass | |κ| well depth | Accumulated log-integrity |
| Tidal Force | d²Γ/dω² | Cost curvature (spaghettification) |
| Gravitational Lensing | Δ = F − IC | Heterogeneity gap deflects trajectories |
| Ascent/Descent Asymmetry | Cost ratio → ∞ | Escape cost / infall cost diverges near horizon |
| Hawking Temperature | T_H = 1/(8π|κ|) | Inverse well depth — shallow wells radiate hotter |
| Gravitational Redshift | z = Γ/(1+Γ) | Photon energy loss escaping the well |
| ISCO | ω ≈ 0.50 | Innermost stable circular orbit — tidal destabilization |
| Photon Sphere | ω ≈ 0.65 | Unstable circular photon orbits |
| Frame-Dragging | ω_drag(ω, a*) | Lense-Thirring: a*·Γ(ω)/(1+Γ)² |
| Ergosphere | ω > ω_ergo | Region where frame-dragging exceeds orbit speed |
| Penrose Process | η_P(a*) | 1 − √((1+√(1−a*²))/2) — up to 29.3% |
| BH Entropy | S_BH = 4πκ² | Bekenstein analog — area law from log-integrity |
| Orbital Precession | δφ(ω) | 6πΓ(ω)/(1+Γ)² — perihelion advance |
| GW Strain | h(ω, C, d) | Γ²·C/d — quadrupole radiation |
| Proper Time Dilation | dτ/dt | √(1 − 2Γ/(1+Γ)) — clocks freeze at horizon |
| Radiative Efficiency | η_rad(a*) | 1 − E_ISCO — up to 42% at extremal spin |
3D Gravity Well
Puteus gravitatisThe budget surface Γ(ω) rendered as a 3D funnel. The center plunges toward the event horizon where cost diverges and τ_R = ∞_rec. The luminous band is the accretion disk; particles spiral inward on Kepler-like trajectories. Drag to rotate · Scroll to zoom · Double-click to reset.
Gravitational Infall
Casus gravitatisSlide ω toward 1.0 to fall toward the event horizon. Drift cost Γ, tidal force d²Γ/dω², and escape ratio all diverge at the boundary. The three regime zones are shaded: Stable (ω < 0.038), Watch (0.038 ≤ ω < 0.30), and Collapse (ω ≥ 0.30).
Budget Surface
Superficies impensaeThe full cost landscape over (ω, C). Height = Γ(ω) + α·C. Objects deeper in the potential well have higher ω; the dark ridge at ω → 1 is the event horizon. Entity positions are marked with regime-colored dots.
Entity Positions
Color Scale
Entity Kernel Comparison
Comparatio entiumEach spacetime entity passes through the GCD kernel K: [0,1]ⁿ × Δⁿ → (F, ω, S, C, κ, IC). The six invariants determine the regime; the heterogeneity gap Δ = F − IC is the lensing diagnostic. Cards update when you change the scenario above.
Arrow of Time
Sagitta temporisAscent cost vs. descent cost as a function of depth. Near the horizon, escape cost diverges relative to infall cost — this asymmetry is time dilation in GCD. The ratio grows without bound as ω → 1.
Tidal Force
Vis discerptionisThe second derivative d²Γ/dω² measures the rate of change of the gravitational field — the spaghettification diagnostic. Tidal force is negligible far from the horizon and catastrophic near it.
Gravitational Lensing
Deflexio luminisThe heterogeneity gap Δ = F − IC determines lensing morphology. Small Δ produces an Einstein ring; large Δ yields weak distortion. The deflection angle θ = 4|κ|/(Δ+ε) is the GCD analog of Einstein's deflection formula.
Frame-Dragging
Tractio referentiae — Lense-ThirringA spinning black hole drags the local inertial frame. In GCD, ω_drag = a*·Γ(ω)/(1+Γ)² grows with both drift and spin. At a* = 0 (Schwarzschild) there is no dragging; as a* → 1 the ergosphere swells. Use the spin slider above to see the effect.
Penrose Process — Energy Extraction
Processus PenroseThe Penrose process extracts rotational energy from a spinning black hole's ergosphere. Efficiency η_P = 1 − √((1+√(1−a*²))/2) reaches ≈29.3% at extremal spin (a* → 1). Radiative efficiency η_rad from accretion onto the spin-dependent ISCO is also shown, peaking at ≈42% for a* → 1.
Proper Time Dilation & Entropy
Moratio temporis propriiThe proper-to-coordinate time ratio dτ/dt approaches zero at the horizon — clocks freeze for a distant observer. Black hole entropy S_BH = 4πκ² follows the Bekenstein area law: deeper wells (larger |κ|) have higher entropy.
Orbital Precession & Gravitational Waves
Praecessio et undae gravitatisOrbital precession δφ = 6πΓ(ω)/(1+Γ)² grows super-linearly near the horizon. Gravitational wave strain h = Γ²·C/d radiates the curvature channel as quadrupole waves — only asymmetric systems (C > 0) emit. The observer distance slider above scales h inversely.