Black Hole Simulation
A black hole in GCD is the Γ(ω->1) pole -- the point where drift cost diverges and return becomes impossible. The event horizon is not imported from general relativity; it is derived independently as the structural boundary where τ_R = inf_rec.
Collapsus generativus est; solum quod redit, reale est. -- At the horizon, nothing returns.
Launch Immersive Space Simulator ->GR <-> GCD Translation
| GR Concept | GCD Analog | Formula |
|---|---|---|
| Event Horizon | ω -> 1 pole | Γ(ω) = ω^3/(1-ω+ε) -> inf |
| Gravitational Field | dΓ/dω | Gradient of drift cost |
| Mass | |κ| well depth | Accumulated log-integrity |
| Tidal Force | d^2Γ/dω^2 | Cost curvature (spaghettification) |
| Gravitational Lensing | Δ = F - IC | Heterogeneity gap deflects trajectories |
| Time Dilation | Ascent/descent asymmetry | Cost(ascent) / Cost(descent) -> inf near horizon |
| Hawking Temperature | T_H = 1/(8pi|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: ω*a**(1-ω^2)^-1.5 |
| Ergosphere | ω > ω_ergo | Region where frame-dragging exceeds orbit speed |
| Penrose Process | η_P(a*) | Energy extraction: 1 - sqrt(1 - (a*/2)^2) |
| BH Entropy | S_BH = κ^2/(4pi) | Bekenstein analog from log-integrity squared |
| Orbital Precession | δφ(ω) | 6piω^2/(1-ω)^2 -- perihelion advance near well |
| GW Strain | h(ω,C,d) | 4ω^2C/(d+ε) -- quadrupole radiation |
| Time Dilation | dτ/dt | sqrt(1 - ω) -- proper time slowdown near horizon |
| Radiative Efficiency | η_rad(a*) | 1 - sqrt(1 - 2/(3r_ISCO)) -- accretion luminosity |
3D Gravity Well
The budget surface Γ(ω) rendered as a 3D funnel. The event horizon (ω -> 1) plunges to the center -- where cost diverges and τ_R = inf_rec. The bright band is the accretion disk. Orbiting particles spiral inward following Kepler-like trajectories. Drag to rotate * Scroll to zoom * Double-click to reset
Gravitational Infall
Drag ω toward 1.0 to "fall into" the black hole. Watch cost, tidal force, and escape ratio diverge at the event horizon. The three regime boundaries are marked.
Budget Surface
The full cost landscape over (ω, C). Height = Γ(ω) + alpha*C. Entity positions are marked -- objects deeper in the potential well have higher ω. The dark ridge at ω -> 1 is the event horizon.
Entity Positions
Color Scale
Entity Kernel Comparison
Each spacetime entity through the GCD kernel. The trace vector becomes six invariants; the invariants determine the regime. The heterogeneity gap Δ = F - IC is the lensing diagnostic.
Arrow of Time
Descent cost vs ascent cost as a function of depth. Near the horizon, escape cost diverges relative to infall cost -- this IS time dilation in GCD. The asymmetry ratio grows without bound as ω -> 1.
Tidal Force (d^2Γ/dω^2)
The second derivative of Γ(ω) measures the rate of change of the gravitational field -- the "spaghettification" diagnostic. Tidal force is small far from the horizon and catastrophic near it.
Gravitational Lensing
The heterogeneity gap Δ = F - IC determines lensing morphology. Small Δ -> Einstein ring. Large Δ -> weak distortion. The deflection angle theta = 4|κ|/(Δ+ε) mirrors Einstein's formula.
Frame-Dragging (Lense-Thirring)
A spinning black hole drags spacetime around it. In GCD, frame-dragging ω_drag = ω*a**(1-ω^2)^-1.5 diverges as ω -> 1 -- the ergosphere where dragging exceeds orbital velocity. At a* = 0 (Schwarzschild), there is no dragging; at a* -> 1 (extremal Kerr), the ergosphere fills the space.
Penrose Process -- Energy Extraction
The Penrose process extracts rotational energy from a spinning black hole. In GCD, efficiency η_P = 1 - sqrt(1 - (a*/2)^2). Maximum extraction at a* = 1 gives η_P ~= 13.4%. Also shown: radiative efficiency η_rad from accretion onto the ISCO, which reaches ~42% for extremal spin.
Proper Time Dilation (dτ/dt)
The ratio of proper time to coordinate time: dτ/dt = sqrt(1-ω). As ω -> 1, clocks freeze and time dilation becomes infinite. This is the GCD analog of gravitational time dilation at the Schwarzschild radius. Also shown: BH entropy S_BH = κ^2/(4pi), the Bekenstein analog.
Orbital Precession & Gravitational Waves
Orbital precession δφ = 6piω^2/(1-ω)^2 grows super-linearly near the horizon -- the GCD analog of Mercury's perihelion advance. Gravitational wave strain h = 4ω^2C/(d+ε) from the inspiral radiates the curvature channel as quadrupole waves.