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Simulations & Visualizations

Interactive simulations for Black Hole and Quantum Chaos phenomena.

OTOC Decay & Scrambling

Visualization of Out-of-Time-Order Correlator F(t) showing exponential chaos growth before scrambling time, then saturation.

F(t) ≈ 1 - (ε/N) eλLt
1.0
100
Lyapunov Exponent λL 2π T
Scrambling Time t* --

Black Hole Parameter Calculator

Calculate black hole properties from its mass: Schwarzschild radius, Hawking temperature, Bekenstein-Hawking entropy, and evaporation time.

10 M
Schwarzschild Radius rs --
Hawking Temperature TH --
Bekenstein-Hawking Entropy S --
Evaporation Time tevap --
Surface Gravity κ --

Page Curve Visualization

Evolution of Hawking radiation entropy. Initially rises (thermal), then falls after Page time (information exits) - demonstrating unitarity.

100
Page Time tPage --
Maximum Srad --

Kerr Black Hole Structure

Visualization of horizons and ergosphere for a rotating black hole (Kerr). Spin parameter a determines the ergosphere shape.

0.50
Outer Horizon r+/M --
Inner Horizon r-/M --
ISCO rISCO/M (prograde) --

λ MSS Chaos Bound

Comparison of Lyapunov exponents across various systems with the Maldacena-Shenker-Stanford bound. Black holes (and SYK model) saturate the bound.

λL ≤ 2πkBT/ℏ
MSS Bound (at T=1) λmax = 2π ≈ 6.28
Black Hole Saturates bound ✓
SYK Model Saturates bound ✓

∿ Quasinormal Mode Spectrum

Quasinormal mode frequencies for Schwarzschild black hole. Real part (oscillation) and imaginary part (decay rate).

ωn = ωR - i(n + ½)|λ|
l = 2

Penrose Diagram (Schwarzschild)

Conformal diagram for Schwarzschild black hole. Each point represents a 2-sphere. 45° lines = null geodesics (light).

Region I (Exterior) Region II (Interior) Region III (White Hole) Singularity (r=0) Singularity (r=0) Event Horizon (r = 2M) i⁺ (future timelike) i⁻ (past timelike) i⁰ (spatial) ℐ⁺ (future null) ℐ⁻ (past null) Observer
Regions I (ext), II (BH), III (WH)
Null infinity ℐ⁺, ℐ⁻

Complexity Growth

Computational complexity growth for black holes. Linear growth until exponential time, then saturation (Poincaré recurrence).

dC/dt = 2M/πℏ (Lloyd's bound)
5