Black Hole Simulator

A physically accurate 3D black hole visualization and sonification system using C++ and OpenGL, implementing real general relativity physics with scientifically accurate audio generation.

Features

Physics Accuracy

• Dual Physics Engines: Factory pattern supporting both Schwarzschild (non-rotating) and Kerr (rotating) black holes
• Schwarzschild Metric: Accurate spacetime curvature around non-rotating black holes
• Kerr Metric: Full rotating black hole physics with frame dragging and ergosphere effects
• Ergosphere Physics: Regions where spacetime itself is dragged, enabling energy extraction
• Dual Event Horizons: Outer and inner horizons for rotating black holes with spin-dependent geometry
• Geodesic Ray Tracing: Light paths calculated using Einstein's field equations
• 4th-Order Runge-Kutta Integration: Numerical solution of geodesic differential equations
• Adaptive Step Sizes: Smaller integration steps near the event horizon for accuracy

Visual Effects

• Gravitational Lensing: Realistic light bending around the black hole
• Event Horizon: True black region where light cannot escape (dual horizons for Kerr black holes)
• Photon Sphere: Critical orbital radius at 1.5 × Schwarzschild radius
• Ergosphere Visualization: Frame-dragging region where spacetime is dragged around rotating black holes
• Relativistic Jets: Highly collimated plasma streams with synchrotron emission and relativistic beaming
• Accretion Disk: Hot plasma with temperature gradient and Doppler shift effects
• Background Stars: Procedural star field showing gravitational lensing distortion

Music & Sonification

• Scientific Sonification: Real-time audio generation from physics calculations
• Gravitational Wave Audio: Convert orbital frequencies using Kepler's laws (f = (1/2π)√(GM/r³))
• Electromagnetic Sonification: Temperature-based frequencies from accretion disk (f = k_B T/h)
• Relativistic Effects: Gravitational redshift, Doppler shifts, and frame-dragging audio
• Musical Scales: Physics frequencies mapped to pleasant musical scales (pentatonic, chromatic, harmonic series)
• Multi-Physics Audio: Simultaneous sonification of jets, tidal forces, and Hawking radiation
• Spatial Audio: Stereo positioning based on frame-dragging and observer position
• Real-time Processing: 44.1kHz audio with professional-grade EQ, compression, and de-essing

Technical Implementation

• Factory Pattern Architecture: Modular physics engines with runtime switching between Schwarzschild and Kerr
• Real-time Ray Marching: GPU-accelerated fragment shader ray tracing
• Camera System: Precision orbital camera with emergency escape controls
• Performance System: Adaptive quality scaling, turbo mode, and automatic FPS optimization
• Performance Logging: Real-time monitoring and analysis of rendering performance
• Cross-platform: macOS native with OpenGL 3.3 Core Profile
• Modular Audio Engine: Separate sonification system with EQ, compression, and spatial audio

Physics Background

Schwarzschild Metric (Non-rotating Black Holes)

Event Horizon: rs = 2GM/c² = 2M (in geometric units) Photon Sphere: rph = 1.5rs = 3M ISCO: risco = 3rs = 6M

Metric Components (exact implementation):

g_tt = -(1 - rs/r)
g_rr = 1/(1 - rs/r) 
g_θθ = r²
g_φφ = r²sin²θ

Geodesic Equations (actual implementation):

d²t/dλ² = -2rs/(r(r-rs)) × dr/dλ × dt/dλ
d²r/dλ² = -rs(r-rs)/(2r³)(dt/dλ)² + rs/(2r(r-rs))(dr/dλ)² + (r-rs)[(dθ/dλ)² + sin²θ(dφ/dλ)²]  
d²θ/dλ² = -2/r × dr/dλ × dθ/dλ + sinθcosθ(dφ/dλ)²
d²φ/dλ² = -2/r × dr/dλ × dφ/dλ - 2cosθ/sinθ × dθ/dλ × dφ/dλ

Kerr Metric (Rotating Black Holes)

Boyer-Lindquist Coordinates with spin parameter a: Horizons: r± = M ± √(M² - a²)
Ergosphere: r_ergo = M + √(M² - a²cos²θ) Critical Functions:

• Δ = r² - 2Mr + a²
• Σ = r² + a²cos²θ
• A = (r² + a²)² - a²Δsin²θ

Metric Components:

g_tt = -(1 - 2Mr/Σ)
g_rr = Σ/Δ
g_θθ = Σ
g_φφ = (A sin²θ)/Σ
g_tφ = -2Mar sin²θ/Σ  // Frame-dragging term

Frame-Dragging Angular Velocity:

ω = -g_tφ/g_φφ = 2Mar/(r² + a²)² - a²Δsin²θ)

4th-Order Runge-Kutta Integration

Implementation (both Schwarzschild and Kerr):

k1 = h × f(x_n, y_n)
k2 = h × f(x_n + h/2, y_n + k1/2)  
k3 = h × f(x_n + h/2, y_n + k2/2)
k4 = h × f(x_n + h, y_n + k3)
y_{n+1} = y_n + (k1 + 2k2 + 2k3 + k4)/6

Sonification Physics (Actual Implementations)

Orbital Frequency to Audio:

f_orbital = (1/(2π)) × √(GM/r³)    [Kepler's Law]
audio_freq = scale × log(f_orbital/1e-6) / log(1e6)

Electromagnetic to Audio:

f_em = c/λ
log_audio = log_min + (log_em - 6)/(18-6) × (log_max - log_min)
audio_freq = 10^log_audio

Temperature to Audio (Planck's relation):

f_thermal = k_B × T / h
audio_freq = scale × √(f_thermal/1e12)

Synchrotron Emission (relativistic jets):

f_sync = (eB/2πm_e) × γ²    [Cyclotron × Lorentz²]
audio_freq = scale × log(f_sync/1e9) / log(1e6)

Relativistic Jets Physics

Beaming Factor (exact implementation):

δ = 1/(γ(1 - β·n̂))    [Relativistic Doppler]
γ = 1/√(1 - β²)       [Lorentz factor]
β = v/c               [Normalized velocity]

Synchrotron Power:

P_sync ∝ B²γ²β²sin²α   [Larmor formula]
Temperature: T = 1e9 × (B/B_0)^(2/3)  [Equipartition]

Beaming Cone Angle:

θ_cone ≈ 1/γ          [Natural beaming cone]

Musical Scale Mapping

Physics-to-Audio Conversion:

log_normalized = (log₁₀(f_physics) - log₁₀(f_min)) / (log₁₀(f_max) - log₁₀(f_min))
f_musical = A₄ × 2^((scale_note - 69)/12)  [Equal temperament]

GPU Shader Calculations (Real-time Ray Tracing)

Einstein Deflection Angle (weak-field approximation):

α ≈ 4GM/(c²b) = 2rs/b    [Impact parameter b]
deflectionAngle = 2.0 * schwarzschildRadius / max(impactParameter, 0.1 * rs)

Strong-Field Deflection (Darwin's exact formula):

δφ ≈ 4Rs/b + 15πRs²/(4b²) + 128Rs³/(3b³) + ...  [Higher-order terms]

Impact Parameter Calculation:

impactParameter = distanceToBlackHole × sin(acos(dot(rayDir, toBlackHole)))

Coordinate Transformations (Boyer-Lindquist ↔ Cartesian):

r = √(0.5 × (ρ² + √(ρ²⁺ + 4a²z²)))  where ρ² = x² + y² + z² - a²
x = √(r² + a²) × sinθ × cosφ
y = √(r² + a²) × sinθ × sinφ  
z = r × cosθ

Adaptive Integration Methods

Schwarzschild Step Size:

if (r < 2rs):     h = 0.001         [Event horizon region]
if (2rs < r < 5rs): h = linear interpolation  [Transition zone] 
if (r > 5rs):     h = 0.1           [Far field]

Kerr Step Size (includes ergosphere):

if (r < 1.1r₊ || inErgosphere): h = 0.001    [Critical regions]
if (r < 2r_ergo):                h = interpolated  [Near ergosphere]
if (r > 2r_ergo):                h = 0.1      [Safe region]

Performance-Adaptive Quality:

if (frameTime > 100ms): performance_scale = 0.2  [Emergency mode]
if (frameTime > 50ms):  performance_scale = 0.4  [Severe degradation] 
zoom_speed = base_speed × (1 - 0.95 × journey_progress²)  [Dramatic slowdown]

Strong-Field Deflection (exact Darwin series):

u = M/b                          [Dimensionless parameter]
α₀ = 4u                         [Weak field (Newton)]
α₁ = (15π/4)u²                  [Post-Newtonian correction]
α₂ = (128/3)u³                  [Strong field correction] 
α₃ = (3465π/64)u⁴               [Higher-order relativistic]
α_total = α₀ + α₁ + α₂ + α₃      [Complete deflection]

System Requirements

• macOS 10.15+ (optimized for Apple Silicon M4 Max)
• OpenGL 3.3 Core Profile support
• CoreAudio framework (built-in on macOS)
• Audio output device (headphones recommended for spatial audio)
• CMake 3.20+
• C++17 compatible compiler (Clang recommended)

Building

Quick Build

./build.sh

Manual Build

mkdir build && cd build
cmake -DCMAKE_BUILD_TYPE=Release ..
make -j$(sysctl -n hw.ncpu)

Running

cd build
./BlackHoleSimulator

Controls

Navigation Controls

• SPACEBAR: Toggle auto-movement through the black hole (enable/disable autopilot)
• Mouse Wheel: Zoom in/out (allows extremely close approach to event horizon)
• W/S: Zoom in/out (hold Ctrl for fine control)
• Q/E: Precision zoom control (ultra-fine steps)
• BACKSPACE: Emergency reverse to safe distance (instant escape)

Visual Controls

• 1-3: Change render modes
• L: Toggle light ray visualization
• J: Toggle relativistic jets on/off
• U/I: Increase/decrease jet Lorentz factor
• O/P: Increase/decrease jet opening angle
• V: Show dramatic visual effects status
• F: Test jump to close distance (1.5 Rs)
• ESC: Exit

Performance Controls

• Q: Toggle optimization mode (OPTIMIZED ↔ STANDARD)
• R: Reset performance settings to defaults
• T: TURBO mode (ultra performance for smooth movement)
• Y: QUALITY mode (all features enabled)
• E: Toggle control panel

Audio Controls

• S: Toggle black hole sonification on/off
• +/-: Adjust audio volume
• M: Toggle spatial audio effects
• N: Toggle relativistic audio effects

Physics Parameters

Schwarzschild (Non-rotating) Black Hole

Default configuration with mass equal to our Sun:

• Mass: 1 Solar Mass (1.989 × 10³⁰ kg)
• Spin Parameter: a = 0.0 (no rotation)
• Event Horizon: ~2,954 meters (single horizon)
• Photon Sphere: ~4,431 meters (exact calculation using Newton's method)
• ISCO (Innermost Stable Circular Orbit): ~8,862 meters

Kerr (Rotating) Black Hole

Configurable spin parameter up to near-extremal rotation:

• Spin Parameter Range: -0.998 ≤ a ≤ 0.998 (dimensionless)
• Outer Event Horizon: r₊ = M + √(M² - a²) (depends on spin)
• Inner Event Horizon: r₋ = M - √(M² - a²) (Cauchy horizon)
• Ergosphere: r_ergo = M + √(M² - a²cos²θ) (frame-dragging region)
• ISCO: Spin-dependent, ranges from ~6M (a=0) to ~1.2M (a→1)
• Frame Dragging: Spacetime rotation effects with angular velocity Ω

Audio Parameters

• Frequency Range: 20 Hz - 20 kHz (typical hearing range)
• Sample Rate: 44.1 kHz professional audio standard
• Default Musical Scale: Major Pentatonic (most consonant)
• Dynamic Range: 60 dB with professional audio compression
• Spatial Audio: Stereo positioning based on frame-dragging effects

Performance

Optimized for real-time rendering on M4 Max MacBook:

• Adaptive Ray Marching: Variable step sizes based on gravitational field strength
• GPU Fragment Shaders: Parallel ray computation
• Efficient Integration: 4th-order Runge-Kutta with optimized step selection
• Auto-FPS Optimization: Dynamic quality scaling to maintain target framerate
• Turbo Mode: Ultra-performance mode for smooth extreme proximity navigation
• Emergency Controls: Instant escape mechanisms for safe exploration

Software Architecture

Factory Pattern Physics Engine

• Modular Design: Separate implementations for Schwarzschild and Kerr physics
• Runtime Switching: Dynamic selection between physics engines based on spin parameter
• Interface Abstraction: Clean separation between physics calculations and rendering
• Extensible Framework: Easy addition of new black hole types (e.g., charged Reissner-Nordström)

Component System

• BlackHolePhysics: Core physics calculations and metric computations
• RayTracer: GPU-accelerated ray marching with geodesic integration
• BlackHoleSonification: Real-time audio generation from physics parameters
• RelativisticJets: Synchrotron emission and relativistic beaming calculations
• LightSystem: Photon sphere and gravitational lensing effects
• PerformanceLogger: Real-time monitoring and adaptive optimization
• MelodiousCalculator: Musical scale mapping for pleasant sonification

Audio Pipeline

• Multi-Physics Sources: Orbital dynamics, thermal emission, jets, and gravitational waves
• Professional Processing: EQ, compression, de-essing, and spatial audio
• Musical Mapping: Physics frequencies converted to consonant musical scales
• Real-time Generation: 44.1 kHz sample rate with low-latency CoreAudio integration

Scientific Accuracy

This simulation implements exact solutions with validated numerical methods:

General Relativity Implementation

1. Schwarzschild Metric: Complete implementation with exact metric components g_μν
2. Kerr Metric: Full Boyer-Lindquist coordinates including frame-dragging term g_tφ
3. Geodesic Integration: 4th-order Runge-Kutta with adaptive step sizes near horizons
4. Event Horizon Detection: Exact calculation using r < r_± for Kerr, r < 2M for Schwarzschild

Physics Calculations

1. Orbital Mechanics: Kepler's third law f = (1/2π)√(GM/r³) with exact GM computation
2. Photon Spheres: Newton's method for r = 1.5rs (Schwarzschild) and spin-dependent Kerr orbits
3. ISCO Computation: 3rs for Schwarzschild, complex spin-dependent formula for Kerr
4. Frame Dragging: Angular velocity ω = 2Mar/((r²+a²)²-a²Δsin²θ) from metric

Relativistic Jets

1. Synchrotron Emission: Exact f = (eB/2πm_e)γ² with realistic magnetic field profiles
2. Relativistic Beaming: Doppler factor δ = 1/(γ(1-β·n̂)) with proper Lorentz transformations
3. Magnetic Field Evolution: B(r) ∝ (r_h/r)^1.5 with ergosphere amplification for rotating BHs
4. Temperature Profiles: Equipartition T = 10⁹(B/B₀)^(2/3) K with realistic cooling

Audio Sonification Accuracy

1. Physics Frequencies: All derived from fundamental constants (ℏ, k_B, c, G, e, m_e)
2. Logarithmic Mapping: Proper frequency scaling from physics ranges to audible spectrum
3. Musical Quantization: Equal temperament f = A₄ × 2^((n-69)/12) with consonant scale selection
4. Relativistic Effects: Gravitational redshift √(-g_tt_obs/-g_tt_emit) and Doppler corrections

Numerical Methods

1. Adaptive Integration: Step size h ∝ 1/√(|acceleration|) near strong-field regions
2. Coordinate Transformations: Exact spherical ↔ Cartesian with singularity handling
3. Metric Evaluation: Cache-optimized computation of Σ, Δ, A functions for Kerr
4. Professional Audio: 44.1 kHz sampling with anti-aliasing and musical EQ processing

References

Physics & General Relativity

• Schwarzschild, K. (1916). "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie"
• Kerr, R. P. (1963). "Gravitational field of a spinning mass as an example of algebraically special metrics"
• Chandrasekhar, S. (1983). "The Mathematical Theory of Black Holes"
• Luminet, J.-P. (1979). "Image of a spherical black hole with thin accretion disk"

Sonification & Musical Theory

• Hermann, T. (2008). "Taxonomy and definitions for sonification and auditory display"
• Kramer, G. (1994). "Auditory Display: Sonification, Audification, and Auditory Interfaces"
• Dubus, G. & Bresin, R. (2013). "A systematic review of mapping strategies for the sonification of physical quantities"
• Milne, A. (2013). "A computational model of the cognition of tonality"

Future Enhancements

Visual Features

• Gravitational wave visualization ripples in spacetime
• Multiple black hole systems and binary mergers
• Charged black holes (Reissner-Nordström metric)
• Wormhole traversability visualization
• VR/AR support for immersive exploration

Physics Extensions

• Tidal force visualization on extended objects
• Hawking radiation particle animation
• Frame-dragging field line visualization
• Plasma magnetohydrodynamics in jets

Audio Features

• MIDI output for external synthesizers
• Real-time gravitational wave data sonification (LIGO/Virgo data)
• Interactive physics parameter control via audio feedback
• Binaural beats from orbital resonances and beat frequencies
• Machine learning-optimized pleasant frequency mapping
• Sonification of gravitational memory effects

License

MIT License - See LICENSE file for details.

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"The most beautiful thing we can experience is the mysterious." - Albert Einstein