Black Hole Gothic

Singularity Typography

Singularity

Where Light Cannot Escape

r → 0 | ρ → ∞ | T → 0

Event Horizon

Point of No Return

Event Horizon

Pure black sphere where v_escape > c

R_s = 2GM / c²

Accretion Disk

Superheated plasma spiraling at v ≈ 0.3c

T ≈ 10⁶ K | λ_peak ≈ 3 nm

Photon Ring

Gravitational lensing: r_ph = 3√3 GM/c²

θ_E = √(27GM/ c²d)

Feynman Diagrams

Electron Scattering e⁻ + γ → e⁻ + γ

Photon Exchange γ mediation

Beta Decay n → p + e⁻ + ν̄_e

Particle Trajectories

Geodesic Paths

Spacetime Curvature

Curved Geometry

Einstein Field Equations

G_μν + Λg_μν = 8πT_μν / c⁴

∇_μ G^μν = 0 (conservation)

R_μν - ½Rg_μν = 8πG T_μν

Where spacetime tells matter how to move

Quantum Observables

∞

Singularity

ρ → ∞

c

Light Speed

299,792,458 m/s

ħ

Planck Constant

1.054×10⁻³⁴

G

Gravitational

6.674×10⁻¹¹

k_B

Boltzmann

1.381×10⁻²³ J/K

σ

Stefan-Boltzmann

5.67×10⁻⁸

λ_c

Compton

ħ/mc

Event Horizon Radius

R_s = 2GM / c²

Schwarzschild Metric

For M_⊙: R_s ≈ 2.95 km

For 10⁶ M_⊙: R_s ≈ 2.95×10⁶ km

Metric Tensor

g₀₀

g₀₁

g₀₂

g₁₀

g₁₁

g₁₂

g₂₀

g₂₁

g₂₂

ds² = -(1-2M/r)dt² + (1-2M/r)⁻¹dr² + r²dΩ²

Geodesic Orbits

Photon Sphere r = 3GM/c²

Unstable circular photon orbits

Gravitational Lensing

Hawking Radiation

T_H = ħc³ / (8πGMk_B)

Black hole thermal emission (quantum tunneling at event horizon)

For M_⊙: T_H ≈ 62 nK (negligible)

Lifetime: t ≈ (5120πG²M³) / (ħc⁴)

Bekenstein-Hawking Entropy

S_BH = Akc³ / (4Għ)

Entropy proportional to horizon area (not volume)

A = 4πR_s² = 16πG²M² / c⁴

S/M ratio: s = S/M = 4πk_BGM / (ħc)

Kerr Metric (Rotating)

R_± = GM/c² ± √[(GM/c²)² - (J/Mc)²]

Rotating black holes have inner (R_-) and outer (R_+) horizons

Spin parameter: a = J/(Mc) | Dimensionless spin: a* = a/(GM/c²)

Ergosphere radius: r_E = GM/c² + √[(GM/c²)² - (J/Mc)²cos²θ]

Philosophical Note

At the event horizon, time ceases to flow as we understand it. Space and time exchange their roles. The fabric of reality itself curves beyond comprehension. Yet mathematics offers us a language to describe the indescribable.