Black Holes: Singularity, Event Horizon & Hawking Radiation

1. Formation & Classification

Black holes form through gravitational collapse when massive stars (>25 M☉) exhaust nuclear fuel. Stellar-mass black holes (3–100 M☉) form from core-collapse supernovae. Intermediate-mass black holes (10²–10⁵ M☉) may form through runaway stellar mergers in dense clusters. Supermassive black holes (10⁶–10¹⁰ M☉) reside in galactic centers — Sagittarius A* has 4.15 million solar masses. The Schwarzschild radius defines the event horizon: r_s = 2GM/c². For Earth's mass, this equals ~9mm.

2. Spacetime Geometry

The Schwarzschild metric describes non-rotating black holes: ds² = -(1-r_s/r)c²dt² + (1-r_s/r)⁻¹dr² + r²dΩ². The Kerr metric adds angular momentum: rotating black holes drag spacetime (frame-dragging). The ergosphere exists between the outer event horizon and the static limit. Penrose process extracts rotational energy — up to 29% of mass-energy for maximally spinning holes. Kerr-Newman metric includes charge. Extremal Kerr holes have a/M = 1 with ring singularity.

3. Hawking Radiation

Stephen Hawking (1974) showed black holes radiate thermally via quantum effects near the horizon. Virtual particle pairs near the event horizon: one falls in (negative energy), one escapes. Temperature: T = ℏc³/(8πGMk_B) — inversely proportional to mass. A solar-mass black hole radiates at 60 nanokelvin — negligible. Micro black holes (<10¹² kg) would evaporate rapidly. Evaporation time: t ∝ M³. The information paradox: does quantum information survive? Recent proposals: soft hair, island formula, ER=EPR conjecture.

4. Observational Evidence

Event Horizon Telescope (2019) imaged M87* — 6.5 billion solar masses, photon ring diameter ~42 μas. In 2022, EHT imaged Sgr A*. LIGO/Virgo detected gravitational waves from black hole mergers: GW150914 (36+29 M☉), GW190521 (85+66 M☉ — intermediate mass). X-ray binaries (Cygnus X-1) show accretion signatures. Quasars powered by supermassive BH accretion reach 10⁴⁷ erg/s luminosity. Tidal disruption events occur when stars approach too closely.

5. Open Problems

The black hole information paradox remains unsolved — unitarity vs. Hawking's calculation. Firewall paradox (AMPS 2012): complementarity vs. quantum mechanics vs. equivalence principle. Singularity resolution requires quantum gravity. Loop quantum gravity suggests "Planck stars" — bouncing cores. String theory provides microstate counting matching Bekenstein-Hawking entropy for extremal holes. Primordial black holes could constitute dark matter if they exist in the 10¹⁷–10²³ g range.

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