Фізика чорних дір

⚫ Schwarzschild metric

Static black hole

Spherically symmetric and uncharged. The metric is given by: ds² = -(1 - 2GM/(rc²))c²dt² + dr²/(1 - 2GM/(rc²)) + r²(dθ² + sin²θdφ²).

Event horizon

r = r_s = 2GM/c². The light cone is tilted, marking the causality boundary known as the Schwarzschild radius.

Singularity

At r = 0: curvature → ∞. Spacetime becomes incomplete, and the Einstein equations break down. Quantum gravity theories are needed to describe this region.

Geodesics

Radial plunge: r(t) → 0 as t → ∞ while proper time τ remains finite. This behavior is of interest in particle physics.

🌪️ Kerr Metric

Rotating black hole

M (mass), J (angular momentum). a = J/(Mc) is the spin parameter. |a| < GM/c². Spacetime rotation.

Ergosphere

Ergosphere: r_+ < r < r_+ + √(r_+² - a²). Frame dragging: −dt/dφ > 0. Negative energy.

Horizons

Outer: r+ = GM/c² + √((GM/c²)² - a²). Inner: r-. S = 0 for extremal (a = GM/c²).

Penrose process

Energy extraction: negative-energy particle in ergosphere. Black hole spin ↓, energy out.

⚡ Hawking Radiation

Quantum mechanics

Quantum fluctuations at the horizon: particle-antiparticle pairs. One falls in, one escapes.

Temperature

T_H = ℏc³/(8πGMk_B) ≈ 6×10⁻⁸ (M☉/M) K. Stellar BH: T_H ~ 10⁻⁷ K. Blackbody radiation.

Decay time

t ~ M³ ~ 10⁶⁹ years for stellar BH. Mass loss: dm/dt ~ -M⁻². Final burst.

Information paradox

Hawking radiation thermal: information loss? Unitarity violation. Black hole firewall? ER=EPR?

🔬 Thermodynamics

Bekenstein-Hawking entropy

S_BH = A/(4Gℏ) = 4πGM²/(ℏc). A = 16π(GM/c²)². Massive entropy: ~10⁷⁸ for stellar BH.

Laws

0: T_H constant. 1: dM = T_H dS + Ω dJ. 2: dS ≥ 0. 3: T_H → 0 for extremal.

Area theorem

Hawking: A always increases. Merger: A_merged ≥ A₁ + A₂. Information content.

Microscopic states

S_BH ~ ln(# microscopic states). String theory, loop quantum gravity calculations. Entropy matching.

🔍 Formation

Stellar collapse

M > 3 M☉: neutron star → BH. Core collapse supernova. Spinning accretion.

Binary mergers

GW150914+: merging BHs. 10-100 M☉. Galactic vs primordial origin.

Supermassive BHs

SMBH: 10⁶-10¹⁰ M☉. Galaxy centers. AGN accretion. Growth: mergers + accretion.

Primordial BHs

Early Universe: collapse ~M_Pl. Dark matter candidate? Yet to detect.

🌌 Observations

Event Horizon Telescope

EHT: Sgr A*, M87*. Radio interferometry. Shadow: ≈5 r_s diameter. Kerr metric tests.

Accretion disks

Thin disks: thermal emission. Broad Fe Kα lines: redshift, broadening. Reverberation mapping.

Quasars

AGN: SMBH accretion. Luminosity L ~ L_Edd = 4πGMm_pc/σ_T. High z (z ~ 7).

Gravitational waves

GW150914+: BH mergers. Masses, spins. Rate: ~100/year. Population studies.

📊 Graphs and Diagrams

Schwarzschild radius

Event horizon:

Black holes: M_crit for stellar (M > 3 M☉), supermassive (10⁶-10¹⁰ M☉).

Hawking radiation

Temperature and decay:

Stellar BH: t ~ 10⁶⁹ years. SMBH: t ~ 10¹⁰⁰ years.

Sun: r_s ≈ 3 km Earth: r_s ≈ 9 mm Sgr A*: r_s ≈ 10⁷ km

🧪 Practical Examples

Example 1: Sgr A*

Milky Way center: ~4×10⁶ M☉. r_s ≈ 10⁷ km. EHT: shadow ≈5 r_s. Kerr tests.

Example 2: GW150914

BH merger: 30+20 M☉. Final ~50 M☉, spin ~0.7. Energy ~3 M☉c². LIGO detection.

Example 3: Hawking radiation

Stellar BH (10 M☉): T_H ~ 10⁻⁷ K. Very cold. Decay ~10⁶⁹ years. Unobservable.

Example 4: M87*

EHT (2019): shadow image. ~6×10⁹ M☉. Kerr metric fit. Jet formation.

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Black holes: gravitational singularities