Black Hole Entropy

Hawking radiation, Bekenstein-Hawking entropy, and the information paradox
10 M☉
10
Mass (M☉)
29.5 km
Schwarzschild Radius
6.2 nK
Hawking Temperature
--
Entropy (k_B)
--
Evaporation Time

Bekenstein-Hawking Entropy

S = A·c³ / (4Gℏ)

Entropy is proportional to horizon area, not volume — a radical departure from thermodynamics. For a 10 M☉ black hole: S ≈ 1078 k_B. Each Planck area (ℓ_P² ≈ 2.6×10⁻⁷⁰ m²) carries exactly ln(2) bits.

Hawking Temperature

T = ℏc³ / (8πGMk_B)

A 10 M☉ black hole radiates at ~6 nanokelvin — far colder than the CMB. As mass decreases, temperature rises — a runaway evaporation ending in a burst. Hawking radiation is the black body spectrum of virtual pair production at the horizon.

Information Paradox

S_Page ≈ S_BH at half-evaporation

Hawking 1974: radiation appears thermal → pure state → mixed state (information lost). Page 1993: unitarity requires information return by Page time. Maldacena 2001 (AdS/CFT): information preserved. Penington/Almheiri 2019: island formula resolves paradox via replica wormholes.