Definition
Black hole entropy is the entropy a black hole carries by virtue of its event horizon, equal to one-quarter of the horizon area measured in Planck units: S = kA / 4ℓ_P².
This formula — known as the Bekenstein-Hawking entropy — was the first concrete bridge between thermodynamics, quantum mechanics, and general relativity.
Why it matters
How it works
Jacob Bekenstein argued in 1972 that a black hole must have entropy, because otherwise dropping a hot cup of tea into one would erase entropy from the universe and violate the second law. He proposed S ∝ A, the horizon area, since the area never decreases (Hawking's earlier area theorem already showed this).
Hawking initially resisted the idea — thermodynamic entropy implies temperature, and a "black" hole was supposed not to radiate. But when he computed the quantum field theory near a horizon in 1974, he found exactly the thermal spectrum required to make Bekenstein's entropy a true thermodynamic quantity. The proportionality constant came out to S = A / 4 in Planck units.
The scaling is profound. For ordinary matter, entropy scales with volume — bits per cubic meter. For black holes, entropy scales with surface area — bits per square meter. A black hole the size of the observable universe would have an entropy of order 10¹²², dwarfing the entropy of all the stars, gas, and radiation inside it. Any complete theory of quantum gravity — string theory, loop quantum gravity, or other contenders — must reproduce this number by counting actual microstates.