Bekenstein–Hawking entropy

The Bekenstein–Hawking entropy is the thermodynamic entropy carried by a black hole, given by S = k_B c³ A / 4Għ = (k_B/4)(A/ℓ_P²), where A is the area of the event horizon and ℓ_P = √(ħG/c³) ≈ 1.6 × 10⁻³⁵ m is the Planck length. In words: the entropy equals one quarter of the horizon's surface area when that area is measured in Planck-sized cells, with each cell contributing k_B/4. It is the only physical formula in which the gravitational constant G, the quantum of action ħ, the speed of light c, and Boltzmann's constant k_B all appear together — making it the most concrete known bridge between gravity, quantum mechanics, and thermodynamics.

Jacob Bekenstein proposed in 1972–73 that horizon area should be interpreted as entropy, fixing the proportionality only up to an order-unity constant. Stephen Hawking's 1974 derivation of black-hole temperature, T = ħκ/2πk_Bc, pinned the constant at exactly 1/4. Unlike ordinary entropy, which scales with the volume of a system, this entropy scales with the boundary area: S ∝ M² for a Schwarzschild hole, so a solar-mass black hole carries about 10⁷⁷ k_B — roughly twenty orders of magnitude more than the gas that formed it. Black holes are therefore the most entropic objects the universe permits for a given size.

The formula implies a universal limit, the Bekenstein bound: no region of space can hold more entropy than would fit on a black hole exactly filling it, S ≤ A/4ℓ_P². This caps the information content of any volume by the area of its boundary, the founding statement of the holographic principle. What the entropy actually counts — which microstates of a black hole it enumerates, given that the no-hair theorem leaves only three macroscopic parameters — remains a central open problem, partly answered for special supersymmetric holes by the 1996 Strominger–Vafa string-theory calculation.

Introduced by Jacob Bekenstein in 1972–73 from an information argument and given its exact 1/4 coefficient by Stephen Hawking's 1974 temperature calculation; the entropy formula is engraved on Bekenstein's tombstone in Jerusalem.