Bekenstein-Hawking Entropy

A black hole, classical general relativity insists, is the simplest object in the universe: the no-hair theorem strips it down to three numbers — mass $M$ , angular momentum $J$ , charge $Q$ — and nothing else. That very simplicity looks like a catastrophe for thermodynamics. Drop a canister of hot gas across the horizon and its entropy disappears from the outside world, so the entropy of the accessible universe falls and the second law is broken. Jacob Bekenstein refused the loophole. In 1972 he argued that a black hole must itself carry entropy, and that the one horizon quantity Hawking's area theorem forbids from ever decreasing — the area $A$ — is what that entropy must be built from. Dimensional analysis then forces $S\propto A/\ell_P^2$ , entropy measured in Planck areas. The proportionality became an equation when Hawking computed, in 1974, that a black hole formed by collapse is not black at all but glows with a thermal spectrum at temperature $T_H=\hbar\kappa/2\pi k_B c$ , set by the surface gravity $\kappa$ of its Killing horizon. A real temperature turns the four laws of black hole mechanics from a suggestive analogy into literal thermodynamics, and matching the mechanical first law to $\mathrm{d}E=T\,\mathrm{d}S$ pins the coefficient to exactly one quarter: $S_{\mathrm{BH}}=k_B c^3 A/4G\hbar$ . The number is staggering — a solar-mass hole holds $\sim10^{77}k_B$ , some nineteen orders of magnitude more than the star it came from — and, uniquely among thermodynamic systems, it scales as area rather than volume. That single fact, that the maximum information storable in a region is set by its boundary, is the seed of the holographic principle and our deepest clue to the microscopic structure of quantum gravity.

Key takeaways

The Bekenstein-Hawking entropy of a black hole is one quarter of its horizon area measured in Planck units, $S_{\mathrm{BH}}=k_B A/4\ell_P^2$ with $\ell_P^2=G\hbar/c^3$ — the only entropy in physics that scales with area, not volume.
Bekenstein (1972-73) proposed $S\propto A$ to rescue the second law when matter falls in; Hawking (1974-75) fixed the coefficient at $1/4$ by finding the horizon radiates at $T_H=\hbar\kappa/2\pi k_B c$ .
The first law of black hole mechanics $\mathrm{d}(Mc^2)=\tfrac{\kappa c^2}{8\pi G}\,\mathrm{d}A+\Omega_H\,\mathrm{d}J+\Phi_H\,\mathrm{d}Q$ becomes ordinary thermodynamics the instant $T=T_H$ and $S=S_{\mathrm{BH}}$ are identified.
A one-solar-mass Schwarzschild hole has $S\approx1.0\times10^{77}k_B$ and $T_H\approx6.2\times10^{-8}\,\mathrm{K}$ — colder than the cosmic microwave background and vastly more entropic than its progenitor star.
The generalized second law $\delta(S_{\mathrm{BH}}+S_{\mathrm{matter}})\ge0$ repairs thermodynamics, and demanding it hold implies the Bekenstein bound $S\le2\pi k_B RE/\hbar c$ on any ordinary system.
String theory (Strominger-Vafa 1996) and loop quantum gravity both reproduce $A/4$ by counting microstates, evidence that $S_{\mathrm{BH}}$ is genuine statistical-mechanical entropy, not a formal analogy.
Key idea: a horizon of area $A$ carries entropy $S=k_B A/4\ell_P^2$ and temperature $T=\hbar\kappa/2\pi k_B c$ , forced together so that the first law of black hole mechanics becomes literal thermodynamics.

The core insight

A black hole horizon of area $A$ carries entropy $S=k_B A/4\ell_P^2$ and temperature $T=\hbar\kappa/2\pi k_B c$ ; these are forced together so that the first law of black hole mechanics $\mathrm{d}(Mc^2)=(\kappa c^2/8\pi G)\,\mathrm{d}A+\Omega_H\,\mathrm{d}J+\Phi_H\,\mathrm{d}Q$ becomes ordinary thermodynamics.

Where this lives today

Black hole entropy has moved from paradox to precision tool. The Event Horizon Telescope's images of M87 and Sagittarius A are portraits of the two largest entropy reservoirs we can point at — M87* alone holds $\sim4\times10^{96}k_B$ , more than all the starlight ever emitted in the observable universe — and its shadow fixes the horizon area the formula turns into that number. In string theory the Strominger-Vafa counting of D-brane states reproduces $A/4\ell_P^2$ for near-extremal holes, and loop quantum gravity recovers the same law by counting spin-network punctures. It is realized exactly in AdS/CFT, where a horizon area equals $4G$ times the entropy of a thermal state of the dual boundary theory, and Ryu-Takayanagi extends $A/4$ to boundary entanglement. The deepest tension it exposes — whether evaporation is unitary, so $S_{\mathrm{BH}}$ counts recoverable microstates — remains the sharpest open problem in fundamental physics, driving the recent "island" calculations that reproduce the expected Page curve.

The physical picture

Picture lowering a box of high-entropy gas slowly toward the horizon on a string. As you pay out the rope the box's energy, measured at infinity, is redshifted away — you can extract almost all of $mc^2$ as useful work before letting go, so the mass the hole gains is tiny. Yet whatever entropy the gas carried is now sealed behind the horizon, gone from the outside universe. If nothing compensates, the external second law is violated outright. Bekenstein's resolution: the horizon area must grow, and that growth is not bookkeeping but entropy — the black hole pays back, in area, at least the entropy you hid. Only the area can play this role, because Hawking's area theorem guarantees it never decreases classically.

Figure

A box of hot gas of entropy S_matter falls through the horizon and vanishes from the outside universe, yet the area A grows so that S_BH = k_B A / 4ℓ_P² rises by at least as much — the generalized second law δS_BH + δS_matter ≥ 0. The horizon behaves as if tiled in Planck cells of area ≈ 4ℓ_P², one nat of hidden entropy per cell

Now think of surface gravity $\kappa$ as the redshifted acceleration a static observer needs to hover just above the horizon. An observer accelerating through the quantum vacuum sees a thermal bath (the Unruh effect), and the black hole version of that bath is Hawking radiation at $T_H\propto\kappa$ . Once the horizon has both an energy $Mc^2$ and a temperature, thermodynamics is unavoidable: $\mathrm{d}S=\mathrm{d}(Mc^2)/T_H$ integrates to an entropy equal to $A/4$ in Planck units. The temperature is what upgrades Bekenstein's proportionality into an exact law and fixes the elusive coefficient. The most radical feature is how little entropy fits in how much space. An ordinary gas of radius $R$ has entropy $\sim R^3$ ; a black hole of radius $R$ has entropy $\sim R^2$ , yet it is the maximum-entropy object that fits — anything denser would already have collapsed to a larger hole. The ceiling on information in a region is therefore set by its bounding area, not its volume. The second figure makes this quantitative: across every black hole we know of, the entropy tracks $M^2$ over more than fifty orders of magnitude.

Figure

Bekenstein-Hawking entropy versus mass for real black holes on a log-log scale: S ∝ M² with slope 2, from a primordial hole of 2×10¹¹ kg up to M87* at 6.5×10⁹ solar masses. A solar-mass hole sits at S ≈ 1.0×10⁷⁷ k_B and T_H ≈ 6.2×10⁻⁸ K, colder and vastly more entropic than the star that formed it

The core equations

The central formula assigns an entropy to a horizon purely from its area. It is written two ways: in fundamental constants, and in the compact Planck-area form that shows the area is being counted in units of $\ell_P^2$ .

S_{\mathrm{BH}}=\frac{k_B c^3}{4 G\hbar}\,A=\frac{k_B}{4}\,\frac{A}{\ell_P^2} \qquad (1)

where: $S_{\mathrm{BH}}$ — black hole entropy; $A$ — event-horizon area; $\ell_P=\sqrt{G\hbar/c^3}=1.616\times10^{-35}\,\mathrm{m}$ — Planck length; $k_B$ — Boltzmann constant. Reading: the entropy is one quarter of the horizon area in Planck areas, so one square metre of horizon already stores $\sim10^{69}$ nats — huge because $\ell_P^2$ is so tiny. The coefficient $1/4$ is not free; it is forced by the horizon temperature. Hawking's calculation gives a thermal spectrum whose temperature is proportional to the surface gravity $\kappa$ .

T_H=\frac{\hbar\,\kappa}{2\pi k_B c} \qquad (2)

where: $T_H$ — temperature of the emitted radiation seen at infinity; $\kappa$ — surface gravity, the redshifted acceleration at the horizon (units $\mathrm{m/s^2}$ ). Reading: the $2\pi$ is the fingerprint of the Euclidean-time periodicity that makes the horizon look thermal — the same $2\pi$ as in the Unruh temperature. For a Schwarzschild black hole the surface gravity is fixed entirely by the mass, which is why an uncharged non-rotating hole gets cooler as it grows.

\kappa=\frac{c^4}{4GM}\qquad(\text{Schwarzschild}) \qquad (3)

where: $\kappa$ — surface gravity evaluated at $r_+=2GM/c^2$ ; $M$ — mass. Reading: $\kappa\propto 1/M$ , so $T_H\propto 1/M$ — bigger holes are colder, a hallmark of the negative heat capacity that makes black hole thermodynamics unusual. The four laws of black hole mechanics are summarized by the first law, a purely geometric identity relating changes in mass, area, spin and charge.

\mathrm{d}(Mc^2)=\frac{\kappa c^2}{8\pi G}\,\mathrm{d}A+\Omega_H\,\mathrm{d}J+\Phi_H\,\mathrm{d}Q \qquad (4)

where: $\Omega_H$ — angular velocity of the horizon; $J$ — angular momentum; $\Phi_H$ — horizon electrostatic potential; $Q$ — charge. Reading: this follows from the Einstein equations alone, no thermodynamics assumed — its resemblance to $\mathrm{d}E=T\,\mathrm{d}S+\cdots$ is what begs for interpretation. Interpreting equation (4) as genuine thermodynamics means reading the area term as heat.

\mathrm{d}E=T_H\,\mathrm{d}S_{\mathrm{BH}}+\Omega_H\,\mathrm{d}J+\Phi_H\,\mathrm{d}Q \qquad (5)

where: $E=Mc^2$ — total energy; $T_H, S_{\mathrm{BH}}$ — Hawking temperature and Bekenstein-Hawking entropy. Reading: equating the first terms of (4) and (5) with the known $T_H$ forces $S_{\mathrm{BH}}=k_B c^3 A/4G\hbar$ — temperature and entropy are two sides of one identification. Consistency of horizon thermodynamics with the outside world produces a universal ceiling on ordinary entropy: the Bekenstein bound.

S\le\frac{2\pi k_B R E}{\hbar c} \qquad (6)

where: $R$ — radius of the smallest enclosing sphere; $E$ — total energy, rest mass included. Reading: no system of size $R$ and energy $E$ can hide more entropy than this, and a black hole of that size saturates it — matter is most information-dense right at collapse. Because horizon entropy can grow while ordinary entropy falls behind it, the correct second law is the generalized one — and beyond it, quantum-gravity calculations agree on a universal logarithmic correction $S=k_B A/4\ell_P^2-\tfrac32 k_B\ln(A/\ell_P^2)+\cdots$ whose subleading coefficient is a sharp cross-check between theories.

\delta S_{\mathrm{tot}}=\delta S_{\mathrm{BH}}+\delta S_{\mathrm{matter}}\ge0 \qquad (7)

where: $S_{\mathrm{matter}}$ — ordinary entropy outside the horizon. Reading: neither term alone is monotonic once quantum effects are on — an evaporating hole has $\delta S_{\mathrm{BH}}<0$ — but the sum never decreases, and that is the law that survives.

Derivation ABekenstein's information argument that entropy scales with area

The goal is to see, without any knowledge of the Hawking temperature, why entropy must scale with horizon area rather than mass or volume, using only three ingredients: a black hole hiding an unknown system conceals at least one bit, a single quantum lowered onto the horizon raises its area by of order one Planck area regardless of the hole's size, and Hawking's area theorem forbids the area from ever shrinking. The argument fixes the form $S\propto A/\ell_P^2$ and shows it automatically obeys a second law, while leaving the coefficient undetermined.

Step 1. Consider dropping a single quantum of energy $E\sim\hbar c/b$ and size $b\sim\hbar/mc$ (its own Compton wavelength) onto a horizon of radius $r_+$ . The mass increase is $\Delta M\sim E/c^2$ , and the first law $\mathrm{d}A=(8\pi G/\kappa c^2)\,\mathrm{d}(Mc^2)$ with $\kappa\sim c^4/GM\sim c^2/r_+$ gives an area increase $\Delta A\sim(8\pi G/c^4)\,b\,(\hbar c/r_+)\times(r_+/b)\sim \ell_P^2$ . The dependence on $M$ and on the particle's own size cancels. Why: the minimum controllable parcel of information you can feed a horizon always costs the same fixed area, one Planck area — this is the atom of horizon entropy.

Step 2. Because each hidden bit costs a fixed minimal area, the total hidden information — the entropy — must be proportional to the total area accumulated: $\mathrm{d}S\propto\mathrm{d}A$ , hence $S=\eta\,k_B A/\ell_P^2$ with $\eta$ a pure number of order unity. Bekenstein's own estimate, counting one bit as $\ln 2$ of entropy per minimal area, gave $\eta\sim\ln2/8\pi$ . Why: proportionality to area, not to mass or volume, is the whole content of the argument — it is what makes black hole entropy non-extensive and ultimately holographic.

Step 3. Invoke Hawking's area theorem: in any classical process obeying the null energy condition, $\mathrm{d}A\ge0$ . Combined with Step 2 this gives $\mathrm{d}S_{\mathrm{BH}}\ge0$ , so the proposed entropy already satisfies a second law of its own. Why: an entropy that could spontaneously decrease would be useless; the area theorem is precisely the classical guarantee that makes the identification thermodynamically consistent. *The order-unity constant $\eta$ is fixed to exactly $1/4$ only once the Hawking temperature is known.*

Go deeper · Derivation B · 1 minFixing the one quarter from the Hawking temperature

Derivation BFixing the one quarter from the Hawking temperature

The goal is to pin down the coefficient Bekenstein's argument left open, by treating a Schwarzschild hole as an honest thermodynamic system with energy $E=Mc^2$ and temperature $T_H$ , then integrating the Clausius relation $\mathrm{d}S=\mathrm{d}E/T_H$ from zero mass. It needs only the Schwarzschild surface gravity and the Hawking formula, and delivers both the closed form $S\propto M^2$ and, rewritten through the area, the celebrated factor of one quarter.

Step 1. Compute the surface gravity. For the Schwarzschild metric with $f(r)=1-2GM/rc^2$ , the surface gravity is $\kappa=\tfrac12 c^2|f'(r_+)|$ evaluated at the horizon $r_+=2GM/c^2$ . Since $f'(r_+)=c^2/2GM$ , this gives $\kappa=c^4/4GM$ . Why: $\kappa$ is the one horizon quantity the temperature depends on, and for Schwarzschild it is set entirely by the mass.

Step 2. Insert $\kappa$ into equation (2) to get $T_H=\hbar c^3/8\pi G M k_B$ . Numerically this is $T_H\approx6.2\times10^{-8}\,\mathrm{K}\times(M_\odot/M)$ — a solar-mass hole is far colder than the $2.7\,\mathrm{K}$ microwave background, so in today's universe astrophysical holes absorb far more than they emit. Why: the explicit $1/M$ scaling is what gives black holes negative heat capacity and drives runaway evaporation once a hole is small enough to be hotter than its surroundings.

Step 3. Integrate the Clausius relation. With $E=Mc^2$ , $\mathrm{d}E=c^2\,\mathrm{d}M$ , so $\mathrm{d}S=c^2\,\mathrm{d}M/T_H=(8\pi Gk_B/\hbar c)\,M\,\mathrm{d}M$ . The integrand is linear in $M$ , and integrating from $0$ gives $S=4\pi Gk_B M^2/\hbar c$ , with the integration constant fixed by $S(M{=}0)=0$ . Why: demanding zero entropy for zero mass is the physical boundary condition — a vanishing horizon hides nothing.

Step 4. Convert to area. The Schwarzschild horizon area is $A=4\pi r_+^2=16\pi G^2M^2/c^4$ , so $M^2=Ac^4/16\pi G^2$ . Substituting, $S=4\pi Gk_B/(\hbar c)\times Ac^4/16\pi G^2=k_B c^3 A/4G\hbar$ . Why: the $16\pi$ from the area and the $2\pi$ from the temperature combine with the $8\pi$ of the Einstein equations to leave exactly $1/4$ — the coefficient is a conspiracy of geometric factors, not an input. *The results $S(M_\odot)=1.05\times10^{77}k_B$ and $T_H(M_\odot)=6.2\times10^{-8}\,\mathrm{K}$ are the standard values.*

Go deeper · Derivation C · 1 minThe generalized second law and the Bekenstein bound

Derivation CThe generalized second law and the Bekenstein bound

The goal is to show that the generalized second law not merely is consistent with, but actually implies, a universal upper bound on the entropy of ordinary matter. The vehicle is Bekenstein's gedankenexperiment of lowering a box — energy $E$ , entropy $s$ , height $R$ — adiabatically toward a horizon and releasing it where its redshifted energy is minimized. Tracking the entropy bookkeeping turns $\delta S_{\mathrm{tot}}\ge0$ into the inequality $s\le 2\pi k_B RE/\hbar c$ .

Step 1. Lower the box until its centre is a proper distance of order its own size $R$ above the horizon, then release. The energy actually added to the hole is the box energy redshifted by the near-horizon factor, $\Delta(Mc^2)\approx(\kappa R/c^2)\,E$ , far below the naive $E$ . Why: the redshift is what let Bekenstein imagine sneaking entropy past the horizon almost for free — the tighter that free lunch, the stronger the bound needed to forbid it.

Step 2. The horizon entropy gain follows from $\delta S_{\mathrm{BH}}=\Delta(Mc^2)/T_H$ . Using $T_H=\hbar\kappa/2\pi k_B c$ and the redshifted energy from Step 1, the surface gravity cancels: $\delta S_{\mathrm{BH}}=(2\pi k_B c/\hbar\kappa)\times(\kappa R/c^2)E=2\pi k_B RE/\hbar c$ . Why: that $\kappa$ drops out is the crucial point — the bound is a property of the box, not of the particular hole used to test it.

Step 3. Apply the generalized second law. The box's entropy $s$ has left the exterior, so $\delta S_{\mathrm{matter}}=-s$ , and $\delta S_{\mathrm{tot}}=\delta S_{\mathrm{BH}}-s\ge0$ demands $s\le2\pi k_B RE/\hbar c$ . A system that violated this bound could be dropped in to decrease total entropy — so the generalized second law forbids such systems from existing. Why: the argument runs backwards from the one law we trust, turning a statement about horizons into a hard limit on how much information any bounded system can hold. *The bound is saturated by a black hole itself, the maximum-entropy object of its size.*

Limiting cases

How the single formula $S=k_B A/4\ell_P^2$ specializes across the standard black hole families and regimes (areas in $\mathrm{m}^2$ , geometrized $G=c=1$ where noted):

Case	Result	Meaning
Schwarzschild	$S=\dfrac{4\pi k_B G M^2}{\hbar c},\quad T_H=\dfrac{\hbar c^3}{8\pi G M k_B}$	Entropy grows as $M^2$ , temperature falls as $1/M$ : big holes are cold with negative heat capacity.
Reissner-Nordström	$S=\dfrac{\pi k_B r_+^2}{\ell_P^2},\quad r_+=\dfrac{GM}{c^2}+\sqrt{\big(\tfrac{GM}{c^2}\big)^2-\tfrac{GQ^2}{4\pi\epsilon_0 c^4}}$	Charge shrinks $r_+$ and the area; the first law gains the work term $\Phi_H\,\mathrm{d}Q$ .
Kerr	$S=2\pi k_B\big(M^2+\sqrt{M^4-J^2}\big)\;(G{=}c{=}1)$	Spin lowers the area; the irreducible mass $M_{\mathrm{irr}}=\sqrt{A/16\pi}$ caps Penrose energy extraction.
Extremal limit	$T_H\to0,\quad S\to S_{\mathrm{ext}}>0$	As $\kappa\to0$ ( $J\to M^2$ or $Q\to M$ ) the temperature vanishes but area and entropy stay finite — the case Strominger-Vafa counted.
de Sitter horizon	$S=\dfrac{k_B A}{4\ell_P^2},\ A=4\pi(c/H)^2,\ T=\dfrac{\hbar H}{2\pi k_B}$	The same $A/4$ applies to a cosmological Killing horizon (Gibbons-Hawking).

Common trapthe coefficient 1/4 follows from the area theorem alone

It does not. Bekenstein's information-theoretic and dimensional argument fixes only the form $S\propto A/\ell_P^2$ , with an order-unity constant he could estimate but not derive. The $1/4$ is nailed down only once Hawking's temperature $T_H=\hbar\kappa/2\pi k_B c$ is computed and the area term of the first law is matched to $T_H\,\mathrm{d}S$ ; the $2\pi$ from the Euclidean periodicity and the $8\pi G$ from the Einstein equations combine to give one quarter. A related trap: $S_{\mathrm{BH}}$ contains $\hbar$ even though the area is a purely classical quantity — the entropy is intrinsically quantum, diverging as $\hbar\to0$ .

Worked examples

1 · A one-solar-mass black hole, end to end (fully worked)

Take a Schwarzschild hole of $M=M_\odot=1.99\times10^{30}\,\mathrm{kg}$ . Find its horizon radius, area, Hawking temperature, and entropy, and compare that entropy to the star that formed it ( $S_\star\sim10^{58}k_B$ ).

Step 1. Horizon radius. $r_+=2GM/c^2=\mathbf{2.95\times10^3\,m}$ , about 3 km — the familiar Schwarzschild radius of the Sun.

Step 2. Horizon area. $A=4\pi r_+^2=\mathbf{1.10\times10^8\,m^2}$ , roughly the area of a mid-sized city.

Step 3. Hawking temperature. $T_H=\hbar c^3/8\pi G M k_B$ : numerator $\hbar c^3=2.84\times10^{-9}$ , denominator $8\pi G M k_B=4.61\times10^{-2}$ , so $T_H=\mathbf{6.2\times10^{-8}\,K}$ — forty-four million times colder than the CMB, so a real solar-mass hole grows rather than evaporates today.

Step 4. Entropy. Using $S/k_B=A/4\ell_P^2$ with $\ell_P^2=2.61\times10^{-70}\,\mathrm{m^2}$ : $S/k_B=1.10\times10^8/(4\times2.61\times10^{-70})=\mathbf{1.05\times10^{77}}$ , equivalently $S=1.45\times10^{54}\,\mathrm{J/K}$ (the cross-check $S=4\pi Gk_BM^2/\hbar c$ gives the identical value).

Step 5. Compare to the progenitor. The star's thermodynamic entropy was $\sim10^{58}k_B$ , so collapse multiplied it by $\sim10^{19}$ . The black hole holds about $10^{19}$ times more entropy than the star — a colossal irreversibility fully consistent with the second law, since the hole is the maximum-entropy configuration of that mass in that space.

2 · A primordial black hole evaporating today

Hawking's power law gives an evaporation lifetime $t_{\mathrm{evap}}\approx10^{67}\,(M/M_\odot)^3\,\mathrm{yr}$ . Which mass has a lifetime equal to the age of the universe ( $1.4\times10^{10}\,\mathrm{yr}$ ), and what are its temperature and entropy?

Step 1. Set $t_{\mathrm{evap}}=1.4\times10^{10}\,\mathrm{yr}$ . Then $(M/M_\odot)^3=1.4\times10^{-57}$ , so $M/M_\odot=1.1\times10^{-19}$ and $M\approx\mathbf{2\times10^{11}\,kg}$ — a modest asteroid's mass, compressed smaller than a proton.

Step 2. Temperature. $T_H=6.2\times10^{-8}\,\mathrm{K}\times(M_\odot/M)=6.2\times10^{-8}\times9.0\times10^{18}=\mathbf{5.5\times10^{11}\,K}$ , corresponding to $k_BT_H\approx\mathbf{48\,MeV}$ . Such a hole is in its final, blazing burst, emitting gamma rays and particles — exactly what MeV-band searches for primordial-black-hole evaporation look for.

Step 3. Entropy. $S/k_B=1.05\times10^{77}\times(1.1\times10^{-19})^2=\mathbf{1.3\times10^{39}}$ . Tiny by black hole standards, yet still astronomically larger than the ordinary matter it formed from would carry — the area law is extravagant at every scale.

Physics of the portrait

The story of black hole entropy is that of a graduate student who trusted a dimensional analysis against the intuition of a generation of relativists — and of the sceptic who set out to prove him wrong and handed him the missing constant instead.

Origins · 1 minJacob Bekenstein and how this was discovered

Jacob Bekenstein

1947-2015 · black hole thermodynamics and the universal entropy bound

Jacob Bekenstein, working under John Wheeler at Princeton, took seriously Wheeler's needling question — what happens to the entropy of a cup of hot tea poured into a black hole? — and in his 1972-73 papers proposed the then-heretical answer that the hole's own entropy rises to compensate, in proportion to its horizon area. Almost everyone objected: a body with entropy has a temperature, and a body with temperature radiates, which a black hole by definition cannot. Stephen Hawking was among the loudest sceptics, and in 1973 he, James Bardeen and Brandon Carter published the four laws of black hole mechanics precisely as a formal analogy, insisting the "temperature" was not real. Then in 1974 Hawking applied quantum field theory to the curved spacetime of a collapsing star and found, to his own surprise, that black holes do radiate — thermally, at exactly $T_H=\hbar\kappa/2\pi k_B c$ . The analogy was real, and his calculation fixed Bekenstein's undetermined coefficient at exactly one quarter; the joint formula $S=k_B A/4\ell_P^2$ has borne both names ever since. Bekenstein went on to prove the generalized second law and derive the universal bound $S\le2\pi k_B RE/\hbar c$ that carries his name, planting the seed from which 't Hooft and Susskind grew the holographic principle. He remained, until his death in 2015, one of the deepest thinkers on the thermodynamic meaning of gravity.

History

1972

— Bekenstein proposes that a black hole carries entropy proportional to its horizon area

1973

— Bardeen, Carter and Hawking publish the four laws of black hole mechanics as a formal analogy

1974

— Hawking shows black holes radiate thermally, making the analogy literal and fixing the coefficient at one quarter

1975

— Hawking's detailed derivation of particle creation cements black hole thermodynamics

1996

— Strominger and Vafa count string-theory microstates and recover the area law exactly

Going further

Texts. Wald, General Relativity (1984), Ch. 12, and Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (1994) · Carroll, Spacetime and Geometry (2004), Ch. 9 · Misner, Thorne & Wheeler, Gravitation (1973), §33 · Hartle, Gravity (2003), Ch. 12 · Schutz, A First Course in General Relativity, 2nd ed., Ch. 11.

Primary sources. Bekenstein, "Black Holes and Entropy", Phys. Rev. D 7, 2333 (1973) · Hawking, "Particle Creation by Black Holes", Commun. Math. Phys. 43, 199 (1975) · Bardeen, Carter & Hawking, "The Four Laws of Black Hole Mechanics", Commun. Math. Phys. 31, 161 (1973) · Strominger & Vafa, "Microscopic Origin of the Bekenstein-Hawking Entropy", Phys. Lett. B 379, 99 (1996). Key numerical checks: the Schwarzschild surface gravity $\kappa=c^4/4GM$ gives $T_H=\hbar c^3/8\pi GMk_B\approx6.2\times10^{-8}(M_\odot/M)\,\mathrm{K}$ ; integrating $\mathrm{d}S=c^2\,\mathrm{d}M/T_H$ yields $S=4\pi Gk_BM^2/\hbar c$ , with $S(M_\odot)=1.05\times10^{77}k_B=1.45\times10^{54}\,\mathrm{J/K}$ ; the area relation $A=16\pi G^2M^2/c^4$ converts this to $k_Bc^3A/4G\hbar$ ; and the primordial-hole mass $\approx2\times10^{11}\,\mathrm{kg}$ follows from $t_{\mathrm{evap}}\approx10^{67}(M/M_\odot)^3\,\mathrm{yr}$ , giving $k_BT_H\approx48\,\mathrm{MeV}$ . The Planck length is $\ell_P=1.616\times10^{-35}\,\mathrm{m}$ . Note: the solar-mass entropy is $1.05\times10^{77}k_B$ , not $1.5\times10^{77}k_B$ — the latter is the value in $\mathrm{J/K}$ ( $1.5\times10^{54}$ ) miscopied.

Prerequisites. Four Laws of Black Hole Mechanics · Killing Horizons and Surface Gravity · Event and Apparent Horizons · Schwarzschild Black Hole · No-Hair Theorem. Leads to. Holographic Principle · Black Hole Information Paradox · Reissner-Nordström Charged Black Hole · Kerr Metric and Rotating Black Holes.

Ask the tutor: Ask me to walk through why matching the first law of black hole mechanics to $\mathrm{d}E=T\,\mathrm{d}S$ forces the coefficient to be exactly one quarter, or how string theory recovers it by counting microstates.

Does a black hole really have a temperature, or is it just an analogy?

It really has one. Until 1974 the "temperature" $\kappa/2\pi$ in the laws of black hole mechanics was regarded as a formal analogy — Bardeen, Carter and Hawking explicitly said so. Hawking's quantum calculation overturned that view: a black hole formed by collapse emits a genuinely thermal spectrum at $T_H=\hbar\kappa/2\pi k_B c$ , indistinguishable from an ordinary hot body at that temperature. Once the temperature is physical, so is the entropy, and black hole thermodynamics becomes literal rather than metaphorical. The temperature is fantastically low for astrophysical holes — which is why it was never seen — but it is real.

If the horizon area is classical, why does the entropy contain ħ?

Because entropy counts microstates, and microstates are a quantum notion. The area $A$ is a purely classical quantity, but to turn it into a dimensionless count of states you must divide by a fundamental area — the only one available is the Planck area $\ell_P^2=G\hbar/c^3$ , which carries $\hbar$ . Physically the $\hbar$ says the entropy exists only because spacetime is quantum: in the classical limit $\hbar\to0$ the entropy diverges, the correct statement that a truly classical black hole would hide unlimited information behind its horizon.

Why is the entropy proportional to area and not volume?

Every other thermodynamic system — a gas, a star — has entropy proportional to its volume, because you can subdivide it into independent pieces. A black hole breaks this: its entropy scales as horizon area, $\sim R^2$ , not enclosed volume, $\sim R^3$ . The reason is that a fixed parcel of information lowered onto the horizon always costs one Planck area, so the total hidden information tracks the total area. Since a black hole is also the densest way to pack entropy into a region, area — not volume — becomes the true ceiling on information, which is the origin of the holographic principle.

What is the generalized second law, and why do we need it?

The ordinary second law fails around black holes: throw entropy across the horizon and exterior entropy drops. The classical area theorem $\mathrm{d}A\ge0$ patches this — but it too fails once quantum effects are on, because an evaporating hole shrinks. The generalized second law is what survives everything: the sum $S_{\mathrm{BH}}+S_{\mathrm{matter}}$ never decreases. When matter falls in, $S_{\mathrm{matter}}$ falls but $S_{\mathrm{BH}}$ rises more; when the hole evaporates, $S_{\mathrm{BH}}$ falls but the Hawking radiation carries away at least as much. It has been proved under a range of assumptions and is the correct thermodynamic law for gravitating systems.

Has anyone actually derived S = A/4 by counting microstates?

Yes, for certain black holes. In 1996 Strominger and Vafa described a class of extremal charged black holes as string-theory bound states of D-branes. Counting those states with ordinary statistical mechanics gave $\ln\Omega=A/4\ell_P^2$ , matching Bekenstein-Hawking exactly, coefficient and all — the first microscopic derivation. Loop quantum gravity independently reproduces the same area law by counting the ways spin-network edges puncture the horizon. These agreements, including the shared $-\tfrac32\ln A$ correction, are the strongest evidence that $S_{\mathrm{BH}}$ counts real quantum-gravitational microstates.

Does black hole entropy destroy information when the hole evaporates?

That is the unresolved information paradox. Hawking's original calculation makes the emitted radiation exactly thermal, depending only on mass, charge and spin — so if a hole forms from a book and evaporates completely, the book's information appears to vanish, violating the unitarity of quantum mechanics. Most physicists now believe unitarity holds and the information is subtly encoded in correlations within the radiation; recent "island" calculations reproduce the Page curve that unitary evaporation requires. But exactly how the information gets out, and what that says about the microstates $S_{\mathrm{BH}}$ counts, remains the sharpest open question at the boundary of gravity and quantum theory.