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Bekenstein-Hawking entropy

Bekenstein-Hawking entropy quantifies the entropy of a black hole as proportional to its event horizon area.
106 related articles · 6 formulas · 2 who worked here

Description

Black holes seem like bottomless pits where everything falls in and disappears. But they have a special memory — entropy, a measure of disorder. The more falls in, the larger the area of their 'surface' — the event horizon — becomes. The Bekenstein-Hawking formula links this area to entropy: it's enormous! Imagine a black hole is an ideal draft document that can't be read from the outside, but the amount written can be judged by the size of the crumpled paper.

History

Jacob Bekenstein realized in 1973 that without attributing entropy to black holes, the second law of thermodynamics would be violated when matter falls in. He proposed proportionality to horizon area. Stephen Hawking, initially skeptical, in 1974 used quantum field theory in curved spacetime to derive that black holes radiate and independently confirmed the 1/4 factor. Their work gave birth to modern black hole thermodynamics.

How it works

Used in astrophysics, for calculating the information capacity of black holes, in string theory, and in AdS/CFT correspondence. Applicable to any equilibrium black holes, but is semiclassical: quantum corrections become significant in the final stages of evaporation. For extremal black holes, entropy is zero, requiring separate interpretation.

Fun fact

Stephen Hawking initially argued with Bekenstein and even made a bet, but later helped derive the exact formula himself! The story of the two geniuses reads like a scientific detective novel.

Concept graph →

Formulas

$$S = \frac{k c^3 A}{4 G \hbar}$$
Black hole entropy is proportional to its event horizon area.
What is in the formula
SvariableEntropy of the black hole — entropy joule per kelvin
AvariableArea of the event horizon — area square metre
kconstantBoltzmann constant — boltzmann constant = 1.380649e-23 joule per kelvin
cconstantSpeed of light in vacuum — speed of light = 299792458 metre per second
GconstantNewtonian gravitational constant — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
ħconstantReduced Planck constant — reduced planck constant = 1.054571817e-34 joule second
s=\frac{a}{4} · 2604.18669
s=\frac{a}{4} · 2605.22172
s=\frac{πr_h^2}{g} · 2607.03749
s_{{bh}}=\frac{a}{4g} · 2607.06582
$$S_{\text{gen}} = \frac{A}{4G} + S_{\text{matter}}$$
Generalized entropy includes horizon area and matter entropy.
What is in the formula
S_genvariableGeneralized entropy — entropy joule per kelvin
AvariableArea of the horizon — area square metre
S_mattervariableEntropy of matter outside the horizon — entropy joule per kelvin
GconstantNewton's gravitational constant, determining the strength of gravity — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
s_{{gen}}=\frac{{area}}{4g}+s_{{vn}} · 2605.05326v1
$$S \propto A$$
Entropy of a black hole is proportional to its horizon area.
What is in the formula
SvariableBlack hole entropy — entropy joule per kelvin
AvariableBlack hole horizon area — area square metre
s\proptoa^{\frac{2+α}{2α}} · 2606.22431
$$S_A = rac{ ext{Area}( ext{minimal surface})}{4G_N}$$
Entanglement entropy of a boundary region equals its minimal surface area in the bulk.
What is in the formula
S_AvariableEntanglement entropy of boundary region A — entanglement entropy
Area(minimal surfacevariableArea of the minimal surface in the bulk homologous to A — area square metre
G_NconstantNewton's gravitational constant in the bulk — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
s_a=\frac{{area}(\gamma_a)}{4g} · 2605.02446v2
$$S = k_B \ln W$$
Entropy is proportional to logarithm of states.
What is in the formula
SvariableEntropy of the system — entropy joule per kelvin
WvariableNumber of microstates — number of microstates
k_BconstantRelates energy to temperature at the particle level — boltzmann constant = 1.380649e-23 joule per kelvin
lnoperatorNatural logarithm — natural logarithm
δs=k_b\logd · 2607.02188
$$\eta = \frac{n_B}{s}$$
Ratio of baryon number density to entropy density.
What is in the formula
\etavariableBaryon to entropy ratio — baryon to entropy ratio
n_BvariableBaryon number density — baryon number density per cubic metre
svariableEntropy density — entropy density joule per cubic metre per kelvin
\eta=\frac{n_b}{s} · 2605.02008

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