concept

black hole thermodynamics

Black hole thermodynamics describes the thermal properties and entropy of black holes through analogies with classical thermodynamics.
43 related articles · 5 formulas

Description

Black hole thermodynamics is a framework that treats black holes as thermodynamic systems with temperature, entropy, and energy, despite their classical description as objects that only absorb. The key results are that a black hole's entropy is proportional to its event horizon area (Bekenstein-Hawking entropy) and its temperature is inversely proportional to its mass (Hawking temperature). This theory bridges general relativity, quantum mechanics, and thermodynamics, revealing deep connections between gravity and statistical mechanics.

History

The framework was developed in the early 1970s by Jacob Bekenstein, who proposed entropy proportional to area, and Stephen Hawking, who derived black hole radiation in 1974, completing the thermodynamic picture.

How it works

The analogy works because black holes obey laws mathematically identical to the laws of thermodynamics: the zeroth law defines a constant surface gravity (temperature), the first law relates mass change to area change (energy and entropy), and the second law states that horizon area never decreases. Hawking's quantum calculation showed that black holes emit thermal radiation, giving the temperature a real physical meaning: smaller black holes are hotter and radiate more intensely.

In practice

It guides theoretical research in quantum gravity, holography (AdS/CFT correspondence), and cosmological models, and helps interpret observations of black hole mergers and shadows.

Fun fact

A black hole with the mass of the Moon would have a temperature of about 0.1 kelvin, while a solar-mass black hole is far colder than the cosmic microwave background at 60 nanokelvin.

Concept graph →

Formulas

$$T = \frac{\hbar c^3}{8\pi G M k_B}$$
Black hole temperature is inversely proportional to its mass.
What is in the formula
TvariableHawking temperature of the black hole — temperature kelvin
MvariableMass of the black hole — mass kilogram
\hbarconstantReduced Planck constant, fundamental quantum of action — reduced planck constant = 1.054571817e-34 joule second
cconstantSpeed of light in vacuum, universal speed limit — speed of light = 299792458 metre per second
GconstantNewtonian constant of gravitation, sets strength of gravity — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
k_BconstantBoltzmann constant, relates temperature to energy — boltzmann constant = 1.380649e-23 joule per kelvin
t\sim\frac{1}{m} · 2603.02493
t_h=\frac{ħc^3}{8πgmk_b} · 2605.13901v1
t_{bh}=\frac{m_{pl}^2}{8πm_{bh}} · 2606.03316v2
t_{bh}=\frac{1}{8πm}\sqrt{1-\frac{3(γ-3)}{2}} · 2606.26166
$$S = \frac{k c^3 A}{4 G \hbar}$$
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
$$S \propto A$$
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
$$T_{\rm bb} \propto M^{-a} \dot{m}^{-b}$$
Blackbody temperature scales with mass and accretion rate.
What is in the formula
T_bbvariableBlackbody temperature — temperature kelvin
MvariableMass of the accreting object — mass kilogram
avariablePositive exponent for mass scaling
dot_mvariableMass accretion rate — mass flow rate kilogram per second
bvariablePositive exponent for accretion rate scaling
∝operatorProportional to — proportionality
t_{\rmbb}=5.8*10^3m_6^{-2/11}(\dot{m}_3/2)^{-15/11}{\rmk} · 2607.17448v1
$$T_H = \frac{\hbar \kappa}{2\pi k_B}$$
Black hole temperature depends on surface gravity and spin.
What is in the formula
T_HvariableHawking temperature of the black hole — temperature kelvin
κvariablesurface gravity of the black hole — surface gravity metre per second squared
ħconstantfundamental constant equal to h/(2π), relating angular frequency to energy — reduced planck constant = 1.054571817e-34 joule second
k_Bconstantfundamental constant relating temperature to energy — boltzmann constant = 1.380649e-23 joule per kelvin
t_h=\frac{ħc^3}{8πgmk_b}\frac{\sqrt{1-a_*^2}}{1+\sqrt{1-a_*^2}} · 2607.18395v1

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