concept

quantum black hole thermodynamics

Thermodynamic properties and quantum effects of black holes, including temperature and information.
17 related articles · 6 formulas

Description

Quantum black hole thermodynamics extends classical black hole thermodynamics by incorporating quantum field theory effects. It describes black holes as thermodynamic systems with temperature (Hawking radiation), entropy (proportional to horizon area), and other state variables, while also addressing puzzles like the information paradox and the microscopic origin of entropy. This framework connects gravity, quantum mechanics, and thermodynamics, offering insights into the nature of spacetime at the quantum level.

History

The foundations were laid by Jacob Bekenstein in the 1970s, who proposed black hole entropy, and Stephen Hawking, who derived black hole temperature in 1974. Subsequent developments include the holographic principle and string theory calculations of microstates.

How it works

Near a black hole's horizon, quantum vacuum fluctuations become polarized, leading to the emission of thermal radiation (Hawking radiation) with a temperature inversely proportional to the black hole's mass. The entropy, given by the Bekenstein-Hawking formula, counts the microscopic quantum states of the horizon. Quantum corrections modify these relations, for example by adding logarithmic terms, and can affect how black holes evaporate or interact with surrounding fields.

In practice

These concepts guide research in quantum gravity, including attempts to resolve the information paradox, and are used to model black hole evaporation and its potential observational signatures in gravitational wave data.

Fun fact

A black hole with the mass of the Moon would have a temperature of about 0.001 kelvin, colder than the cosmic microwave background, so it would absorb more radiation than it emits.

Concept graph →

Formulas

$$T_{\mathrm{BKT}} = \frac{\pi}{2} \rho_s(0)$$
Temperature of vortex unbinding in 2D superfluids.
What is in the formula
T_BKTvariableBKT transition temperature — temperature kelvin
ρ_svariableSuperfluid density at zero temperature — superfluid density kilogram per metre
t_{\rmbkt}=\frac{π}{8}\sqrt{\detd_s} · 2607.02267
$$T_{\rm bb} \propto M^{-a} \dot{m}^{-b}$$
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
$$T = \frac{\hbar c^3}{8\pi G M k_B}$$
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
$$T_{\text{eff}} = \left( \frac{1}{\sigma} \int F_\lambda \, d\lambda \right)^{1/4}$$
Temperature of blackbody with same total flux.
What is in the formula
T_effvariableEffective temperature — temperature kelvin
F_lambdavariableSpectral flux density per unit wavelength — spectral flux density watt per metre cubed
lambdavariableWavelength — wavelength metre
sigmaconstantRelates total blackbody flux to temperature — stefan boltzmann constant = 5.670374419e-8 watt per metre squared per kelvin to the fourth
∫operatorIntegration over all wavelengths — integral
t_{\rmeff}=(\frac{1}{σ_{\rmsb}}\intf_{\rmp,surf,λ}dλ)^{1/4} · 2607.01316v1
$$\frac{\Delta T}{T} = -2 \int \dot{\Phi} \, d\eta$$
CMB temperature shift from time-varying gravitational potentials.
What is in the formula
ΔT/TvariableRelative temperature fluctuation of CMB — temperature fluctuation
ΦvariableGravitational potential perturbation — gravitational potential metre squared per second squared
ηvariableConformal time — conformal time second
∫operatorIntegrates over conformal time along the photon path — integral
dotoperatorDerivative with respect to conformal time — time derivative
\frac{δt}{t}=-2\int\dot{\phi}d\eta · 2607.19648v1

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