property

black hole spin

Black hole spin refers to the angular momentum of a rotating black hole.
29 related articles · 6 formulas

Description

Black hole spin is the angular momentum of a rotating black hole, a fundamental property that affects the geometry of spacetime around it. The spin is characterized by a dimensionless parameter a* (ranging from 0 for non-rotating to 1 for maximally rotating). It influences the size and shape of the event horizon, the innermost stable circular orbit (ISCO), and the energy extraction mechanisms like the Penrose process.

History

The Kerr solution was discovered by Roy Kerr in 1963, and the concept of spin as a measurable parameter was developed in subsequent decades. Observations of X-ray binaries and, more recently, gravitational waves have provided measurements of black hole spins.

How it works

In general relativity, a rotating black hole is described by the Kerr metric. The spin parameter determines the frame-dragging effect, where spacetime itself is dragged around the black hole. This affects the orbits of matter and light, leading to phenomena like the Lense-Thirring precession and the existence of an ergosphere. The spin also sets the maximum efficiency of energy extraction from accretion disks, which can be up to 42% for a maximally spinning black hole.

In practice

Measuring black hole spin is crucial for understanding the formation and evolution of black holes, testing the no-hair theorem, and interpreting observations from X-ray telescopes (like NuSTAR) and gravitational wave detectors.

Fun fact

The spin of a black hole can be measured from the shape of its shadow, as seen by the Event Horizon Telescope, because the shadow is slightly asymmetric for rotating black holes.

Concept graph →

Formulas

$$A = 8\pi M \left(M + \sqrt{M^2 - a^2}\right)$$
The area of a Kerr black hole depends on mass and spin.
What is in the formula
AvariableArea of the event horizon — area square metre
MvariableMass of the black hole — mass kilogram
avariableSpecific angular momentum (angular momentum per unit mass) — specific angular momentum metre
πconstantRatio of a circle's circumference to its diameter — pi = 3.14159
a=8πm^2(1+\sqrt{1-a^2}) · 2606.06592v1
$$r_{\rm inf} = \frac{G M_{\rm BH}}{\sigma^2}$$
Radius where black hole gravity dominates stellar motions.
What is in the formula
r_{\rm inf}variableInfluence radius — radius metre
M_{\rm BH}variableBlack hole mass — mass kilogram
\sigmavariableVelocity dispersion of stars — velocity dispersion metre per second
GconstantNewtonian gravitational constant — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
r_{{inf}}=\frac{gm_{{bh}}}{σ^2} · 2606.06620v1
$$a = \frac{Jc}{GM^2}$$
Spin parameter of a black hole, normalized by mass.
What is in the formula
avariableDimensionless spin parameter — dimensionless spin parameter
JvariableAngular momentum of the black hole — angular momentum kilogram metre squared per second
MvariableMass of the black hole — mass kilogram
cconstantSpeed of light in vacuum, a fundamental constant of nature. — speed of light = 299792458 metre per second
GconstantNewtonian constant of gravitation, determining the strength of gravity. — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared
a=\frac{jc}{gm^2} · 2607.06902
a_*=\frac{jc}{gm^2} · 2607.14218
$$\Delta(r) = r^2 - 2Mr + a^2$$
Horizon function for Kerr black holes with mass M and spin a.
What is in the formula
\Delta(r)variableHorizon function of the Kerr metric — horizon function square metre
rvariableRadial coordinate (Boyer-Lindquist) — radial coordinate metre
MvariableBlack hole mass — mass kilogram
avariableSpecific angular momentum (spin parameter) — specific angular momentum metre
δ_{{dm}}(r)=r^2-2m[1+\frac{\varepsilon}{m}μ_{{dm}}(r)]r+a^2 · 2607.13992
$$r_{\text{ISCO}} = r_g \left[3 + Z_2 \mp \sqrt{(3-Z_1)(3+Z_1+2Z_2)}\right]$$
Radius of the innermost stable circular orbit around a Kerr black hole.
What is in the formula
r_{\text{ISCO}}variableradius of the innermost stable circular orbit — radius metre
r_gvariablegravitational radius (GM/c^2) — radius metre
Z_1variableauxiliary function of spin parameter a
Z_2variableauxiliary function of spin parameter a
\mpoperatorindicates two possible solutions depending on orbital direction — minus plus sign
\sqrt{}operatortakes the square root of the argument — square root
r_{{isco}}=r_g[3+z_2\mp\sqrt{(3-z_1)(3+z_1+2z_2)}],\quadz_1=1+(1-a^2)^{1/3}[(1+a)^{1/3}+(1-a)^{1/3}],\quadz_2=\sqrt{3a^2+z_1^2} · 2607.22418v1
$$T_H = \frac{\hbar \kappa}{2\pi k_B}$$
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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