Black hole spin refers to the angular momentum of a rotating black hole.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.
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.
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.
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.
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.
| A | variable | Area of the event horizon — area square metre |
| M | variable | Mass of the black hole — mass kilogram |
| a | variable | Specific angular momentum (angular momentum per unit mass) — specific angular momentum metre |
| π | constant | Ratio of a circle's circumference to its diameter — pi = 3.14159 |
| r_{\rm inf} | variable | Influence radius — radius metre |
| M_{\rm BH} | variable | Black hole mass — mass kilogram |
| \sigma | variable | Velocity dispersion of stars — velocity dispersion metre per second |
| G | constant | Newtonian gravitational constant — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| a | variable | Dimensionless spin parameter — dimensionless spin parameter |
| J | variable | Angular momentum of the black hole — angular momentum kilogram metre squared per second |
| M | variable | Mass of the black hole — mass kilogram |
| c | constant | Speed of light in vacuum, a fundamental constant of nature. — speed of light = 299792458 metre per second |
| G | constant | Newtonian constant of gravitation, determining the strength of gravity. — gravitational constant = 6.67430e-11 cubic metre per kilogram per second squared |
| \Delta(r) | variable | Horizon function of the Kerr metric — horizon function square metre |
| r | variable | Radial coordinate (Boyer-Lindquist) — radial coordinate metre |
| M | variable | Black hole mass — mass kilogram |
| a | variable | Specific angular momentum (spin parameter) — specific angular momentum metre |
| r_{\text{ISCO}} | variable | radius of the innermost stable circular orbit — radius metre |
| r_g | variable | gravitational radius (GM/c^2) — radius metre |
| Z_1 | variable | auxiliary function of spin parameter a |
| Z_2 | variable | auxiliary function of spin parameter a |
| \mp | operator | indicates two possible solutions depending on orbital direction — minus plus sign |
| \sqrt{} | operator | takes the square root of the argument — square root |
| T_H | variable | Hawking temperature of the black hole — temperature kelvin |
| κ | variable | surface gravity of the black hole — surface gravity metre per second squared |
| ħ | constant | fundamental constant equal to h/(2π), relating angular frequency to energy — reduced planck constant = 1.054571817e-34 joule second |
| k_B | constant | fundamental constant relating temperature to energy — boltzmann constant = 1.380649e-23 joule per kelvin |











