Black-hole spin ($a_{\ast }$)

Black-hole spin is the angular momentum of a black hole, commonly expressed by the dimensionless Kerr parameter $a_{\ast }=cJ/(G{M}^2)$. It changes the innermost stable circular orbit, accretion efficiency, and relativistic frame dragging.

An inference of black-hole angular momentum, usually quoted through $a_{\ast }$. Accretion-based techniques use the dependence of the stable inner orbit and relativistic radiation transfer on spin in the Kerr black hole model. Disk truncation, inclination, illumination and atmosphere assumptions can imitate a change of spin and must enter the uncertainty model.

An inference from radiation reflected by an irradiated accretion disc, including iron-line and continuum features. Doppler effects, gravitational redshift and ray propagation determine its profile. Associating the innermost reflecting radius with the innermost stable circular orbit can constrain spin, conditional on illumination, ionization, abundance, inclination and disk structure.

A fit of a thermal accretion-disk spectrum that infers the inner emitting radius and identifies it with the innermost stable circular orbit. A relativistic thin-disk model requires independently constrained mass, distance and inclination, together with spectral hardening and absorption. Use of a suitable disk-dominated state is an essential physical check, not merely a statistical cut.

Coherent prograde gas accretion increases the angular momentum of a black hole. For a zero-torque thin disc, the rest mass $d{m}_0$ entering from the innermost stable circular orbit changes the hole by $dM=E_{\mathrm{ISCO}}d{m}_0$ and $dJ={\ell }_{\mathrm{ISCO}}d{m}_0$. Its dimensionless spin obeys $d{a}_{\ast }/d\log M=c{\ell }_{\mathrm{ISCO}}/(GM{E}_{\mathrm{ISCO}})-2a_{\ast }$. Starting from zero spin and neglecting captured radiation, the formal extremal endpoint has $M_f=\sqrt{6}{M}_i$ and requires rest-mass supply approximately $1.8464M_i$. Photon capture leads instead to the Thorne spin limit.

The Thorne spin limit is the equilibrium spin produced by competition between coherent thin-disc matter accretion and capture of disc photons by a Kerr black hole. Preferential capture of counterrotating photon angular momentum prevents the standard radiatively efficient model from reaching $a_{\ast }=1$. The commonly quoted limit is approximately $0.998$, with some dependence on emission and accretion assumptions. Thorne's original calculation derives this result.