Robustness of the Thorne limit: the asymptotic spin and accretion efficiency of a black hole growing by disk accretion under realistic magnetic, radiative, and geometric conditions

Thorne (1974) computed the spin evolution of a black hole fed by a thin, equatorial, stress-free accretion disk and found that photons captured by the hole halt the spin-up at a limiting dimensionless spin of about 0.998, with a mass-to-radiation efficiency near 30%, while warning - most seriously through his referee James M. Bardeen - that magnetic stresses could cause large deviations from circular orbits in the innermost disk, change the energy-angular-momentum balance of the accreting matter by an amount of order unity, and make the true limiting spin considerably smaller. This problem asks for the limiting spin and efficiency of an accreting black hole once the neglected effects are included: magnetic stresses coupling the inner disk and the plunging region to the horizon, photon recapture by the disk, disk-hole misalignment, finite disk thickness or thick/supercritical and low-angular-momentum inflow, and counterrotating (retrograde) accretion. Fifty years of later work has confirmed the concern and produced quantitative, regime-dependent answers: a refined thin-disk limit near 0.99 from plunging-region emission sourced by magnetohydrodynamic stresses (Mummery 2025), spin equilibrium near 0.9 in magnetized thick tori (Gammie, Shapiro and McKinney 2004), about 0.3 in thin luminous magnetically arrested disks and about 0.07 in thick magnetically arrested disks (Lowell et al. 2024, 2025). What remains open is the predictive, first-principles determination of the equilibrium spin as a continuous function of horizon magnetic flux, disk thickness, luminosity and tilt across these regimes, together with the observational question of which real accreting holes actually approach any of these limits.

Background

An astrophysical black hole is described, in general relativity, by the Kerr metric, characterized only by its mass $M$ and angular momentum $J$; the dimensionless spin $a_\ast = cJ/GM^2$ ranges from 0 (nonrotating, Schwarzschild) to 1 (extremal Kerr). Gas falling onto such a hole - from a companion star in an X-ray binary, or from the interstellar medium in the nucleus of a galaxy - usually carries enough angular momentum to settle into a rotating disk before spiraling in. The standard model is the relativistic thin disk (the Novikov-Thorne model, the relativistic completion of the Shakura-Sunyaev disk): an optically thick, geometrically thin disk in which the gas moves on nearly circular geodesic orbits while a weak turbulent stress drives a slow inward drift. The inner edge sits at the innermost stable circular orbit (ISCO), the radius inside which no stable circular orbit exists in the Kerr metric; it moves from $6GM/c^2$ at $a_\ast = 0$ toward the horizon as $a_\ast \to 1$. The model assumes the turbulent stress vanishes at and inside the ISCO, so the plunging gas delivers exactly the energy and angular momentum of the last stable orbit, $E_{\rm ms}$ and $L_{\rm ms}$. The radiative efficiency $1 - E_{\rm ms}$ - the fraction of accreted rest mass converted to radiation - rises from 5.7% for a nonrotating hole to 42% for an extremal one, which is why the spin of an accreting black hole controls its luminosity.

Bardeen (1970) first posed the evolution question: how do $M$ and $J$ change as the hole swallows disk gas? Page and Thorne (1974) supplied the time-averaged disk structure (Paper I), and Thorne (1974) computed the evolution (Paper II). Matter alone spins the hole up to extremality after accreting only about 1.5 times its initial mass. However, a fraction of the photons emitted by the hot inner disk is captured by the hole, and the captured photons carry, on average, retrograde angular momentum, so the radiation exerts a braking torque. Including this effect, Thorne found the celebrated limiting state $a_\ast \approx 0.998$ (0.9978 for isotropic emission, 0.9982 for an electron-scattering angular pattern) with a maximum efficiency of about 30%.

Thorne's calculation rests on idealizations: a razor-thin, equatorial disk; zero stress at and inside the ISCO; no magnetic coupling between disk and hole; and no recapture of emitted photons by the disk (in fact roughly 30% of inner-disk photons would be reabsorbed by the disk). In Section IV he estimated that perturbations changing the delivered energy and angular momentum by less than about 10% would move the limit only within $0.995 \lesssim a_\ast \lesssim 0.999$ and the efficiency within 0.29-0.34. But he then quoted his referee, James M. Bardeen, warning that magnetic stresses could cause large deviations from circular orbits in the very inner part of the disk and change the energy-angular-momentum balance of the accreting matter by an amount of order unity, in which case the limiting value of $a/M$ may be considerably less than 0.998 (echoing earlier private warnings of Ya. B. Zel'dovich and V. F. Schwartzman). Thorne also noted that the extension of his calculation to counterrotating (retrograde) disks would be straightforward, but did not carry it out.

Later developments make the caveat concrete. It is now understood that real disks are magnetized and turbulent, driven by the magnetorotational instability (MRI), the instability of a weakly magnetized, differentially rotating plasma that generates the turbulence responsible for angular-momentum transport in ionized disks. General relativistic magnetohydrodynamic (GRMHD) simulations show that the magnetic stress does not vanish at the ISCO and that magnetic field lines can thread the event horizon itself; when the accumulated vertical magnetic flux becomes dynamically strong, the flow enters the magnetically arrested disk (MAD) state, in which a Blandford-Znajek jet - the process by which magnetic field lines threading the horizon extract the hole's rotational energy - can remove angular momentum and energy faster than accretion supplies them. Each of these effects alters precisely the torque balance that fixes Thorne's limit, so the true limiting spin and efficiency of an accreting black hole, and whether real holes approach them, are live questions with direct observational stakes: measured spins of X-ray binaries and active galactic nuclei, jet powers, and gravitational-wave spin distributions are all interpreted against some assumed limiting spin.

Problem Statement

Determine the asymptotic ('limiting' or equilibrium) dimensionless spin $(a_\ast)_{\rm lim}$, and the associated accretion efficiency, of a black hole that grows by disk accretion, once the idealizations of the 1974 thin-disk calculation are removed, where the accretion efficiency is defined as the total fraction of accreted rest mass converted into energy, comprising the radiative efficiency - the fraction radiated by the disk, equal to the thin-disk value $1 - E_{\rm ms}$ in Thorne's baseline and the quantity against which Thorne's canonical $\approx 30\%$ is measured - and the outflow efficiency, the fraction of accreted rest mass carried away as jet and wind energy. All five effects that Thorne (1974) flagged are to be included: (i) magnetic stresses that couple the innermost disk and the plunging matter to the hole - the effect Bardeen warned could cause large deviations from circular orbits and order-of-unity changes to the energy-angular-momentum balance of the accreting matter; (ii) recapture of emitted photons by the disk (about 30% in the inner regions) and the resulting changes to the emission pattern and torques; (iii) misalignment between the disk plane and the hole's equatorial plane, with precession and modified torques; (iv) finite disk thickness, super-Eddington or destroyed inner disks, and inflow with insufficient angular momentum to circularize; and (v) counterrotating (retrograde) disk accretion - the extension Thorne described as straightforward but did not carry out. The objective has two parts. The first is a first-principles predictive account of the equilibrium spin and the associated efficiency as a continuous function of horizon magnetic flux, disk thickness, luminosity and tilt - in particular by how much the limiting spin lies below the canonical 0.998 - across the full range from weakly magnetized thin disks to magnetically arrested states; the account must incorporate all five effects (i)-(v) above quantitatively and reproduce each of the per-regime spin limits cited above to within about $\pm 0.05$ in $a_\ast$. The second part is the observational assessment, as the other side of the same objective, of whether real accreting black holes (X-ray binaries, active galactic nuclei, gravitational-wave sources) actually reach these limits.

References

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