Spin inversion for disk-mediated black-hole rotational energy extraction under partial extraction, accretion-fed evolution, and magnetized tori

Pugliese and Stuchlík (2021) introduced a model-independent inversion that determines the spin of a Kerr black hole from the energy of an outflow powered by extraction of the hole's rotational energy, expressed through the parameter $\xi=1-M_{\mathrm{irr}}/M$ (with $M_{\mathrm{irr}}$ the irreducible mass and $M$ the ADM mass of the initial state) and organized by metric bundles and light surfaces in the extended Kerr plane. The analysis rests on idealizations: total extraction down to a Schwarzschild final state, no accretion feeding of the black-hole mass and angular momentum, stationary perfect-fluid tori with constant specific angular momentum, no magnetic fields, and negligible torus self-gravity and runaway mass transfer. The open problem is whether this outflow-energy-to-spin inversion survives realistic relaxation of these assumptions: partial extraction to a spinning final state with simultaneous accretion-driven changes of $M$ and $J$ , non-constant angular-momentum profiles with viscosity and alignment, and magnetohydrodynamic flows. Later work advanced several related directions (GRMHD simulations of double tori around a Schwarzschild hole, viscous ringed-disk evolution, horizon-replica observables in shadows and redshifted emission, super-spinning attractor signatures) but the corrected inversion relations for the realistic regime, and the identifiability of the mapping itself, remain unestablished.

Background

A Kerr black hole (the unique stationary vacuum black-hole solution of general relativity) is characterized by mass $M$ and angular momentum $J=aM$ . Its energy splits into an irreducible part, tied to the horizon area through the Christodoulou–Ruffini mass formula $M^2=M_{\mathrm{irr}}^2+J^2/(4M_{\mathrm{irr}}^2)$ , and a rotational (reducible) part $M-M_{\mathrm{irr}}$ , at most $1-1/\sqrt{2}\approx29\%$ of the total mass for an extremally spinning hole. Penrose-type processes extract this rotational energy; in astrophysical settings the extraction is expected to be mediated by accreting material and magnetic fields, e.g. the Blandford–Znajek mechanism (electromagnetic extraction of spin energy by fields threading the horizon), and to power jets and outflows in active galactic nuclei. Empirically, Daly and collaborators developed an 'outflow method' that estimates supermassive-black-hole spins from measured outflow (beam) powers under the hypothesis that the outflow is powered by the black hole's rotational energy. Pugliese and Stuchlík (2021) gave this idea a spacetime-geometric formulation. They model the orbiting matter as geometrically thick, pressure-supported perfect-fluid tori ('Polish doughnuts'), which are stationary configurations of constant specific angular momentum $\ell=-u_\phi/u_t$ solving the relativistic Euler equation (Boyer problem), and whose inner edge terminates in a cusp through which matter accretes. A 'ringed accretion disk' (RAD) is an aggregate of such tori, possibly alternating corotating and counter-rotating, orbiting one central Kerr black hole. The extraction is quantified by $\xi=1-M_{\mathrm{irr}}/M$ evaluated at the initial state (so $0\le\xi\le\frac{1}{2}(2-\sqrt2)$ ), and the work assumes complete extraction, i.e. the initial spinning hole transitions to a final Schwarzschild state. The geometric machinery consists of 'light surfaces' — the surfaces, defined by the condition $\mathcal{L}\cdot\mathcal{L}=0$ for the Killing field $\mathcal{L}=\partial_t+\Omega\,\partial_\phi$ associated with a stationary observer of angular velocity $\Omega$ , on which such an observer would have to move at the speed of light — and 'metric bundles': curves of constant limiting photon-orbit frequency $\omega$ in the extended $(r,a)$ plane (the plane extended beyond the Kerr black-hole bound $a\le M$ into the naked-singularity sector), each tangent to the horizon curve. Bundles relate different spacetime states before and after extraction, and predict 'horizon replicas' — orbits, equatorial or polar, whose angular velocity equals the Killing-horizon angular frequency $\omega_H$ of some black-hole state — as candidate observables linking the pre- and post-extraction geometries. Within this idealized setting the analysis identifies potentially significant spins ( $a\approx0.94M$ , $0.7M$ , $0.3M$ ) and formation bounds on corotating ( $\ell/a\ge2$ ) and counter-rotating ( $\ell/a\le-22/5$ ) tori. The authors explicitly state, however, that the analysis 'made use of some simplificative assumptions' planned 'for future analysis': the total-extraction hypothesis; neglect of mass feeding and of the runaway instability (the self-amplifying mass transfer from a thick torus into the hole, an iterative disk–black-hole process studied since Abramowicz et al. and Font–Daigne in the test-fluid limit); torus self-gravity; the constant- $\ell$ idealization versus realistic $\ell(r)$ profiles, viscous transport, and Bardeen–Petterson-type warping and alignment (the Lense–Thirring-driven tearing and alignment of a misaligned disk); and the absence of magnetic fields and magnetospheres, including Blandford–Znajek and magnetic-Penrose channels. The observational feasibility of the replica predictions, by contrast, is presented in the paper as an affirmative outlook rather than a deferral — the characteristic frequencies of the bundle 'can be measured with the replicas orbits', and the authors claim it 'should be possible to observe the replicas along the axis of the black hole'. They also defer the physical interpretation of bundle features originating in the naked-singularity sector of the extended plane.

Problem Statement

Establish — or refute — the physically robust version of the spin inversion of Pugliese & Stuchlík (2021). Given the energy of an outflow attributed to disk-mediated extraction of Kerr black-hole rotational energy, possibly together with disk-state observables (torus cusp/inner-edge locations, horizon-replica frequencies), determine the initial black-hole spin $a$ , where the idealized analysis assumes: (i) complete extraction of the rotational energy, the final state being Schwarzschild; (ii) no accretion feeding of the black-hole mass and angular momentum during the process (fixed background); (iii) stationary perfect-fluid tori with constant specific angular momentum $\ell$ ; (iv) no magnetic fields or magnetospheres; and (v) negligible torus self-gravity and runaway mass transfer. The problem is to derive the corrected inversion relations (the generalized mapping that replaces $\xi\mapsto a$ ) when these assumptions are relaxed — at minimum partial extraction to a final state of nonzero spin together with accretion-driven changes of $M$ and $J$ , non-constant angular-momentum profiles with viscosity and alignment, and magnetized (GRMHD) disk physics — or to demonstrate that the inversion is not identifiable in stated model classes that satisfy the same minimum relaxation bundle and a stated operational definition of extraction efficiency (separating black-hole-powered from accretion- and disk-powered contributions to the outflow), e.g. by explicit families of distinct initial spin configurations that produce identical outflow observables. The common objective uniting all requested results is a physically defensible, observationally testable mapping from outflow energy to black-hole spin for the disk-mediated extraction scenario; each component (bookkeeping relations, disk-model realism) is a load-bearing piece of that single mapping rather than an independent target.

References

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