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The conserved total angular momentum is ${\mathbf{J}}_t=(0,0,2)$, and the black-hole spin magnitude isThe initial hole is noncollinear with the total vector, so persistent positive alignment torque leads to the positive $z$ direction. The final vectors areThe final disk magnitude is $2-\sqrt{3}$, smaller than its initial magnitude one, and its direction is parallel to the hole. Their sum remains $(0,0,2)$.

Here the conserved total angular momentum is ${\mathbf{J}}_t=(0,0,1)$, while the black-hole spin magnitude remains $\sqrt{3}$. The hole again aligns with the positive $z$ direction in the persistent-torque regime. ThusSince $1-\sqrt{3}<0$, the disk points in the negative $z$ direction and has magnitude $\sqrt{3}-1$. This realizes counteralignment of a black hole and accretion disk even though the hole itself is aligned with the conserved total vector. The black hole and disk do not always finish with aligned spins.

Interpret $\dot{m}$ as the rest-mass supply rate through a coherent disc; the hole's gravitating mass grows more slowly because radiation carries away binding energy. Initially the hole is a Schwarzschild black hole, whose innermost stable circular orbit is at $R_{\ast }=6G{M}_0/c^2$. The relativistic orbital constants there areThey follow by extremizing the timelike geodesic effective potential for a circular orbit and imposing marginal stability. Assume negligible torque inside the inner edge, so the gas carries this specific angular momentum into the hole. Its initial spin-up torque isfor $M_0=10^8{M}_{\odot }$ and $\dot{m}=0.1M_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$. In SI units this is $9.7\times 10^{41}\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-2}$. A purely Newtonian estimate $\ell \sim \sqrt{G{M}_0{R}_{\ast }}$ gives a similar order of magnitude, but the relativistic value is appropriate at the innermost stable circular orbit.

The maximal Kerr black hole angular momentum at the initial mass is $J_{\max ,0}=G{M}_0^2/c\simeq 8.8\times 10^{64}\mathrm{g}\mathrm{c}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. Holding both that mass and the initial torque fixed would giveInitially ${\dot{M}}_{\mathrm{BH}}=E_{\mathrm{ISCO}}\dot{m}\simeq 0.094M_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$, so the corresponding frozen-coefficient estimate of gravitating mass gain is about $2.7\times 10^7{M}_{\odot }$. These are useful initial spin-growth scales, not a self-consistent time and mass increase for reaching extremality: the target angular momentum itself grows as $M^2$, and the inner orbit changes as the spin increases.

A consistent endpoint estimate is obtained from black-hole spin-up by coherent accretion. Let $a_{\ast }=cJ/(G{M}^2)$ and $r=R_{\mathrm{ISCO}}{c}^2/(GM)$, where $M$ is the current mass. On the prograde Kerr black hole inner-orbit branch, $1\le r\le 6$, the orbital constants may be writtenIgnoring photon capture, an accreted rest mass $d{m}_0$ changes the hole by $dM=Ed{m}_0$ and $dJ=\ell d{m}_0$. Differentiating the spin definition givesSubstitution of the orbital constants gives $dr/d\log M=-2r$, henceAt the formal extremal limit $r=1$, the result isFor constant rest-mass supply, use $r=6(M_0/M{)}^2$ to integrate the elapsed time:Thus the idealized coherent-disc endpoint isIf the quoted $0.1M_{\odot }\mathrm{y}{\mathrm{r}}^{-1}$ were instead the actual gravitating mass-growth rate, the corresponding endpoint time would be $\Delta M_{\mathrm{BH}}/{\dot{M}}_{\mathrm{BH}}\simeq 1.45\times 10^9\mathrm{yr}$. The two rate conventions cannot be interchanged.

The exact extremal endpoint is an idealization. Capture of disc photons exerts a counteracting torque and gives the Thorne spin limit, $a_{\ast }\simeq 0.998$, for the standard radiatively efficient thin-disc model. Therefore physical disc accretion approaches a large spin below unity rather than producing an exactly extremal hole. Thorne's original spin-evolution calculation describes that correction. The independently integrated expressions above explain why the frozen-mass estimate is too short for the ideal endpoint.

The qualitative conclusion is robust: a hole that gains most of its mass through a persistent, coherently corotating Shakura--Sunyaev thin disk is expected to have substantial black-hole spin. Very low spins require a different angular-momentum history, such as short episodes with changing orientations, counterrotating accretion, or mergers. Coherent feeding, not merely the existence of a disc during each episode, is the decisive assumption.

The Soltan argument offers a population-level observational test. From the bolometric luminosity function of active galactic nuclei, infer the total energy radiated per comoving volume over cosmic history,Compare it with the increase $\Delta {\rho }_{\mathrm{BH}}$ in supermassive black hole mass density attributed to radiative accretion. For an effective radiative efficiency of black-hole accretion ${\eta }_{\mathrm{eff}}$,In a zero-torque relativistic thin disc, $\eta =1-E_{\mathrm{ISCO}}$ rises from about $0.057$ at zero spin to $0.423$ in the photon-free prograde extremal limit. The measured effective efficiency therefore constrains population spin models under the thin-disc assumption. Soltan's original population argument explains how integrated quasar light constrains accumulated black-hole mass.

This is an accreted-rest-mass-weighted efficiency constraint, not a direct measurement of the arithmetic mean spin of every black hole. Obscured emission, bolometric corrections, seed masses, nonradiative growth, and disc orientation introduce uncertainty, and the efficiency-spin relation is nonlinear. These qualifications are essential when interpreting an inferred average spin for the whole population.

At radius $r$, radiation of luminosity $L$ gives matter of flux-mean opacity $\kappa$ the outward accelerationEquating this with the gravitational acceleration $G{M}_{\mathrm{BH}}/r^2$ gives the Eddington luminosity

Let $L=f{L}_{\mathrm{Edd}}$ and let $\mathcal{M}(r)$ be the radiative force multiplier relative to the Compton or electron-scattering acceleration. A steady spherical wind obeys mass conservation ${\dot{M}}_w=4\pi r^2\rho v$ and the radial Euler momentum equationWhen gas pressure is negligible, radiation accelerates the wind locally if $f\mathcal{M}>1$. Electron scattering alone has $\mathcal{M}\simeq 1$, so it launches a wind only near or above the Eddington ratio $f=1$. Bound-free absorption, spectral lines, or dust can give $\mathcal{M}\gg 1$ and allow a sub-Eddington source to launch a wind, provided the gas is not so highly ionized that those extra opacities disappear. A wind that has already acquired sufficient kinetic energy may coast through a region where $f\mathcal{M}<1$, but such a region cannot launch a cold wind from rest.

Write the innermost stable circular orbit asEstimating the emitting area as $4\pi R_{\mathrm{ISCO}}^2$, the Stefan–Boltzmann law givesand henceThus $T_{\mathrm{BB}}\propto M_{\mathrm{BH}}^{-1/4}$ at fixed $f$ and opacity: a larger black hole radiates from an area growing as $M_{\mathrm{BH}}^2$, faster than its Eddington luminosity grows. Prograde black-hole spin moves the ISCO inward and raises the characteristic temperature, whereas retrograde spin moves it outward and lowers the temperature. Relativistic transfer and the radial temperature profile change the numerical coefficient, not these leading scalings.