= Solution
{c}
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 $R_*=6GM_0/c^2$. The relativistic orbital constants there are
$$
E_{\rm ISCO}=\sqrt{\frac89},\qquad
\ell_{\rm ISCO}=2\sqrt3\frac{GM_0}{c}.
$$
They 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 is
$$
\boxed{\dot J_0=\dot m\,\ell_{\rm ISCO}
=2\sqrt3\frac{GM_0\dot m}{c}
\simeq9.7\times10^{48}\,\mathrm{g\,cm^2\,s^{-2}}}
$$
for $M_0=10^8M_\odot$ and $\dot m=0.1M_\odot\,\mathrm{yr^{-1}}$. In SI units this is $9.7\times10^{41}\,\mathrm{kg\,m^2\,s^{-2}}$. A purely Newtonian estimate $\ell\sim\sqrt{GM_0R_*}$ 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 $J_{\max,0}=GM_0^2/c\simeq8.8\times10^{64}\,\mathrm{g\,cm^2\,s^{-1}}$. Holding both that mass and the initial torque fixed would give
$$
\boxed{t_{\rm spin,rough}\sim\frac{J_{\max,0}}{\dot J_0}
=\frac{M_0}{2\sqrt3\dot m}\simeq2.9\times10^8\,\mathrm{yr},
\qquad\Delta m_{\rm rest,rough}\sim2.9\times10^7M_\odot.}
$$
Initially $\dot M_{\rm BH}=E_{\rm ISCO}\dot m\simeq0.094M_\odot\,\mathrm{yr^{-1}}$, so the corresponding frozen-coefficient estimate of gravitating mass gain is about $2.7\times10^7M_\odot$. 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 $a_*=cJ/(GM^2)$ and $r=R_{\rm ISCO}c^2/(GM)$, where $M$ is the current mass. On the prograde <Kerr black hole> inner-orbit branch, $1\leq r\leq6$, the orbital constants may be written
$$
a_*(r)=\frac{\sqrt r}{3}\left(4-\sqrt{3r-2}\right),
\qquad
E(r)=\sqrt{1-\frac{2}{3r}},
\qquad
\frac{c\ell(r)}{GM}=\frac{2}{3\sqrt3}\left(1+2\sqrt{3r-2}\right).
$$
Ignoring photon capture, an accreted rest mass $dm_0$ changes the hole by $dM=E\,dm_0$ and $dJ=\ell\,dm_0$. Differentiating the spin definition gives
$$
\frac{da_*}{d\log M}=\frac{c\ell}{GME}-2a_*.
$$
Substitution of the orbital constants gives $dr/d\log M=-2r$, hence
$$
M\sqrt r=M_0\sqrt6.
$$
At the formal extremal limit $r=1$, the result is
$$
\boxed{M_f=\sqrt6\,M_0\simeq2.45\times10^8M_\odot,
\qquad\Delta M_{\rm BH}=(\sqrt6-1)M_0\simeq1.45\times10^8M_\odot.}
$$
For constant rest-mass supply, use $r=6(M_0/M)^2$ to integrate the elapsed time:
$$
\Delta m_0=\int_{M_0}^{\sqrt6M_0}\frac{dM}{E(M)}
=\int_{M_0}^{\sqrt6M_0}\frac{dM}{\sqrt{1-M^2/(9M_0^2)}}
=3M_0\left[\arcsin\sqrt{\frac23}-\arcsin\frac13\right]
\simeq1.8464M_0.
$$
Thus the idealized coherent-disc endpoint is
$$
\boxed{t_{\rm extremal,ideal}=\frac{\Delta m_0}{\dot m}
\simeq1.85\times10^9\,\mathrm{yr}.}
$$
If the quoted $0.1M_\odot\,\mathrm{yr^{-1}}$ were instead the actual gravitating mass-growth rate, the corresponding endpoint time would be $\Delta M_{\rm BH}/\dot M_{\rm BH}\simeq1.45\times10^9\,\mathrm{yr}$. 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>, $a_*\simeq0.998$, 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. https://articles.adsabs.harvard.edu/pdf/1974ApJ...191..507T[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,
$$
U_{\rm AGN}=\int dt\int L\,\Phi(L,t)\,dL.
$$
Compare it with the increase $\Delta\rho_{\rm BH}$ in <supermassive black hole> mass density attributed to radiative accretion. For an effective <radiative efficiency of black-hole accretion> $\eta_{\rm eff}$,
$$
\Delta\rho_{\rm BH}=\frac{1-\eta_{\rm eff}}{\eta_{\rm eff}}
\frac{U_{\rm AGN}}{c^2},\qquad
\boxed{\eta_{\rm eff}=\frac{U_{\rm AGN}}
{U_{\rm AGN}+\Delta\rho_{\rm BH}c^2}.}
$$
In a zero-torque relativistic thin disc, $\eta=1-E_{\rm ISCO}$ 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. https://academic.oup.com/mnras/article/200/1/115/2893514[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.
Back to article page
