= Black-hole spin-up by coherent accretion
Coherent prograde gas accretion increases the <angular momentum> of a <black hole>. For a zero-torque thin disc, the rest mass $dm_0$ entering from the <innermost stable circular orbit> changes the hole by $dM=E_{\rm ISCO}dm_0$ and $dJ=\ell_{\rm ISCO}dm_0$. Its dimensionless spin obeys $da_*/d\log M=c\ell_{\rm ISCO}/(GME_{\rm ISCO})-2a_*$. Starting from zero spin and neglecting captured radiation, the formal extremal endpoint has $M_f=\sqrt6M_i$ and requires rest-mass supply approximately $1.8464M_i$. Photon capture leads instead to the <Thorne spin limit>.
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