= Black-hole and disk alignment under internal torque
{title2=$\dot{\mathbf J}_h=-K_1\mathbf J_h\times\mathbf J_d-K_2\mathbf J_h\times(\mathbf J_h\times\mathbf J_d)$}
For the opposite internal torques on a <black hole> and its <accretion disk>, the total <angular momentum> $\mathbf J_t$ and the black-hole spin magnitude $J_h$ are constant. If $A=K_2|\mathbf J_h\times\mathbf J_d|^2$, then $d(\mathbf J_h\cdot\mathbf J_t)/dt=A$ and $dJ_d^2/dt=-2A$. The $K_1$ term produces <precession>; the positive $K_2$ term increases the alignment of the hole with the total vector while decreasing the disk angular-momentum magnitude. The exact direction equation is $\dot c=K_2J_hJ_t(1-c^2)$ for $c=\cos\angle(\mathbf J_h,\mathbf J_t)$.
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