= Counteralignment of a black hole and accretion disk
{title2=$J_t<J_h\quad\Longleftrightarrow\quad\cos\theta_0<-J_{d,0}/(2J_h)$}
After persistent dissipative alignment of the hole with the conserved total <angular momentum>, the final vectors are $\mathbf J_{h,f}=J_h\widehat{\mathbf J}_t$ and $\mathbf J_{d,f}=(J_t-J_h)\widehat{\mathbf J}_t$. The disk is therefore antiparallel to the hole if $J_t<J_h$. Squaring $J_t^2=J_h^2+J_{d,0}^2+2J_hJ_{d,0}\cos\theta_0$ gives the displayed initial-angle criterion. The disk and hole need not finish aligned with each other even though the hole aligns with the total. Equality $J_t=J_h$ leaves zero final disk angular momentum rather than a defined disk spin direction.
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