Counteralignment of a black hole and accretion disk ($J_t<J_h⟺\cos {\theta }_0<-J_{d,0}/(2J_h)$)

After persistent dissipative alignment of the hole with the conserved total angular momentum, the final vectors are ${\mathbf{J}}_{h,f}=J_h{\hat{\mathbf{J}}}_t$ and ${\mathbf{J}}_{d,f}=(J_t-J_h){\hat{\mathbf{J}}}_t$. The disk is therefore antiparallel to the hole if $J_t<J_h$. Squaring $J_t^2=J_h^2+J_{d,0}^2+2J_h{J}_{d,0}\cos {\theta }_0$ 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.