Aligning spinning black holes and accretion discs

We consider the alignment torque between a spinning black hole and an accretion disc whose angular momenta are misaligned. This situation must hold initially in almost all gas accretion events on to supermassive black holes, and may occur in binaries where the black hole receives a natal supernova kick. We show that the torque always acts to align the hole’s spin with the total angular momentum without changing its magnitude. The torque acts dissipatively on the disc, reducing its angular momentum, and aligning it with the hole if and only if the angle θ between the angular momenta Jd of the disc and Jh of the hole satisfy the inequality cosθ > −J d/2J h. If this condition fails, which requires both θ > π/2 and J d < 2J h, the disc counteraligns.