Spin-flip

A black hole spin-flip occurs when the spin axis of a rotating black hole undergoes a sudden change in orientation due to absorption of a second (smaller) black hole. Spin-flips are believed to be a consequence of galaxy mergers, when two supermassive black holes form a bound pair at the center of the merged galaxy and coalesce after emitting gravitational waves. Spin-flips are significant astrophysically since a number of physical processes are associated with black hole spins; for instance, jets in active galaxies are believed to be launched parallel to the spin axes of supermassive black holes. A change in the rotation axis of a black hole due to a spin-flip would therefore result in a change in the direction of the jet.

A spin-flip is a late stage in the evolution of a binary black hole. The binary consists of two black holes, with masses ${\displaystyle M_{1}}$ and ${\displaystyle M_{2}}$, that revolve around their common center of mass. The total angular momentum ${\displaystyle J}$ of the binary system is the sum of the angular momentum of the orbit, ${\displaystyle {L}}$, plus the spin angular momenta ${\displaystyle {S}_{1,2}={S}_{1}+{S}_{2}}$ of the two holes. If we write ${\displaystyle \mathbf {M_{1}} ,\mathbf {M_{2}} }$ as the masses of each hole and ${\displaystyle \mathbf {a_{1}} ,\mathbf {a_{2}} }$ as their Kerr parameters, then use the angle from north of their spin axes as given by ${\displaystyle \theta }$, we can write,

$${\displaystyle \mathbf {S} _{1}=\{\mathbf {a} _{1}\mathbf {M} _{1}\cos(\pi /2-\theta ),\mathbf {a} _{1}\mathbf {M} _{1}\sin({\tfrac {\pi }{2}}-\theta )\}}$$ $${\displaystyle \mathbf {S} _{2}=\{\mathbf {a} _{2}\mathbf {M} _{2}\cos({\tfrac {\pi }{2}}-\theta ),\mathbf {a} _{2}\mathbf {M} _{2}\sin({\tfrac {\pi }{2}}-\theta )\}}$$ $${\displaystyle \mathbf {J} _{\text{init}}=\mathbf {L} _{\text{orb}}+\mathbf {S} _{1}+\mathbf {S} _{2}.}$$

If the orbital separation is sufficiently small, emission of energy and angular momentum in the form of gravitational radiation will cause the orbital separation to drop. Eventually, the smaller hole ${\displaystyle M_{2}}$ reaches the innermost stable circular orbit, or ISCO, around the larger hole. Once the ISCO is reached, there no longer exists a stable orbit, and the smaller hole plunges into the larger hole, coalescing with it. The final angular momentum after coalescence is just

$${\displaystyle \mathbf {J} _{\text{final}}=\mathbf {S} ,}$$

the spin angular momentum of the single, coalesced hole. Neglecting the angular momentum that is carried away by gravitational waves during the final plunge—which is small—conservation of angular momentum implies

$${\displaystyle \mathbf {S} \approx \mathbf {L} _{\text{ISCO}}+\mathbf {S} _{1}+\mathbf {S} _{2}.}$$

${\displaystyle S_{2}}$ is of order ${\displaystyle (M_{2}/M_{1})^{2}}$ times ${\displaystyle S_{1}}$ and can be ignored if ${\displaystyle M_{2}}$ is much smaller than ${\displaystyle M_{1}}$. Making this approximation,