What is the relationship between the spin of a galaxy and the spin of its corresponding black hole?
Associated questions:
Do they always have the same axis of rotation?
Do they always spin in the same direction?
Does galaxy angular momentum increase due to frame dragging from black hole, causing continuing increase in galaxy angular velocity?
How does Hawking radiation impact rotational coupling?
Answer
The definitive answer is that we don't know how the spins are connected. And, unfortunately, it might be a pretty long time before we have reliable observational evidence.
Measurements
Black hole spins are very difficult to measure. Two methods have very recently started to be used, but both are approximate - and there is a lot of skepticism about the reliability of their results (especially since they often don't agree). Both methods rely on inferring how close the accretion disk comes to the black hole to infer the Innermost Stable Circular Orbit (ISCO) --- which then implies a spin magnitude and axis (but not which direction along that axis). The first method uses the temperature of the accretion disk (hotter means it comes closer to the BH), and the second method uses broadening of an iron emission line (the broader it is, the faster it is rotating and the more redshifted it is by strong-gravity -- thus the closer to the BH it is).
Both of these methods require fitting noisy data... The main uncertainty seems to be how well the inclination of the system (relative to the observer) can be determined. I.e. if the disk is edge on, could we distinguish that from face-on? How does that affect the measurement?
(Note that both of these methods have mostly been applied to stellar-mass BH systems, within our own galaxy. Massive Black-Holes in other galaxies is much harder.)
Theory
Even [super-]massive black-holes ([S]MBH) are very small compared to the sizes of galaxies they live in; like, 5-8 orders of magnitude smaller. This means it's very hard to simulate/calculate the connection between these size-scales. Specifically, we don't understand very well how MBHs get their gas from the surrounding galaxy. For that reason we don't have very good guesses on how galaxy and BH spins should be related. That being said, both simulations and people's theories suggest the spins should be correlated, but far from directly linked. Because of chaotic effects like galaxy and black-hole mergers, we can say with good certainty that in the 100's of billions of galaxies out there, lots of them will have completely opposite spins. We actually do know of some galaxies that have disk-like populations of stars which rotate in the opposite direction as the bulk of the galaxy... so this suggests the same could happen with BH accretion disks, and thus BH spins.
I can give a couple of definitive answers, however. Both frame-dragging and Hawking radiation from the black-hole definitely have zero effect on the galaxies spin, and evolution. Hawking radiation is completely negligible in all known astrophysical systems, and should only be relevant in micro-black-holes (if they exist[ed]). Frame-dragging is definitely an observable effect in the strong-gravity regime (near the event horizon), but because of the size-scale differences - it will have no way of coupling-to or affecting the galaxy as a whole.
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