Is momentum conserved when dark matter falls into a black hole?

Question

Since dark matter does not interact with any other forces (unless you consider the gravitational force), I would like to know what happens to the momentum of the dark matter when falling into a black hole.

So: what happens to that momentum?

Update: I just read that normally there will be an accretion disk, inner disk heats up due to friction and speeds up the outer disk. This transferring of momentum cause mass to fall closer to the black hole but now I like to know what role does dark matter play for this case and also for another case considering if there is no accretion disk and assume static black hole? I know this is a bad question because the physics of dark matter is still new and on-going so please use whatever we have at hand then maybe someday I will fill in those blanks.

Answer 1

According to the No-hair theorem (or, more generally, conjecture), linear momentum is conserved in a black hole along with angular momentum, mass-energy, and electric charge. Also, nothing in the theory of dark matter contradicts the conservation of momentum as a Noether current due to the translation symmetry of space (see the Noether theorem) plus the Stationary action principle that holds in General Relativity. Therefore the momentum of the infalling dark matter transfers to the momentum of the black hole.

Answer 2

The dark matter (dm) falls into the hole because of gravity, with which (supposedly) the dark matter only interacts. The black hole (bh) has momentum in spacetime and so does the black matter.

It depends on how the dm falls into the hole. When it has (wrt to the bh) no side-momentum (falling straight in), then you have the old problem of what happens to a mass falling into a bh? You can look the answer to that up. For example here. It's observer-dependent.

When it's not falling straight, in essence, the answer is the same. Some angular momentum (for the observer falling along with the dm) angular momentum will be added to the bh.