Could black holes interact with dark matter at distances far greater than matter?

Question

I watched this Nova episode on super massive black holes and learned that the speed at which the outer stars in a galaxy orbit correspond to the size of the black hole at the galaxy's center. They also mention how the gravity of the black holes at the center of galaxies could not possibly exerting any gravitational influence over these distant stars. At the same time, I know that the reason scientists believe the outer stars orbit at the speeds they do is due to dark matter. So, is it possible that black holes in particular, and gravity in general, interact differently with dark matter?

I know dark matter is believed to be made up of an unknown particle(s). However, I have never heard a theory about these particles interacting differently with other particles/forces. The explanations I have been given about how dark matter does what it does, is simply that it is more abundant and massive than regular matter. With this possible scenario, I am wondering if the reason dark matter seems so much more abundant is simply because it interacts and behaves differently than normal matter. And, if this is possible, would dark matter require a different force and force carrier?

Answer 1

Like most proposals, it is possible of course; in physics we must ultimately test proposals experimentally. In the meantime (i.e. in this case whilst waiting for experimental observation and study of dark matter here on Earth), one must resort to assessing plausibility in the light of what we already know.

There are two ways your proposal, if true, could fit with current physics:

1. If you are proposing gravity would act differently, then it would falsify the weak equivalence principle, since dark matter and "normal" matter would fall differently in Galileo's Tower of Pisa experiment. As a none-too-small byproduct, your proposal would thus falsify General Relativity to its very foundations. Dark matter's gravitational mass would be different from its inertial mass, gravity would depend on composition and thus would be much more like the other three fundamental forces (which act on a composition-dependent coupling strength i.e. "charge" of the actor in question) and the simple all-geometrical Einsteinian picture would be much harder to uphold;

2. Alternatively, you raise the possibility of a different force carrier between black holes and dark matter. In effect, you are proposing something separate from gravity: i.e. you are proposing a fifth fundamental force;

In contrast, there are two explanation paradigms commonly discussed:

1. The dark matter proposal is conventionally raised as an explanation of the form of the galatic rotation curve that is in keeping with General Relativity (and Newtonian Theory as an approximation);

2. On the other hand, some theorists explore alternative theories of gravity, such as MOND (Modified Newtonian Dynamics) and its relativistic cousins such as TeVeS (Tensor Vector Scalar gravity).

Your proposal goes further than either of the above: it says that there is something unseen there and it either violates known laws of gravity or calls on a new force. It is therefore a more complicated proposal. Occam's Razor, and the finite resources for physics research, would seem to suggest that we should explore the two simpler alternatives above and rule them out before moving to yours first!

Answer 2

Rod Vance's answer explains why your proposed explanation seems unlikely to many. I'd like to explain what the observations you allude to really show.

The correlation between black hole mass and stellar velocities is known as the M-sigma relation. First, note that it does not involve the outer stars in the galaxy. While it is true that the outer stars (and indeed the motion of the gas beyond them) tells us a lot about the distribution of dark matter, this is a separate issue. The correlation is with central bulge stars. Even in spiral galaxies, the central stars aren't orbiting in an orderly fashion -- they are flying in all directions.

Sigma ($\sigma$) is the symbol used for velocity dispersion. Imagine a bunch of stars moving randomly in a cluster (or imagine a bunch of gnats zipping around). They might not all be going at the same speed (the stars that is; I don't know about gnats). If you plot a distribution of speeds, it would have some width, and we call that $\sigma$.

In fact, $\sigma$ is pretty easy to measure even if you can't see individual stars, and that is part of the reason it is used. Each star imprints specific narrow lines in the spectrum of light emitted. However, if stars are moving at different velocities, these features will be redshifted and blueshifted to us based on how fast each star is moving away from or toward us at the moment. The result from our perspective is a measurable Doppler broadening of the lines. While technically this only tells us about the velocity dispersion along our line of site, a justifiable assumption of isotropy/spherical symmetry (the cluster of stars doesn't have a preferred viewing angle) can bootstrap us to a 3D velocity dispersion.

Now velocity dispersion can be affected by/correlated with a number of things. In particular, if there are more stars, you expect there to be larger velocities in the central region. This is a result of the virial theorem -- if the stars are in thermodynamic equilibrium, their kinetic energy scales proportional to the gravitational potential energy of the cluster.

In some sense then it's not terribly surprising that this relation exists. Centers of galaxies with more mass would be expected to harbor larger black holes ($M$) and at the same time have higher velocity dispersions ($\sigma$). We don't need the stars to actually be orbiting the black hole, and indeed even within a bulge the black hole doesn't dominate the gravitational potential.