Is there a way that this empirical relationship could come about without exotic forms of matter or alterning Einstein's laws of general relativity?
I think that there is, with the help of a phenomena called "ultra-fast outflows" (aka black hole barf) which puts matter just were inferred dark matter distributions need it to go. This post explains that conjecture.
In this view, the large scale structure of a galaxy is largely driven by the properties of its central black holes which is largely one dimensional (see for example, some references here). The radius of a black hole's event horizon is approximately 2.95 times it mass in units of the mass of Earth's sun, adjusted by a factor of up to two for angular momentum (which varies considerably between black holes) and electromagnetic charge (which in practice is almost always almost electrically neutral like the stars that collapse to form them). Black holes of the same mass and angular momentum and charge are for all practical purposes identical, so it makes sense that they galaxies that they would attract to surround them would also be similar (and why we only start seeing certain phenomena associate with black holes at the galactic level).
The "mass of a galaxy's central black hole and the velocity of stars in a vast, roughly spherical structure known as its bulge" are closely related. So is the structure of the galaxy. Elliptical galaxies almost always have black holes in one size range, while spiral galaxies almost always have black holes in another size range. Every type of galaxy has a particular sized black hole associated with it and galaxies of all kinds tend to have some disk of rotation.
If galactic structure is a function of black hole size, that might not be the only thing that is a function of its size.
Active black holes acquire their power by gradually accreting -- or "feeding" on -- million-degree gas stored in a vast surrounding disk. This hot disk lies within a corona of energetic particles, and while both are strong X-ray sources, this emission cannot account for galaxy-wide properties. Near the inner edge of the disk, a fraction of the matter orbiting a black hole often is redirected into an outward particle jet. Although these jets can hurl matter at half the speed of light, computer simulations show that they remain narrow and deposit most of their energy far beyond the galaxy's star-forming regions. . . .
At the centers of some active galaxies, X-ray observations at wavelengths corresponding to those of fluorescent iron show that this radiation is being absorbed. This means that clouds of cooler gas must lie in front of the X-ray source. What's more, these absorbed spectral lines are displaced from their normal positions to shorter wavelengths -- that is, blueshifted, which indicates that the clouds are moving toward us.
In two previously published studies, [Francesco] Tombesi and his colleagues showed that these clouds represented a distinct type of outflow. In the latest study, which appears in the Feb. 27 issue of Monthly Notices of the Royal Astronomical Society, the researchers targeted 42 nearby active galaxies using the European Space Agency's XMM-Newton satellite to hone in on the location and properties of these so-called "ultra-fast outflows" -- or UFOs, for short. The galaxies, which were selected from the All-Sky Slew Survey Catalog produced by NASA's Rossi X-ray Timing Explorer satellite, were all located less than 1.3 billion light-years away.
The outflows turned up in 40 percent of the sample, which suggests that they're common features of black-hole-powered galaxies. On average, the distance between the clouds and the central black hole is less than one-tenth of a light-year. Their average velocity is about 14 percent the speed of light, or about 94 million mph, and the team estimates that the amount of matter required to sustain the outflow is close to one solar mass per year -- comparable to the accretion rate of these black holes.
"Although slower than particle jets, UFOs possess much faster speeds than other types of galactic outflows, which makes them much more powerful," Tombesi explained. "They have the potential to play a major role in transmitting feedback effects from a black hole into the galaxy at large."
By removing mass that would otherwise fall into a supermassive black hole, ultra-fast outflows may put the brakes on its growth. At the same time, UFOs may strip gas from star-forming regions in the galaxy's bulge, slowing or even shutting down star formation there by sweeping away the gas clouds that represent the raw material for new stars. Such a scenario would naturally explain the observed connection between an active galaxy's black hole and its bulge stars.
Tombesi and his team anticipate significant improvement in understanding the role of ultra-fast outflows with the launch of the Japan-led Astro-H X-ray telescope, currently scheduled for 2014. In the meantime, he intends to focus on determining the detailed physical mechanisms that give rise to UFOs, an important element in understanding the bigger picture of how active galaxies form, develop and grow.So, black holes, rather than merely absorbing everything that comes their way are constantly spewing X-rays, particles and ultra-fast outflows at relativistic speeds away from the central black hole, more of less on the axis perpendicular to the galaxy's plane of rotation.
Now, an axis of relativistic matter perpendicular to a spinning disk of matter can get very long relative to the galactic disk travelling at 14%-50% of the speed of light, or the speed of light itself in the case of X-rays over billions of years, with nothing else in the way but the tug of the galaxy's own gravity. The Milky Way galaxy, for example, has a radius of about 50,000 to 60,000 light years, a blink of an eye for a body with 13.2 billion year old stars in it.
A very long axis of matter relative to the galactic disk through is focus at the black hole creates an acceleration of point masses in the galactic disk in the form of GM/r (where G is the gravitational constant, M is the mass of the axis, and r is the distance from the black hole of the object in the galactic disk). The black hole itself and the central bulge, meanwhile, will exert a force like a point mass in the form GM/r^2.
To replicate the MOND modification to gravity, where gravity starts acting like a 1/r force (just like the magnetic field from a long wire) instead of a 1/r^2 force of a point source, where the transition takes place at a fixed acceleration a-zero in all galaxies (with a-zero being approximately 1.2*10^-10 ms^-2), the gravitational mass of the material spewed by the central black hole of every galaxy (in the last five million years or so) must be approximately proportional to (but multiple orders of magnitude smaller than) the square root of the central bulge's mass for almost all galaxies. In particular, it must be equal to the square root of a-zero times the square root of the central bulge's mass times a constant with units of length to assure that the formula is dimensionally consistent. Also note that for these purposes, relativistic momentum in the particles and the energy in the X-rays count as mass.
If this relationship holds true, the ordinary X-rays, and ordinary Standard Model matter in the particle flows and ultra-fast outflows from a black hole (mostly non-luminous) over thirteen billion years or so. Actually, any lengthy of time much longer than the radius of the galaxy in light years, perhaps by a factor of fifty or a hundred, so more like 5 million years or more for a Milky Way sized galaxy (about 0.2% of its age), will have almost the same effect since the far edges of the axis mass don't influence the outcome by too much. For example, the mass of the Milky Way galaxy is about 10^12 solar masses. So, the axial mass would need to be 10^6 solar masses, times 10^-5 (i.e. 10 solar masses), times a constant to fix the proper unit conversions between solar masses and the square root of a-zero as express in meters per second squared. In a galaxy that is spewing out a bit more than one solar mass a year into its axis for many millions of years, the order of magnitude may be about right even if the unit conversion factor is quite large (say in the hundreds of thousands).
The inapplicability of the MOND law in galactic clusters, which unlike ordinary galaxies, aren't structured around a single black hole, also immediately makes sense if this explains dark matter. But, balancing the ledgers in these systems, if you don't need exotic dark matter for the rest of the galaxies, isn't hard. As previously explained in two sets of quotes from people more expert than I which I merge below from this post:
Clusters you certainly could fit just with baryons. They’re rare systems. If that is the only place we need dark baryons, then do the integrals. You can satisfy the residual mass discrepancy in clusters in MOND without making much dent in the BBN missing baryon budget.
Do I *like* such a solution? Certainly not. Neither do I like that fact that clusters are the only systems that come close to having the right baryon content in LCDM. Why are galaxies missing more than half of their baryons? Dwarfs > 90%? . . .
90% of all cosmic baryons are presently undetected, right? Only a fraction of the baryonic matter we see directly is in clusters (O(a few percent), let’s say 10%) So why can’t a small fraction, say O(2%), of all the cosmic dark baryons be in the form of e.g. jupiters in the central parts of clusters? They and stars would then dominate the cluster mass and be dissipationless —> no problem with the bullet cluster in MOND. . . . In [Sander's paper] http://arxiv.org/abs/astro-ph/0703590 he states about cluster dark matter in MOND: “For example, there are more than enough undetected baryons to make up the missing dark component; they need only be present in some non-dissipative form which is difficult to observe.”Conveniently, for example, new observations have discovered a high abundance of low luminosity starts in galactic clusters relative to regular galaxies, explaining some of their seemingly dark matter. Likewise, estimates of the amount of atomic hydrogen have been underestimates as have estimates of the amount of dim matter in elliptical galaxies.
Ultra-fast outflows may be just the thing to explain where some more of the missing baryons have gone - into the axis.
Note also, that since we are only concerned about dark matter-MOND effects very close to the outer rim of the galactic disk and beyond in that plane where we can observe them, rather than spherically, not nearly as much dim axial matter is require to generate the observed effects.
Also, this dim axial matter, rather than being an exotic relic of the early universe that must be explained with some new kind of baryongenesis or the equivalent has a know, recent and dynamic origin.
The MOND effect could also partially be due to the effects in general relativity, but not in Newtonian gravity, of the rotation of the galactic plane itself, an angular momentum that gravitates in general relativity but not in Newtonian gravity. This too should have a 1/r effect, and while it is slight, becomes signficant relative to the point source-like central bulge gravitational effect that falls of like 1/r^2 in the outer fringe of the galaxy that would be added to the axial matter effects.
As in other posts with this tag, the ideas contained in this post are my personal conjectures based on back of napkin class estimates, which while not coming from nowhere, also do not necessarily have any wide currency in the scientific community.
Also, as a footnote it is worth noting that any model of non-collisionless dark matter with particles that weigh less than 45 GeV and does not have electromagnetically charged dark matter particles, just like MOND, requires a new force law of some kind to apply to self-interactions in the dark sector. We know that such particles can't interact via the weak force or they would have been detected in W and Z boson decays.