In the late 19th century, Newton’s theory of gravity had hit a crisis mode. It had been incredibly successful in predicting the orbits of all the planets, save one, the planet Mercury. Mercury, named for the Roman messenger god, closest to the Sun, had an anomaly, first noticed in 1845, in its orbit of precisely 42 arcseconds per Julian century, a tiny amount that nevertheless hinted at a much bigger problem.
Ad hoc solutions were proposed. Some posited an additional inner planet that was tugging on the orbit, invisible mass unaccounted for. Others suggested that we should slightly modify Newton’s laws to make the problem go away. Yet, no extra planets were found. Modifications to Newton’s laws introduced more problems than they solved.
Meanwhile, Albert Einstein, in attempting to explain time and space and how it relates to matter, created a theory that not only resolved the discrepancy but reduced to Newton’s theory when you made assumptions of weak gravitational fields and slow speeds.
Because Mercury was so close to the Sun, the Sun’s warping of space itself caused Mercury’s orbit to move, an effect Newton’s theory had not the philosophical underpinnings to explain.
The theory of gravity was challenged again in the 1930s, however, with the discovery of flat galactic rotation curves that exceeded that which would be predicted by the matter we could see. We know this by observing the visible discs of distant galaxies and comparing them to observations of starlight and radio waves emitted by hydrogen, the most ubiquitous substance in the galaxy. All these tell us that galaxies don’t rotate like solar systems do, with the inner parts much faster than the outer. Rather, they rotate much more rigidly, like wheels, with the outer parts rotating with similar angular velocity to the inner or even faster.
If Solar systems seem to obey Newton’s laws almost perfectly, and even small groups of stars, why would something as large as a galaxy deviate from that?
The answer that astrophysicists came up with was to propose a kind of matter invisible to light called Dark Matter. There are various reasons why we think that Dark Matter can’t be like ordinary matter, baryonic matter. It has to be something very different, something we haven’t discovered yet.
Quite a few candidates for Dark Matter have been proposed from axions to sterile neutrinos to even more bizarre and mysterious candidates like Weakly Interacting Massive Particles (WIMPs) which had been a favorite of string theorists. Nevertheless, particle theorists do not have a clear candidate.
Others, however, are not convinced that there is any matter at all. Rather, they believe that we are on the cusp of a revolution not unlike that which took place when Einstein published his theory in 1915. The laws of gravity need to be rewritten.
While Einstein’s gravity deviates more and more from Newton’s with increasingly massive bodies and increasing speeds, galactic rotation curves need something different. They need a modification that changes with increasing distance. The reason is that bodies in galaxies are not much more massive or fast on average than bodies in solar systems, and we have confirmed Newton’s and Einstein’s laws within the Solar system to a degree that would make it impossible for distance not to play a roll. For some reason Newton’s gravitational force must violate the inverse square law at long distances.
One of the simplest approaches to modifying Newton’s laws is called Modified Newtonian Dynamics (MOND). Invented by an Israeli scientist, Mordehai Milgrom, in 1983 MOND proposes to alter acceleration of gravity depending on how strong the acceleration is. For large accelerations, which applies to anything within the Solar system, the theory is the same as Newton’s which means that the force on an object is proportional to its acceleration, F=ma. For very tiny accelerations, however, the force is proportional to more like the square of the acceleration. What this means is that for objects that are very distant from a galactic center, the force on them is independent of the distance and thus their rotational curve is flat.
This agrees with what is called the Tully-Fisher relation that the visible mass of a galaxy is related to the fourth power of its velocity.
While MOND easily gives the observed prediction, Dark Matter theories of galaxies based on a density profile called the Navarro-Frenk-White (NFW) profile struggle to model the Tully-Fisher law. In order to model this accurately, DM has to form a halo around the galaxy that changes density as it goes from the center outward, with only the middle density explaining the rotational curves. Because the Tully-Fisher relation relates to the visible mass of the galaxy but the halo that explains it is not visible, these have to be fine tuned together.
This becomes even more apparent in low surface brightness (LSB) galaxies, where dark matter should dominate the entire structure. Yet we see that this is not the case. Rather, the centers appear mostly Newtonian. Some sort of feedback mechanism that we don’t understand would have to be balancing the DM with the visible matter.
Meanwhile, cosmological simulations of dark matter halo formation suggest that there is a hierarchy of DM halos with many small ones forming for each big one. Yet, we don’t see this in our part of the universe unless those small DM halos are unpopulated.
With our dominant theory for galactic rotation curves resting on a knife’s edge to explain a very stable result and giving rise to more unexplained problems the more we try to understand it, it is not surprising to want to look elsewhere. After all, we have been here before.
MOND solves the following problems:
LSBs have a very high discrepancy between gravitationally inferred mass and visible mass. Dark matter must explain why there is so little in the center influencing the behavior. MOND predicts the behavior well.
LSBs have a rising rotational curve as opposed to higher surface density disc galaxies where the curve rises then falls. MOND also explains this well while again DM has to explain the distribution.
Globular clusters, which have a much higher acceleration than disc galaxies, appear to have no dark matter at all, i.e., they are primarily Newtonian. DM cannot explain why there is none in these structures. MOND doesn’t need to.
Dwarf spheroidal galaxies, like LSB, also have large mass-to-light ratios that MOND can explain.
At least 100 rotational curves have been fitted with MOND and these have been fit as well as DM halos. This is despite the fact that while MOND only needs one parameter to fit correctly DM needs two parameters in order to explain the lack of core DM as well as the density profile around the galaxies.
On the face of it MOND is a great empirical theory but does not satisfy the needs of a theory of space and time. It is not a complete theory by a long shot. After all, it only relates one body to another body and the theory of gravity needs to be able to explain both the interaction between many bodies as well as the curvature of space and time of Einstein’s theory. Moreover, it needs to explain more strangely shaped objects like the “bullet cluster”.
MOND, moreover, can’t explain many galaxies without additional matter we can’t see. Some of this matter has actually been found since its introduction in the form of X-ray emitting gases, but more is still needed. In giant and dwarf elliptical galaxies, MOND does slightly better than DM theories but still can’t entirely explain the observed central velocity dispersion, radius, and surface brightness.
How to connect MOND to a true theory of gravity and motion is an open question. MOND might just be a good explanation of DM distributions. It could be a modification of the laws of inertia and actually applies to all forces including electromagnetism. It might be an explanation of some unknown force. Or it could be a modification to the theory of gravity.
If you take the last approach, you end up with a theory called AQUAL, an amusing acronym that stands for A QUAdratic Lagrangian. A Lagrangian is just a mechanism in physics to create theories that respect conservation laws like energy and momentum. Thus, unlike MOND, which breaks conservation of energy, AQUAL respects it, so it can explain far more.
AQUAL can explain why a star can have Newtonian dynamics internally while it behaves in a galaxy with MONDian dynamics.
AQUAL predicts the external field effect. This effect was first confirmed in 2020 a significant blow to DM theories. In the EFE, a weak acceleration cluster subject to a high acceleration external field has near Newtonian rather than MONDian dynamics.
AQUAL can also explain coupled systems of galaxies like binary galaxies where one orbits the other or colliding systems such as the Bullet Cluster.
AQUAL is a step in the right direction but there is a lot more to gravity. Indeed, the force in Newton’s universal law of gravitation comes from a mathematical object in Einstein’s theory of general relativity called a Christoffel symbol. It turns out that Christoffel symbols, however, are tricky. You can always choose a coordinate system where they vanish. This is the coordinate system that is freely falling, for example, or traveling with an orbiting world.
We on the Earth observe a gravitation force on objects causing them to fall, as Sir Isaac Newton did when he saw an apple fall from a tree. Yet, from Einstein’s interpretation, that force is simply a point of view. For the apple, there is no gravitational force at all. Likewise, from the perspective of an astronaut on the International Space Station, there is no force on her, and the vanishing of the Christoffel symbol bears that out in Einsteinian physics. That is why she appears to be floating despite being close to a massive body, the Earth.
In MONDian dynamics, that won’t work at all. We actually need a force that doesn’t vanish with changes in coordinate system since that force depends on the strength of acceleration. Therefore, we need some way to determine, in absolute terms, how strong acceleration is. In other words, we need to violate one of Einstein’s cardinal rules: the equivalence principle.
Einstein conceived of his equivalence principle long before he came up with his theory of gravity. In a thought experiment, he envisioned a person in a box being pulled at a steady acceleration upward. Would they know that the force on them was not gravity? He answered no and from that derived a theory where force was relative to your point of view.
Later theorists took his ideas much further than even he wanted and applied them to all gravitational regimes, not just the uniformly accelerating ones. The mathematics bore this out. You could always eliminate the force of gravity, the Christoffel symbol, with a change of coordinates.
This is why to make AQUAL fit with Einstein’s theory the force of gravity cannot be exactly the same as Einstein’s and likewise the gravitational potential, the field that creates the force, cannot be exactly the same either. To make that change, another field is introduced that produces a force that does not change with coordinate system, a “scalar” field. This field produces the MOND modification to the Newtonian force emanating from Einstein’s own field. In other words, there is more than one gravitational field at work here.
There are problems with this model. One is that it doesn’t explain the gravitational deflection of light by galaxies. This is, perhaps, one reason why the DM theory is popular. It actually does appear as if there is more mass there and gravitational deflection of light is one way to measure it. The above theory fails to reproduce that and there is no real hope of recovering it.
The solution is to introduce another gravitational field, called a vector potential. A vector potential is what describes electromagnetism, so there is precedent here.
With a vector potential, it turns out that you can reproduce gravitational lensing without DM! Unfortunately, the first attempts to construct such a theory were not truly relativistic and so over the years DM proponents became even more confident that MOND and its generalizations could never work.
In 2004, Jacob Bekenstein introduced his Tensor-Vector-Scalar theory (TeVeS) that was truly relativistic and put such complaints to rest. But problems still exist.
For example, it is unclear that TeVeS can reproduce all Solar system dynamics. Its reduction to a Newtonian theory happens slowly enough that Kepler’s laws may be violated to a measureable degree. Moreover, it does not agree precisely with first order deviations of general relativity from Newton’s laws, so-called post-Newtonian parameterized gravity. While it squeaks by within the margins of error, any day it could be ruled out.
One of the problems with MOND, TeVes, and similar theories is that they are ad hoc and empirical, much like attempts to explain problems with the orbit of the planet Mercury. Dark Matter, meanwhile, is equally ad hoc since particle physics explanations for it have so far failed. In other words, we don’t have an elegant, overarching theory that explains either Dark Matter or MOND/TeVeS. Will some Einstein come along and provide the answer?
Update: In 2018, the first detection of gravitational waves detected at LIGO arriving simultaneously with bursts of light were analyzed. It turned out that gravity waves and light arrived at the same time. This turned out to be a major blow to “dark matter emulator” theories like TeVeS. These model gravity waves and light waves behaving in a fundamentally different way and predict that gravity waves would arrive hundreds of days before light over tens of millions of parsecs (a phenomenon called Shapiro delay). This discovery does not necessarily impact MOND or other generalizations of it that do not model the bending of light waves differently than gravity waves like TeVeS (and Moffat’s Scalar-Vector-Tensor theory). It may be, however, that the nail in TeVeS’s coffin has already been hammered.
Bekenstein, Jacob D. “Tensor–vector–scalar-modified gravity: from small scale to cosmology.” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369.1957 (2011): 5003–5017.
Angus, Garry W., Benoit Famaey, and HongSheng Zhao. “Can MOND take a bullet? Analytical comparisons of three versions of MOND beyond spherical symmetry.” Monthly Notices of the Royal Astronomical Society 371.1 (2006): 138–146.
Chae, Kyu-Hyun, et al. “Testing the Strong Equivalence Principle: Detection of the External Field Effect in Rotationally Supported Galaxies.” The Astrophysical Journal 904.1 (2020): 51.
Boran, Sibel, et al. “GW170817 falsifies dark matter emulators.” Physical Review D 97.4 (2018): 041501.