[part 1]

> ELI ... my understanding

I don't know what level to pitch an ELI at; below is my best guess, and it's probably uneven. I did't want to aim for a reader who is perfectly capable of picking up a textbook and reading the academic literature starting with Famaey & McGaugh 2012. I also don't want to aim so low that it's basically slogans; there are enough of those in forums like this, and they're often inconsistent or outright wrong. I'm also conscious that we are not the only two participants in this particular discussion.

I'm not sure what I can do to improve your understanding and I'm not about to examine you to assess it; as I said there are textbook treatments (any decent galaxy dynamics or galactic astronomy text updated in the last twenty years will touch on the empirical MOND relation, and the standard grad textbook by Binney & Tremaine goes quite deep on MOND) and more cutting-edge literature that has developed since the 1980s. Likewise, I'm not here to help you choose between MOND or dark matter as ways of resolving observed acceleration anomalies, only to try to describe the two approaches and offer some contrasts to the extent that these very different approaches can be contrasted (IIRC McGaugh, of triton station at the top of the discussion here, calls this the theoretical "incommensurateness problem"). There are plenty of blogs by partisans.

I also don't do galactic dynamics; it has little appeal for me. However, MOND vs Newton/Einstein is a debate that arises almost exclusively because of galactic dynamics. I do do weak field General Relativity, which is rarely used in galaxy astrophysics since more tractable approximations are available, and more still are forthcoming. That colours some of what's below, as it is those approximations which are compared with MOND (which one will want to think of as an approximation of some yet unknown alternative to General Relativity).

> naturally

Don't get hung up on that; nature is what selects the distribution. Nature also selects the distribution of stars in our sky, the particles in the standard model zoo, their masses, spins, couplings, dimensionless constants, and so forth. It's not human scientists who decided to make Mizar a double double-star system with their particular orbits. Newtonian gravitation, invented by humans, can describe those orbits well enough, but you must first measure some initial positions and velocities for those four bodies. Heck, only recently (about 1997) did we discover (through such measurement) that Mizar is a four-body system!

(In fact, Mizar is possibly a test of one aspect of MOND as an adaptation of Newtonian gravitation: the External Field Effect ought to produce marginally different orbits compared to Newtonian Kepler orbits because of the system's immersion in the Galaxy's MOND field. General Relativity agrees completely with Newtonian gravitation because of the former's strong equivalence principle and because dark matter, if it exists, is such a small part of the local stress-energy tensor around Mizar that its generation of curvature is negligible. Maybe Mizar ends up as evidence in favour of MOND-like post-Newtonian gravitation because of the external field effect, who knows?)

We don't choose the matter-energy distribution of the moon, but as we've gotten better at measuring it we can now describe very precisely its orbit millions of years in the future, as long as nobody blows it up or feeds Earth to a giant space goat.

> it's all about the distribution of DM

It's all about the initial position and velocity of every teensy mote of matter however you want to think about matter (e.g. as excitations in a dozen or more quantum matter fields, or as individual interacting particles or whatever). That's the case in every one of these classical field theories be it Newton (in the Newton-Poisson sense), modified Newton (ditto, e.g. Bekenstein & Milgrom 1984), or Einstein. We have no hope of specifying all that data much less evolving every mote of it at every tiny step, so we must coarse-grain the initial data. A dark matter halo is a coarse graining of some very fine properties of matter that we do not understand yet and haven't been directly detected much less produced in a laboratory. MOND is a coarse graining of some very low-energy dynamics that somehow arises from matter as we know it today; no laboratory in our solar system is able even in principle to produce orbital accelerations low enough or decoupled enough from the Milky Way's external field to demonstrate those dynamics -- we would need a laboratory at the edge of our galaxy.

We haven't found any hypothesized candidate for quantum-field dark matter ("particle dark matter") yet. But we also haven't found the generator of the dynamics in any candidate relativistic MOND theory either. The failure for one is in no way a success for the other.

We need a relativistic version of MOND because relativistic gravitational effects are readily apparent in gravitational lensing (they are feasible to just see with off-the-shelf telescopes, although professional telescopes and Hubble and Webb can see more of them and in better detail), gravitational radiation (which for now requires a LIGO/Virgo/KAGRA type laboratory or arrays of radiotelescopes and long observation times), the decays of inspiralling orbits, the formation of stellar remnants (or some alternative explanation for pulsars), the spectrum of quasars and blazars, the expansion history of the universe (the Lyman-alpha forest and the cosmic microwave background) and so forth.

> DM theories have to get the distribution ([and dynamics]) right

yes.

But equivalently

> Modified gravity theories need to get the (distribution) and dynamics right

The distribution in the latter case is that of the luminous matter and its self-interactions and couplings with the extra gravitational fields.

In General Relativity there is only one field to which all matter couples identically. That leads to things like the universality of free-fall (cannon balls, feathers, and entire moons free-fall the same way in empty space), the equivalence between uniform acceleration and a uniform gravitational field, the equality between passive and active gravitational masses and the inertial mass, and the Einstein equivalence principle which holds (simplifying somewhat) that you can repeat the same non-gravitational experiment at different times, at different locations, and with different orientations and get the same results. There are good mathematical proofs that afaik nobody has seriously challenged which demonstrate that General Relativity is the unique theory that produces these equivalences. Any other theory will deviate in one or more of them. Relativistic MOND, quite generally, breaks the strong equivalence principle: the gravitational motion of a star within a galaxy depends on how deep in the galaxy it is, rather than on its initial position and velocity. We have excellent evidence of the strong equivalence principle from pulsars in multi-star systems in our galaxy, so this is a puzzle for MOND.

> my understanding

The field equations of Newtonian gravitation has a relativization in the form of the Einstein Field Equations of General Relativity.

The Modified Newtonian field equations of MOND et al do not have such a relativization.

Relativistic effects are mostly negligible in galaxy dynamics, so because of insufficient evidence one has for now a free choice of describing galaxy dynamics with Newton where one modifies the dynamics ("MOND") or with Newton where one modifies the distribution (initial positions, velocities) of matter ("Dark matter").

If relativistic effects are non-negligible -- for instance, if one is interested in how merging galaxies' central black holes evolve -- then neither adaptation of Newton is suitable, you need a relativistic theory. Newton + DM -> GR + DM is the only generally useful theory today.

Moreover, because we know more about mathematics than Einstein did in the early 20th century, we can justify approximations of General Relativity such as linearized gravity. These are one-way justifications: you can pick out a simplification of General Relativity and prove it has compatible dynamics in some limit, but you can't start with some approximation and derive General Relativity. You likewise can't derive a relativistic MOND theory from the non-relativistic MOND -- the latter maps to a vast number of possible relativistic theories, all with practical differences. And there is no relativistic theory from which you can extract MOND as a weak field approximation.

Given that and the non-relativstic-limit free choice of theory (DM has neither been proven nor disproven), if one wants to solve for anomalous accelerations (which are pretty obvious) in galaxy dynamics, one can choose MOND or Newton + DM.

Meanwhile, people wonder about the nature of the anomalous acceleration. Neither MOND nor approximations of General Relativity get it right in all respects, and there isn't a complete solution in the full-theory-of-General-Relativity-in-microscopic-detail available because it's too hard to compute in 2023, and there are aesthetic choices in how one coarse-grains for tractability.

[superfluid in sibling reply]