How is the External Field Effect in MOND conceptually distinct from Newtonian gravity and GR? I’m trying to understand the External Field Effect (EFE) in Modified Newtonian Dynamics (MOND) and how it is conceptually distinct from GR and Newtonian gravity. More specifically, the descriptions I read about the EFE from press releases confuse me with following phrases as I attempt to discern MOND from everything else. Please bear with me, I am trying to learn and I realize I don’t understand these concepts as well as I would like:
“In Newtonian physics, the inertial mass of an object is an inherent property that exists independent of anything around it. In MoND, the inertial mass depends on the gravitational mass of the object as well as the net gravitational pull from the rest of the universe. In other words, inertial mass is an emergent property rather than an inherent one.”
https://www.universetoday.com/149416/new-data-supports-the-modified-gravity-explanation-for-dark-matter-much-to-the-surprise-of-the-researchers/
My confusion:
Inertial mass is considered an emergent property in MOND. I don’t see how the inertial mass in MOND would be different than the inertial mass in Newtonian gravity. For example, in Newtonian gravity, wouldn’t we need to know the masses and positions of everything in the Universe to create a precise gravitational field? For practicality sakes, I know this is impossible and unnecessary. Ie. To first order, everything but the Sun and possible Jupiter can be ignored in our Solar System. I mention this for the sake of argument. Wouldn’t the pull from “everything in the Universe” in the Newtonian sense be the same then as the inertial mass in MOND? On the most precise level, when would we neglect mass from “the rest of the universe” in Newtonian dynamics? Again, assuming we could be this precise in Newtonian dynamics. I hope that makes sense. MOND seems to discuss things as if there is this extra field and I don’t understand what that extra field really is. Shouldn’t all fields be considered? Ie. The outskirts of galaxies experience low acceleration. Why would it be surprising/possibly controversial to consider the influence of external bodies? What am I not understanding here?

“One of the core tenets of general relativity is the strong equivalence principle, which states that the motion of stars in a galaxy should be independent of an external uniform gravitational field.”
More of my confusion:
What would this motion be independent? I don't understand how the SEP negates the concern of a gravitational field of other bodies but I also don't know what is meant by "external" here so I guess we can start there.
Other snippets for context:
“The researchers looked at 153 galaxies and measured the speed of stars within them at different distances from the galactic center. They then looked at the acceleration of each galaxy caused by the gravitational fields generated by other galaxies surrounding it. The strongly accelerated galaxies experienced ten times the acceleration of the most weakly accelerated ones."
“They then selected the two galaxies which felt the most gravitational tug from their surroundings and compared their rotational behavior to two galaxies that were isolated. They found that the outer stars of the galaxies in strong gravitational fields orbited slower than predicted by the behavior of the isolated galaxies. They also studied galaxies with intermediate external gravitational fields and found that the data was consistent with the extreme examples, with the rotational characteristics of each galaxy depending on its surroundings. Their data appears to violate the strong equivalence principle.”
https://www.forbes.com/sites/drdonlincoln/2020/12/20/study-questions-both-dark-matter-and-einsteins-theory-of-relativity/?sh=2b487d402b7a
I guess my main confusion is as follows: Isn’t there always just one gravitational field at a specific point that is dependent on ALL the masses and positions of everything in the universe (at the most precise, albeit impossible and impractical level). What is all this talk of external fields? I have been trying to parse this through on my own and got lost along the way. I appreciate conceptual explanations.
 A: OK, to explain how people come up with MOND and why it is different than Newtonian gravity, it might be best to explain where it came from.  The core idea is that it is an explanation intended to do away with dark matter.  Amongst the puzzles dark matter is intended to explain is galactic rotation curves.
The idea here is that you can look at the velocities of objects rotating around the galaxy, and using Gauss's law:
$$\int {\vec g}\cdot d{\vec A} = 4\pi G M_{inc}$$,
you can compute $g$ as a function of the distance from the galaxy, and from this, you can compute the velocity of the stars at that radius, using $v^{2}/r = g$.
So, from observing the speed with which stars orbit the center of a galaxy, and using Newtonian laws, you can estimate how much mass is inside a radius $r$.
But, the problem found is that these curves fall off far, far too slowly to be explainable in terms of the observed stars.  The common hypothesis to explain this is dark matter, but Milgrom came up with an alternative explantion:  maybe gravity needs to be modified.
In particular, he proposed a modification to the law of inertia at low acceleration, that amended Newton's second law to be $F = \mu(a/a_{0})ma$, where $\mu$ is some function.  He then showed that you could pick a plausible value for this function to explain the rotation curves, and that hadn't yet been tested.
OK, so, now this was at least not experimentally ruled out for a while, but it struggled to explain the other evidence for dark matter that was out there.  it also did not easily generalize to relativistic regimes.  In order to do this, a theory called TeVeS was worked out.  This modified general relativity by adding several dynamical fields to the theory (namely, a vector filed and a scalar field to go along with the tensor metric tensor, hence the T, V and S in TeVeS).  Then, a Lagrangian could be written down that would reduce to MOND in an appropriate limit and could be used to create relativistic generalizations of MOND to attempt to explain things like the bending of light by globular clusters.  This came at the cost of being tremendously complicated as a theory.  These are the auxiliary fields that are likely being talked about above.
Finally, just for completeness, I'd be remiss if I didn't point out that most of the active research on this came to a close with the observation of the Bullet Cluster, which was an observation of two colliding galaxies, which produced a clear seperation of the dark matter from ordinary matter, and seemed to provide relatively definitive evidence against explaining dark matter through dynamical arguments (though the complexity of TeVeS left open the idea that the auxiliary fields could be coupling to each other).
A: The key to understanding the External Field Effect is through the lens of equivalence principle. How is equivalence principle manifested in galaxy settings?
Let's take a look at an object rotating around a galaxy $G_0$. Its Newtonian acceleration is:
$$
a_{Newton} = a_{ext} + \frac{GM}{r^2},
$$
where $M$ is the mass of the galaxy in concern, and $a_{ext}$ is the acceleration from the gravity pull of neighboring galaxies. Provided that the neighboring galaxies are not too close, we can assume that $a_{ext}$ is a constant within the galaxy $G_0$.
Now let's hitchhike Einstein's elevator with acceleration $a_{ext}$. In  the free-fall (in gravity field $a_{ext}$) reference frame, the observer is moving with the center of the Galaxy $G_0$. In the eye of this free-fall observer, the object rotating around a galaxy $G_0$ is experiencing an acceleration of:
$$
a = a_{Newton} - a_{ext}= (a_{ext} + \frac{GM}{r^2}) - a_{ext}=  \frac{GM}{r^2} = a_{Newton}|_{a_{ext} = 0}.
$$
The equivalence principle is working: we don't care about the the (uniform) external acceleration, insofar as we move along with the free-fall reference frame. So far so good.
Now turn to MOND. The same object rotating around the outskirt of the galaxy $G_0$ is purportedly in the deep MOND regime, wherein the acceleration experienced by the object is:
$$
a_{MOND} = \sqrt {a_0 a_{Newton}}= \sqrt {a_0 (a_{ext} + \frac{GM}{r^2})},
$$
where $a_0$ is the MOND acceleration constant.
Can we hail a ride on Einstein's elevator to cancel out $a_{ext}$? No dice! No matter what you try (try any $a_{ref}$) the acceleration in a free fall reference frame would be:
$$
a = a_{MOND} - a_{ref}=  \sqrt {a_0 (a_{ext} + \frac{GM}{r^2})} - a_{ref} \neq a_{MOND}|_{a_{ext} = 0}.
$$
In other words, you can't get rid of the gravity effect from the neighboring galaxies ($a_{ext}$) . And this violation of equivalence principle is what was observed in the quoted paper as a falsification of the Newtonian dynamics as well as Einstein's general relativity (with or without dark matter), which lends further support to MOND.
