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Newton's inverse square law appears in General relativity for small masses . Since MOND uses 1/radius instead of 1/radius squared has General relativity been modified to accommodate this?

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To incorporate MOND changes do need to be made to the theory of relativity. This is because at small accelerations and speeds, the theory of relativity approximates to normal Newtonian mechanics. Indeed one of the successes for relativity is that at low speeds and accelerations, you can use Newtonian mechanics to get a very good approximation to relativity, thus you can, for example, find the path of a spacecraft in the curved spacetime of the solar system to a very high degree of accuracy by doing Newtonian calculations.

Unfortunately, it isn't at all clear how or why to modify General Relativity to give MOND. The general idea is that at very low accelerations (far lower than anything we get in the solar system, for example) Gravity should be 1/r not 1/r2. But gravity still has to be 1/r2 at everyday accelerations (since this is the rule that the planets obey). It is possible to "patch" relativistic mechanics so that it gives the 1/r rule for low accelerations, but the patching is completely arbitrary. (and if done in a naïve way it predicts that gravitational waves will propagate at a different speed to light)

This is a problem. Compare with the anomalous rotation of the perihelion of Mercury. It was shown that it could be explained if the law of gravity was actually 1/r2.00001 (or something, I forget the details) But this patching of Newtonian mechanics is not based on any theory, just fitting a formula to data. Instead, General relativity account entirely for the anomalous rotation, and it does so in a geometrically elegant way. General Relativity makes sense. It may not be intuitive, but it is worked out logically, not by curve-fitting.

In this sense, MOND is more like Kepler's laws. It does fit the data, but there is no good reason why gravity should behave like MOND. Newton's laws explain and justify Kepler's laws. We can say why a planet should orbit in an ellipse in terms of a more basic theory of Gravity. And this theory of gravity makes specific predictions that can be tested on Earth. We need something like that for MOND - something that explains why and isn't just an arbitrary patch applied to General Relativity

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No, because General Relativity is well-tested (especially on solar system scales) and there is no indication it is incorrect in any way. If anything, it is MOND that needs to be tuned such that it yields general relativistic effects, like how the speed of everything we've seen cannot exceed the speed of light, or how light is bent by mass.

That said, you might want to check out TeVeS, which last I saw is the leading relativistic generalization of MOND.

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  • $\begingroup$ MOND would be impressive if it yielded quantum gravity or even hinted at how to get quantum gravity done. $\endgroup$
    – user50623
    Apr 25, 2023 at 12:05
  • $\begingroup$ The link says that a TeVes star is unstable after 2 weeks under Beckensteins proposals.Not very promising. $\endgroup$
    – user50623
    Apr 25, 2023 at 12:13

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