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Space is expanding, and this effect can be only detected at large scales, on the scale of galaxies and larger.

If space is expanding, the expression $1/r^2$ for the law of gravity will be modified. How would this occur? I am phrasing the question as simply as possible on purpose.

In detail, the first part of the question is whether expansion really does influence $1/r^2$ at large distances. If so, the second part of the question is how this happens. Or, asked more specifically: can a simple combination of $1/r^2$ and expansion lead to MOND - modified Newtonian dynamics?

A more extreme question: if two masses are really distant, say 1% or 10% of the Hubble Radius, or even more, does $1/r^2$ still hold?

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Does it happen? Sure, but the effect is small.

How does it happen? I find it easiest to think of it as gravity propagating with the speed of light through space that gets a bit longer as it travels along. As gravity propagates at the speed of light c, space only expands with the Hubble speed, which is $2\times10^{-4}$ of the speed of light (It takes gravity 3 light years to travel across 1MPc, and over those 3 years, this space expanded by 70km/s). So the $1/r^2$ law gets corrected by a small factor $2\times10^{-4}$.

No, this does not give you MOND. No, the effect doesn't get stronger with larger distance, it's always the ratio of the speeds.

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  • $\begingroup$ So expansion weakens Newtonian gravity, whereas MOND increases it - correct? $\endgroup$
    – Gina Martelli
    Commented May 15, 2020 at 14:31
  • $\begingroup$ Yes...ish. The trouble is which "r" you would plug in your Newtonian equation to begin with. One could argue to plug in that distance that includes the Hubble expansion, and thus recover the 1/r^2 law exactly. No weakening. Or you could plug in some instantaneous r, in which case the actual gravity would be weaker. Both is fine, you just need to keep track of which way you're running the calculations. $\endgroup$
    – rfl
    Commented May 15, 2020 at 14:36
  • $\begingroup$ Thank you! To me it seems incredible that MOND should increase gravity. That gravity is "diluted away" at large distances seems ok; but that it increases seems odd. $\endgroup$
    – Gina Martelli
    Commented May 15, 2020 at 17:14
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Space is not expanding on the scale of galaxies. It is not even expanding on the scale of galaxy clusters. Consequently, expansion is not in any way related to MOND, which is a hypothesis applied on the scale of galaxies. This is a difference between general relativity and Newtonian gravity which only makes sense globally. It is a change to the structure of space. Newton's infinite absolute space and absolute time has been replaced by spacetime.

Before putting any credence to MOND one should look at the actual evidence of rotation curves, not simplified evidence in so many popular accounts. See this review by Sofue & Rubin, 2001, Rotation Curves of Spiral Galaxies, Ann.Rev.Astron.Astrophys 39, 137-174.

Flatness is only a general pattern, not a law as would be predicted by MOND. About half of rotation curves have abnormalities. Some fall off even faster than Newtonian gravity. This cannot be explained by MOND or dark matter. The curves mainly show motions of gas outside the galaxy. There is no good reason to treat intergalactic gas as part of galactic structure. This is ignored in almost all studies. I have a chapter on it in my book Structures of the Sky.

Galaxy Rotation Curves

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  • $\begingroup$ 7 billion years ago, the universe was half the present size. Does this have no effect on galaxy clusters, which are about 1 million light years in diameter? Naively, there is only a factor 7000 in size? $\endgroup$
    – Gina Martelli
    Commented May 14, 2020 at 6:22
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    $\begingroup$ There is no effect of expansion on gravitationally bound objects. Galaxy clusters are generally regarded as gravitationally bound, whereas superclusters are not. $\endgroup$
    – Charles Francis
    Commented May 14, 2020 at 6:36
  • $\begingroup$ How is this lack of effect known? Is there a simple way to get convinced that there is no such effect of expansion? After all, even the ideas linking MOND to de Sitter space properties (Verlinde, Smolin) can be seen as relating MOND to expansion. I just want to understand what is fact and what is wishful thinking. Please understand my asking for further detail. $\endgroup$
    – Gina Martelli
    Commented May 14, 2020 at 10:22
  • $\begingroup$ It is known locally by observation of near galaxies, and it is known because it is the clear prediction of general relativity. $\endgroup$
    – Charles Francis
    Commented May 14, 2020 at 11:09
  • $\begingroup$ I added a question to the topic. Milgrom says that probably, $a_0 \approx c^2/R_H$, which makes MOND related to the Hubble radius, and thus to expansion. Milgrom also says that his $a_0$ acceleration scale is related to the cosmological constant. Smolin says something similar. Maybe I misunderstand what you mean when you say that "expansion is not in any way related to MOND". Can you specify the statement? $\endgroup$
    – Gina Martelli
    Commented May 14, 2020 at 12:29

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