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General relativity may induce the so-called post-Newtonian correction to the inverse square law of gravity. For details, please refer to chapter 9 of Weinberg's Gravitation and Cosmology.

However, there are many research papers about testing the inverse square law on small scale (millimeter range). For example, the following two papers.

What are they testing? Where is the correction due to general relativity?

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    $\begingroup$ Corrections due to general relativity are generally of the order $GM/r c^2$, where $M$ is the mass of the object creating the field and $r$ is the length scale of the problem. I don't have access to the papers you've linked, but I would guess that the size of the GR-based corrections was much smaller than the accuracy of the experiment. $\endgroup$ Dec 3 '20 at 12:41
  • $\begingroup$ @MichaelSeifert I have edited the question and added a link to the preprint of the paper. Thanks. $\endgroup$
    – lewton
    Dec 4 '20 at 6:29
  • $\begingroup$ I think the motivation is to test whether GR is correct, and the mm scale is chosen because it is the one where existing observations have the least precision. At least, this is the motivation of some of the work. $\endgroup$ Dec 4 '20 at 9:10
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Newton acknowledged the $1/r^2$ behaviour his laws expected of point masses and outside spherically symmetric densities would experience $1/r^3$ corrections (which primarily matter at short distances) due to factors such as the central mass being spheroid, and he showed this still leads to elliptical orbits; it just makes them process. Prior to general relativity, we knew of several complications, including gravity quadrupoles. All these do is change the overall $1/r^3$ coefficient, and general relativity changes it again, but the only planet in our Solar System for which this mattered observationally in 1915 is Mercury.

Modern tests are instead concerned with very long-range corrections to $1/r^2$. The outermost stars in galaxies haven't been flung out by their own centripetal acceleration, but the known amount of gravity from baryonic matter can't explain that. While dark matter is the preferred explanation, another is that gravity is surprisingly strong on that length scale.

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