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> it should reduce in the Newtonian limit to something that looks like a tidal effect, i.e. a gradient of the Newtonian gravitational field.

Can you please explain to me how you went from"looks like a tidal effect in the Newtonian limit" to "a gradient of the Newtonian Graviational field"?



"Tidal effects" are defined in terms of having different gravitational fields in one place than another (i.e. the tidal bulge near to the moon occurs because the moon's field is stronger there).


That's not quite true, as illustrated by the tidal bulge opposite the moon.

Tidal forces occur much more due to the difference in the direction of gravity than due to the difference in magnitude.




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