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