Sorry to bother you, but I did not get anywhere answer what exactly moves Mercury periapsis. "Sun gravity" or "GR" or "warp of spacetime" are very broad answers, I want to know how they affect this guy. As far as I learn, at first it seems that root of such behavior is speed of gravity, so when Mercury goes away from the Sun, gravity needs more time to travel and thus trajectory somewhat changes. But if so, there must be the reversed effect when Mercury moves closer. If these effects are not equal by absolute value, why?

My second thought was the "descending" path along gravity curvature is not symmetrical to "ascending". Then question is "Why?" again.

If shorter, if there is a difference in Newton's and Einstein's formulas, what's this difference means if said with common words?

• A better first question would be, "Why does Newtonian gravity predict the perihelion shift to be zero?" This is quite a special property and essentially any deviation will lead to a perihelion shift. – mmeent Sep 19 '19 at 8:52
• @mmeent The Newtonian precession of Mercury is not zero, due to the influences of the other planets, but it's half the observed precession. GR predicts the correct value. Please see physics.stackexchange.com/q/26408/123208 – PM 2Ring Sep 19 '19 at 11:14
• @PM2Ring: For sake of brevity, I ignored that. There is even a measurable Newtonian correction due to the fact that the sun is not spherical. – mmeent Sep 20 '19 at 7:03

A simple explanation comes from Bertrand's theorem. It states that the only types of central forces that result in bounded orbits that repeat their tracks (i.e., "closed") are forces that are either proportional to distance ($$F=-kr$$) or are inversely proportional to the square of distance ($$F=-k/r^2$$). While other forms of a central force can produce bounded orbits, they cannot produce closed orbits.