# About Mercury perihelion again

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. 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 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. Sep 20 '19 at 7:03

## 2 Answers

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.

The simple explanation then is that general relativity is not of either of the two forms that result in closed orbits. Non-circular orbits in general relativity cannot be closed orbits.

In the case of the solar system, where velocities are small compared to the speed of light and distances are large compared to the Schwarzschild radius, general relativity can be viewed as being of the form of small perturbations on top of Newtonian gravitation. Because the perturbations are small, the result is orbits that are close to but not quite Newtonian. The orbits are close to being ellipses, but those ellipses rotate.

• "they cannot produce closed orbits" — they actually can (e.g. hard-sphere potential and a regular-polygonal orbit), but not all of the bounded orbits will have this property of closedness in such potentials. Sep 19 '19 at 9:59
• Hi David, thanks for your answer. Ok, if these changes in orbit are perturbations, does an answer why periapsis move prograde and not retrograde exist? Sep 19 '19 at 10:04
• While other forms of a central force can produce bounded orbits, they cannot produce closed orbits. That is not quite what the theorem says. It is perfectly possible for other forms of central force to produce closed orbits. The theorem just tells us that not all bound orbits will be closed. Sep 19 '19 at 10:14

The reason for GR's contribution to Mercury's perihelion precession is nonlinearity of GR. Nonlinearity is absent in Newtonian gravity: to describe the effect of gravity on the motion of an object (say Mercury), you calculate the gravitational potential that the said object experiences as sourced by other objects in its surrounding (say the sun and other planets). Whichever way you position these objects, or add more or remove some of them, you just need to algebraically add the gravitational potentials to calculate the final effect of gravity on Mercury. This has been done for Mercury where the effect of other planets are taken into account, but the numerical value of perihelion precession doesn't match observations. This is because we missed new contributions coming from the nonlinear nature of GR: in a loose sense, the effect of gravity on Mercury is not just a plain sum of effects of gravity of each object around it. GR is a nonlinear theory which explains this missing piece. In field theory language, nonlinearity (in GR) means that gravitons self-interact. Also see my answer here on a related question.