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I've read previous posts on this topic but I am still unclear. I am not able to access the mathematics of GR but I would like to know what is it about the solution to the Einstein field equation that accounts for the additional precession that is observed in Mercury's orbit that Newton's gravitation could not predict?

I am aware the the majority of the precession is caused by the other planets' influence on Mercury but I am interested in the 'extra' bit that GR predicted. Besides GR being about warpage of space-time, is it simply that the solution to the EFE has an additional 2GM/c^2r^3 term in it and so as Mercury is close to the Sun this term has a noticeable effect on the precession? And therefore GR gives a slightly higher field strength for objects than Newtonian gravity?

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  • $\begingroup$ It's not that the field strength is greater. It's that general relativity predicts a deviation from the inverse square law of Newtonian gravity. Perfectly periodic elliptical orbits depend on the force being perfectly proportional to the inverse square of the distance. When there's a slight deviation from inverse square, the result is that the orbit resembles a precessing ellipse. $\endgroup$
    – S. McGrew
    Commented Apr 7, 2020 at 15:37
  • $\begingroup$ thanks for the reply $\endgroup$
    – tomd7824
    Commented Apr 7, 2020 at 20:50

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The orbital equation found from Newton's law is

$$ {d^2u \over d\phi^2} +u = -p^{-1} = \mathrm{const}$$

where $u=1/r$. From the geodesic equation in general relativity we obtain

$$ {d^2u \over d\phi^2} +u - 3 \mu u^2 = -p^{-1}$$

where $\mu=GM$ is the standard gravitational parameter for the solar system ($M$ is mass of Sun). The two equations differ only in a small term, from which Einstein calculated precession as a perturbation of the Newtonian orbit.

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  • $\begingroup$ thanks for taking the time to reply, that clearly shows the 'extra bit' i was after. does this term mean that Einstein's warpage of space-time has a greater effect on the motion of objects than Newton's gravitation? And is this term only noticeable when looking at Mercury (as opposed to all the other planets) because it is so close to the sun? $\endgroup$
    – tomd7824
    Commented Apr 7, 2020 at 20:35
  • $\begingroup$ I don't think I could compare which has most effect on motion in anything other than an arbitrary way. It is most noticeable when looking at Mercury, but it is also measured for Venus, the Earth and the asteroid Icarus (possibly others) and for pulsars, which precess a few degrees per year over millions of orbits. $\endgroup$
    – Charles Francis
    Commented Apr 7, 2020 at 20:40
  • $\begingroup$ that is clear, thanks again for taking the time to reply $\endgroup$
    – tomd7824
    Commented Apr 7, 2020 at 20:50

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