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I have been searching everywhere for the first order GR term that describes the gravitational force of a mass.

Could someone kindly show me the term that describes this?

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    $\begingroup$ I guess, you refer to PostNewton expansion. Nevertheless I don't know what you mean exactly with first order GR-term ? What is the quantity it should be first order of ? It would be also very useful to which metric (Schwarzschild?) you refer to. $\endgroup$ Mar 6 '20 at 12:38
  • $\begingroup$ There are various links given in physics.stackexchange.com/q/814/123208 I won't vote to close this question as a duplicate of that one because most of the answers there are link-only answers, although one answer does give a brief sketch of how to calculate the precession. $\endgroup$
    – PM 2Ring
    Mar 6 '20 at 15:50
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Look up the term "precession of the perehelion of Mercury's orbit" in any general relativity text.

The term "gravitational force" is a thorny one in general relativity, and it does not map well to the formalism of the theory, but as mentioned in the comments, the Post-Newtonian expansion is the simplest way to apply that concept.

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