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Let me provide some insight, I was studiying about General Relativity, and I read that it "encapsulates" Newtonian Gravity. I understand this perfectly, because, Einstein used lots of ideas form Newton's Laws of Motion for this theory. The thing is, if it "encapsulates" Newtons Laws, suchs as: $$ F = m\vec{a} $$ $$ F = G \frac{Mm}{r^2}\hat{r} $$ $$ \nabla \cdot g = -4 \pi G \rho $$

Yes, you have this: $$ \frac{d^2 \vec{x}}{d \tau^2} = -\large \Gamma_{\normalsize \mu \nu}^{\normalsize \alpha} \normalsize \frac{d\vec{x^{\mu}}}{d \tau} \frac{d\vec{x^{ \nu}}}{d \tau} $$

But this equations takes in accounts Relativistic Effects, like time dilatation, redshift, space contraction... and also is common for all Space-Time metrics such us the Minkowski Metric, the Schwarzschild Metric among others.

But would it be possible to derive the corrected Newton's Laws (the same way you get Maxwell's Equations from Quantum Mechanics) if you don't take in account Relaivistic effects, like time dilatation and space contration when you travel at speeds near $c$, or the rotation of a body (like in the Kerr Metric), etc. This would correct Newton's Laws and also give more precise descriptions of orbits and movements in which gravity is involved (like with the preccesion of Mercury's Orbit) without the needs of using complex maths like differential Geometry or Tensor Calculus.

Is this possible? If possible, how you derive the corrected Laws?

Correct me if I'm wrong.

EDIT: I'm not looking for obtaining the exact same equations form Newton's Laws of gravity with General Relativity. I'm asking for a way of deriving the Newton's Laws which will explain things like the precession of Mercury but at the same time you are not travelling near $c$ so you don't have to bother about Special Relativity effects; only General Relativity ones.

Álvaro Rodrigo
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    If you are looking for deriving Newton's law of gravity from Einstein equation, John Rennie's answer in this post https://physics.stackexchange.com/q/222390/ has what you are looking for. – Kksen Jul 27 '21 at 15:32
  • Deriving Newtonian gravity from Einstein equation is a nice exercise that can be found in many GR textbooks. It is quite amazing to see that in the classical limit the geodesic equations are identical to newton equations in the presence of a gravitational potential. – Davide Dal Bosco Jul 27 '21 at 16:05
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    @DavideDalBosco Yet not all that surprising. The fictional potential has to be proportional to the affected mass by the equivalence principle, and hence also to the source mass to honour Newton's third law, and now dimensional analysis says it will scale as $-c^4\kappa m_1m_2/r$. (The minor miracle is that $\kappa$ in $G_{ab}=\kappa T_{ab}$ has the right dimension to get a $1/r$ potential.) Then we just define $G$ in terms of $\kappa$ to agree with Newtonian physics. – J.G. Jul 27 '21 at 16:13
  • Possible duplicates: https://physics.stackexchange.com/q/211930/2451 and links therein. – Qmechanic Jul 27 '21 at 16:35
  • I guess the OP is asking if it is possible to obtain corrected form(s) of Newton's law of gravity, which (i) doesn't take into account any relativistic effects, but (ii) are good enough to explain precession of Mercury. Corrections from geodesic equations in GR are due to relativistic effects, so that is not what OP is looking for. – KP99 Jul 27 '21 at 18:00

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