# How to derive Newton's law of gravitation from general theory relativity [duplicate]

Newton's law of gravitation must be a special case of the general theory of relativity. How can we derive it from field equations of GTR?

If the electric field F/q is equal to E, then does the acceleration due to gravity provide the notion of the gravitational field?

In the equation $$F=GMm/r^2$$ which part is directly linked to the field?

Working with $$c=G=1$$ throughout, let's assume the simplest gravitational source, a point mass without charge or angular momentum. This gives us the Schwarzschild metric$$ds^2=fdt^2-dr^2/f-r^2d\Omega^2,\,f:=1-\frac{2m}{r}$$in the $$+---$$ signature. The geodesic deviation equation implies$$\frac{d^2x^r}{d\tau^2}=-\Gamma^r_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}.$$The low-speed limit is$$\frac{d^2r}{dt^2}=-\Gamma^r_{tt}=\frac12g_{rr}^{-1}\partial_rg_{tt}=-\frac12f\partial_rf=-\frac{mf}{r^2}.$$Many Schwarzschild radii from the source, $$f\approx1$$ so $$\ddot{r}\approx-\frac{m}{r^2}$$. By dimensional analysis, this really means $$\ddot{r}\approx-\frac{Gm}{r^2}$$, which is what Newton said.