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.
