Newton's Law of Gravitation, Gauss Law and GR From One of My Unpublished  Papers
$$\frac{d^2 x^{\alpha}}{d\tau^2}=-\Gamma^{\alpha}_{\beta \gamma}\frac{dx^{\beta}}{d\tau}\frac{dx^{\gamma}}{d\tau} \tag{1}$$ 
For radial motion in Schwarzschild’s Geometry we have,
$$\frac{d^2 r}{d \tau^2}=-\frac{M}{r^2}\left(1-\frac{2M}{r}\right)\left(\frac{dt}{d\tau}\right)^2+\frac{M}{r^2}\left(1-\frac{2M}{r}\right)^{-1}\left(\frac{dr}{d\tau}\right)^2\tag{2}$$
Again from radial motion, we have from Schwarzchild’s metric:
$$d\tau^2=\left(1-\frac{2M}{r}\right)dt^2-\left(1-\frac{2M}{r}\right)^{-1}dr^2\tag{3}$$
Dividing both sides of (3) by $d\tau^2$ we have,
$$1=\left(1-\frac{2M}{r}\right)\left(\frac{dt}{d\tau}\right)^2-\left(1-\frac{2M}{r}\right)^{-1}\left(\frac{dr}{d\tau}\right)^2\tag{4}$$
Using relation (4) in (2), after factoring out $M/r^2$ from the RHS of (2), we obtain:
$$ \frac{d^2 r}{d \tau^2}=-\frac{M}{r^2}\tag{5}$$
The inverse square law should hold accurately if proper time is used. Here $r$ represents the coordinate distance along the radius.
One may use the relations:
$$M ~\rightarrow~ GM/c^2\qquad\text{and}\qquad\tau ~\rightarrow~ c\tau,$$
to obtain the exact "form" of the law of Gravitation.
Query: Is equation (5) indicative of the fact that Gauss law may be used in the same classical "form" in GR?
[We may introduce a symbol $F=m\frac{d^2 r}{d\tau^2}$.]
 A: Assuming the initial direction of the considered geodesic dr/d{tau}=0 (initial velocity equal to zero), you could calculate the value of d^2r/dt^2 (curvature of geodesic projected on rt-plane) directly from equation (2). Consider however that this (projected geodesic) curvature regards the map coordinates (the acceleration measured by the observer situated very far! i.e. in free, flat space). In equation (5) you have r, which corresponds to the distance in the map (NOT LOCAL), and tau - which is connected to the local time (entities of different spaces!). So, this relation doesn't make sense (in my opinion). However, I like very much the idea to involve Gauss law in GR!!!
A: This is all fine, but in local space also the unit of distance (let's call it "rho") is different from "r" of reference coordinates - and not just time. Of course, considering rho as the distance from the origin would not make sense (at r_s it would tend to infinity), but using its first and second derivatives will be correct!
(I mean d^2 rho/dtau^2 - for local acceleration, and drho/dtau - for the initial velocity or component of the initial 'geodesic line direction'.
Using mixed units, i.e. "dr" - in reference coordinates (in flat-space map) over "dtau" - in local time; is a common but misleading misunderstanding!!! From a logical point of view, it makes no sense!!!
It is easier and more natural to consider (and calculate) gravitational acceleration entirely in reference coordinates, i.e. in flat-space, as d^2r/dt^2!!!
Unfortunately, this leads to the conclusion that there is still something wrong with the Schwarzschild coefficients... if you calculate the REAL LOCAL acceleration, you will see that it does not coincide with Gauss' law. Please try to do it!!!
