Newton's Gravity Law
We will now show how Newton's Universal Law of Gravitation which is based on the spherically symmetric gravitational filed of the Sun comes out of the general relativity metric. The Schwarzschild Metric is an exact solution of the gravitational field equations of General Relativity. We will see this later that this solution represents the field of a massive body that, mathematically, is represented as a point. The line element of this metric is written as follows:
(13.1)
G is the gravitational constant. M is the mass of the gravitational field source. The spatial coordinates are 3D spherical polar coordinates chosen because this type of object should produce a spherically symmetric gravitational field. Therefore we expect these coordinates to make all physical relations have the simplest representations.
Note that the above line element is approximately Minkowskian when the following condition is upheld.
(13.2) 


With this inequality in mind we write the metric in a split form as follows:



This, of course, is just the assumed form for the weak field given previously as
(13.4) 
. 

Our result in the Newtonian limit of the geodesic equation was that the 00 component of the metric was related to the Newtonian gravitational potential by the equation
(13.5) 


Reading off the value for from the above matrix split (13.3) we get that
(13.6) 


Hence
(13.7) 


In terms of the gradient equations of motion that we derived in the last section we see that
(13.8) 


In terms of the 3D position vector
(13.9) 


we get the equation of motion in the form
(13.10) 


Using the spherical coordinate version of the gradient this last equation then becomes
(13.11) 


where m is the mass of the test particle that is doing the moving in the gravitational field of the source mass M. From our derivation we see that Newton's equation is valid only when we have a stable gravitational field where and .