Gravitation in Free Fall

Gravitation is in modern physics described by Einstein's General Theory of Relativity GR, and in classical physics by Newton's Mechanics NM including Newton's 2nd Law and Newton's Law of Gravitation. 

GR reduces to NM when matter/mass velocities are small compared to the speed of light, which covers planetary systems (Solar System 30 km/s) and galaxies (Milky Way 100 km/s) compared to 300.000 km/s. 

In coordinate free form NM is based on the following basic principles: 

1. Gravitational force is conservative (zero work from motion in closed loop).
2. Conservation of gravitational force (divergence of force = mass).                     (P)
3. Conservation of momentum (Newton's 2nd Law).
4. Conservation of mass. 

In Eulerian form in Euclidean space-time coordinates $(x,t)$, (P) take the following form assuming only gravitational force (free fall motion): 

1. $F =\nabla\phi$
2. $\nabla\cdot F=\rho$
3. $\dot m+\nabla\cdot (um)+\rho\nabla\phi =0$                          (PE)
4. $\dot\rho +\nabla\cdot m=0$ 

where $\phi$ is gravitational potential, $F=\nabla\phi$ gravitational force per unit mass, $m$ momentum, $\rho$ mass density and $u=\frac{m}{\rho}$ is matter velocity all depending on $(x,t)$. Different space coordinates can be used (spherical, cylindrical et cet), which transforms (PE) but not (P). 

In NM based on (P) there is no limit on the size of matter velocity. The question is if GR contributes something of importance when matter velocity is not small and GR is different from NM?

What can be wrong with NM as being based on (P)? What can be wrong with (P)? 

We compare (PE) with the presentation of GR in the book Gravitation by Misner, Wheeler and Thorne over 1250 dense pages. Is this Gravitation in free fall?

It is essential to understand that real physics cannot depend on specific coordinate systems used to express laws of physics. In particular, the physics of gravitation, that is how matter interacts with gravitational forces, cannot depend on choices of coordinate systems made by humans. The motion of the planets in our Solar system cannot be influenced by the choice of coordinate systems used by scientists to track their motion. But Einstein was obsessed with coordinate systems when setting up both the special and general theory of relativity. 

As recalled in recent posts, GR and NM both satisfy the Equivalence Principle stating that inertial mass is equal to gravitational mass and both have the same speed of light in all inertial systems.