Universality of Newton's Law of Gravitation

Newtonian Gravitation and Quantum Mechanics is a Theory of Everything ToE.

This is a continuation of previous post on the connection between gravitational potential $\phi (x)$ and mass density $\rho (x)$ through the relation $\rho =\Delta\phi$ where $\Delta$ is the Laplacian differential operator with respect to $x$ as Euclidean space coordinate. 

We take here a non-standard view and consider the gravitational potential $\phi$ to be primordial, from which mass density $\rho =\Delta\phi$ is given to matter by differentiation as an instant local action. 

This is to be compared with the standard view with $\rho$ primordial and $\phi$ obtained by solving the equation $\Delta\phi =\rho$ as a global integration/summation process requiring instant global action or instant action at distance.

Here instant local action, like establishment of contact forces, does not pose the problem of instant action at distance, which may be unsolvable. 

Let us now see how the connection $\rho =\Delta\phi$ can be established viewing $\phi$ as primordial. 

The first step is to see that $F=\nabla\phi$ represents gravitational force field and as such is a conservative force, which means the the work required to move a body from one point to another is independent of the path. As an example the work required to lift a body in the gravitational force field of the Earth only depends on change of level and not path with motion without change of level not requiring work. The gravitational force field is thus given as the gradient of a potential.

The next step is to show that $\nabla\cdot F=\rho$, where $\nabla\cdot F$ is the divergence of $F$. We recall from Calculus  that 

• $\int_\Omega \nabla\cdot F\, dx =\int_\Gamma F\cdot n\, ds$

for a volume $\Omega$ with boundary $\Gamma$ with outward unit normal $n$ with $dx$ volume element and $ds$ boundary surface element.  Viewing $F$ as a flux, $F\cdot n$ is the flux out of $\Omega$ and $\nabla\cdot F$ is the corresponding source inside $\Omega$. 

We can now (simply) define gravitational mass density $\rho =\nabla\cdot F=\nabla\cdot\nabla\phi =\Delta\phi$ to be the source of the gravitational field given by the potential $\phi$, and we can from this relation derive Newton's Law of gravitation in its familiar form as did Laplace in his Celestial Mechanics. 

We see that Newton's Law of gravitation reflects (i) observation that a gravitational force field is conservative, together with (ii) definition of gravitational mass density as the source of this field. 

Newton's Law of Gravitation thus comes out from assuming that a gravitational force field is conservative, which is supported by observation, together with definition of gravitational mass. 

Connecting to Newton's 2nd Law of motion with inertial mass the same as gravitational mass, brings universality to cosmic interaction.

We conclude that Newton's Law of Gravitation can be seen much like an a priori statement about physics in the sense of Kant. We compare with the a priori statement that the length of the circumference of a circle with radius $1$ is equal to $2\pi$, which has universal validity. 

This gives Newton's Law of Gravitation universal validity leaving Einstein's General Theory of Relativity without mission. In particular, it resolves the main dilemma of modern physics of the perceived incompatibility between Quantum Mechanics and General Relativity, since Newtonian gravitation is fully compatible with Schrödinger's quantum mechanics as a Theory of Everything ToE.  

We recall Niels Bohr:

• How wonderful that we have met with a paradox. Now we have some hope of making progress.          

How wonderful that we have found incompatibility of Quantum Mechanics and General Realativity. Now we can make some progress...