Mass: Inertial = Passive Gravitational = Active Gravitational

Maybe, it is now time for Newton to return? Maybe I was wrong? In any case: Newton forgive me!

The basic model of Neo-Newtonian Cosmology takes the following form in terms of gravitational potential $\phi$ as primordial, mass density $\rho$, momentum $m$ and $u=\frac{m}{\rho}$ material velocity all depending on a Euclidean space coordinate $x$ and time coordinate $t$ with the dot on top representing differentiation with respect to time:

• $\rho = \Delta\phi$                                    (inverse square law in differential form)
• $\dot\rho +\nabla\cdot m=0$                       (conservation of mass)
• $\dot m +\nabla\cdot (um)+\rho\nabla\phi=0$    (conservation of momentum: Newton's 2nd Law)  

A basic property of this model is conservation of energy in the sense that total energy $KE + GE$  with $KE$ total kinetic energy and $GE$ total gravitational energy, does not change over time: 

• $\frac{d}{dt}(KE+GE) = 0$        (conservation of total energy)

with 

• $KE = \int \rho\frac{u^2}{2} dx$
• $GE = -\frac{1}{2}\int\vert\nabla\phi\vert^2dx$. 

In this model we have identified 

• inertial mass = passive gravitational mass                        (Galileo's Equivalence Principle EP)
• passive gravitational mass = active gravitational mass     (Newton's EP)

with 

• inertial mass = mass $\rho$ in $\rho\frac{u^2}{2}$
• passive gravitational mass = mass $\rho$ in $\rho\nabla\phi$
• active gravitational mass = mass $\rho$ in $\Delta\phi =\rho$. 

In other words, the model has only one form of mass appearing in three different contexts with equality expressed as Galileo's EP and Newton's EP as fundamental features of Newtonian mechanics in the form of Neo-Newtonian Cosmology.

We recall that Einstein used Galileo's EP as a Postulate for his General Relativity (probably also with less exposure Newton's EP). 

Leibniz' Principle of Sufficient Reason shows that Galileo's EP (universality of free fall) cannot be false since sufficient reason is lacking, while violation of Newton's EP would violate conservation of total energy. 

Conclusion1: The basic model of Neo-Newtonian is simple, universal and computable, and must be fundamentally correct. There is no sufficient reason to replace it by General Relativity, which is the opposite of both simple and computable. Recall that a main dilemma of modern physics is the apparent incompatibility/contradiction between Einstein's relativity and Schrödinger's quantum mechanics, while Newton and Schrödinger are fully compatible.

Conclusion2: Kinetic energy KE can increase from decreasing GE from increasing $\vert\nabla\phi\vert^2$ from concentration of mass as the engine of gravitational collapse, which gives an endless source of kinetic energy. 

Question: Neo-Newtonian Cosmology and Einstein's Cosmology/General Relativity both start from Galileo's EP + Newton's EP. Does it mean that Einstein agrees with Newton? Probably not, since the basic idea of modern physics is that Einstein does not. The difference appears to be that Einstein demands Lorentz invariance as his incarnation of a Principle of Relativity, while Newton is happy with Galilean invariance. So we have Einstein-Lorentz against Newton-Galileo. What is your bet? Newton-Galileo reigned from 1600 to 1955, while Einstein-Lorentz took over in the 1970s along with the decline of modern physics into our days. Time for Renaissance of Newton-Galileo?

PS1 Recall that the Lorentz transformation lacks physical meaning and so must Lorentz invariance, while Galilean invariance expresses real Newtonian physics. 

PS2 Recall that the the above model naturally suggests low intensity distributed $\rho =\Delta\phi\ge 0$ as dark matter and naturally extends to $\rho =\Delta\phi$ of variable sign as origin of dark energy.