Can Newton's Universe Model be Wrong?

Newton's Model of a Universe subject to gravitational forces takes the form 

• $\rho =\Delta\phi$                      (1): Newton's Law of Gravitation
• $\frac{Du}{Dt}+\nabla\phi = 0$         (2): Newton's 2nd Law
• $\dot\rho +\nabla\cdot (\rho u) =0$      (3): Conservation of Mass

where $\rho (x,t)$ is mass density, $\phi (x,t)$ gravitational potential, $u(x,t)$ particle velocity, $x$ is a Euclidean space coordinate, $t$ is a time coordinate, the dot expresses differentiation with respect to time, and $\frac{D}{Dt}$ is the time derivative along a particle trajectory. 

Newton's Model allows prediction of gravitational motion of the Universe by time stepping after relaxing (1) into 

• $\epsilon\dot\phi = \Delta\phi -\rho$    

with $\epsilon$ a small positive coefficient. Newton's Model combines simplicity with generality in a formidable demonstration of the power of mathematical thinking in the form of the Calculus developed by Leibniz and Newton.  

Is it possible that Newton's Model does not correctly capture gravitational motion? Before Einstein this question was not posed since it (1)-(3) appeared to be rock solid as ultimately expressing conservation of mass and energy, without which the Universe could not persist over time. 

Einstein built his fame on a claim that Newton's Model is wrong and has to be replaced by Einstein's General Theory of Relativity GR, which has become a consensus of modern physics supposedly based on GR and quantum mechanics. The consensus says that there is experiments agreeing better with GR than Newton's Model, although the discrepancy is extremely small and most delicate to catch.     

So the question remains: Is it possible that Newton's Model is wrong? It would require at least one of (1)-(3) to be wrong which ultimately would require conservation of mass or energy to be wrong. Is that possible? 

Newton's Model is an expression of Leibniz view on space as expressing local coexistence and time as change/succession of local coexistence, which comes to life by explicit time stepping updating $\rho (x,t)$, $\phi (x,t)$ and $u(x,t)$ from one time instant $t$ to a next instant $t+dt$ involving only local coexistence in space.  

To question Newton Einstein thus had to leave Leibniz view with space and time separated and enter a whole new world of space-time with now time and space intertwined, which has become the trademark of modern physics. 

While Leibniz view very much makes sense, Einstein's space-time does not at all. The crisis of modern physics is rooted in this confusion. 

Note that Leibniz view only involves local coexistence and local time and so does not ask for any notion of (supposedly hard-to-explain) absolute space and time attached to Newton in advocacy for GR. 

This post connects to The World as Computation exhibiting the computational aspects of mathematical models including explicit time-stepping as fundamental technique without need of (supposedly hard-to-explain) instant action at distance again attached to Newton in advocacy of GR.