Newton is Back!

Cam Newton is Back!

The main advancements of physics the last 100 years consist of (i) quantum mechanics as physics on small scales, and (ii) Einstein's relativity theory as physics on large scales. This proclaimed success story of modern physics is shadowed by a realisation that (i) and (ii) are inconsistent/incompatible, and so cannot both be true.  

While quantum mechanics opens classical physics to new atomistic scales, Einstein's relativity theory represents a definite break with Newton's theory of gravitation as the crown jewel of classical physics. The key question concerns the connection between mass density $\rho$ and gravitational potential $\phi$, which is represented by Newton's equation 

• $\Delta\phi (x,t)=\rho (x,t)$.                  (N1)

where $\Delta$ is the Laplacian differential operator with respect to a Euclidean space coordinate $x$ and time coordinate $t$, or by Einstein's equation in coordinate free form.  (N) is the form of Newton's inverse square law given by the mathematician Laplace. 

Newton's equation is well understood and for given mass density $\rho (x)$ as given data, the gravitational potential $\phi$ as solution to (N1) can quickly be computed on a computer. On the other hand, Einstein's equation is poorly understood and virtually impossible to solve. The common view is that solutions to Einstein's equation reduce to solutions of Newton's equations except for very extreme situations of black holes beyond the range of both equations. 

The question is then why Newton's theory of gravitation has to be replaced by Einstein's? What is wrong with (N1) if it covers everything of interest? The standard answer is that (N1) requires instant action at distance since the solution of the differential equation (N1) in analytic form is given by the integral equation  

• $\phi (x,t) = -\frac{1}{4\pi }\int\frac{\rho (y,t)}{\vert x-y\vert }dy$,      (N2)

connecting $\phi (x,t)$ to $\rho (y,t)$ for all $y$ without time delay since the time $t$ on the left is the same as on the right side. The solution formula (N2) expresses that the process of solving is global seemingly instantly connecting $x$ with all $y$. 

So do we then have to accept that Newton's theory of gravitation requires instant action at distance, which is impossible? Do we have to give up all the wonderful physics Newton offers to humanity, and instead rely on Einstein, who is very difficult to understand and use? 

Not necessarily, since it is possible to view (N1) with a different perspective, which does not ask for instant action at distance. The first option is to turn (N1) around to instead

• $\rho (x,t)=\Delta\phi (x,t)$                 (N3)

which thus specifies $\rho (x,t)$ in terms of data $\phi (x,t)$ by the local operation of differentiation, which does not ask for global instant action, only local instant action. 

A second option is to relax (N1) into the equation

• $\epsilon\dot\phi (x,t) = \Delta\phi (x,t)-\rho (x,t)$,       (N4)

with $\epsilon$ a small positive number, and the dot signifies differentiation with respect to time. This is referred to as parabolic relaxation turning (N4) in a heat equation, where $\phi (x,t)$ can be updated by time stepping without instant action at distance. Since $\epsilon$ is small, solutions of (N4) closely agree with solutions of (N3). 

Both (N3) and (N4) connect to Leibniz' principle of pre-established harmony connecting $\rho$ and $\phi$ according to both (N1) and (N3) like two entities playing both roles of data and solution in full harmony.

This may give new light to the basic idea of modern physics of force carrier particles supposed to transmit forces over distance, named gravitons in the case of gravitational force. But no gravitons have been detected, which questions the utility of that idea, and so Leibniz may tell us something even today.

The above argument extends to electromagnetism captured in Maxwell's equations with charge connected to electric potential/field by Coulombs Law.  

Summary:  It is possible to argue that Newton's law of gravitation does not necessarily require instant action at distance in a real physical sense, and so there is no real reason to replace Newton's theory of gravitation by Einstein's. If we allow Newton to come back, then modern physics may be relieved from  the inconsistency/incompatibility trauma brought by Einstein:  

• Newton/Maxwell + quantum mechanics = harmony. 
• Einstein + quantum mechanics = disharmony!  

We are then back to the old idea that all interaction ultimately boils down to "instant touching" like touching the chin of your beloved, in full harmony.