Newton Back! Einstein Out?

Modern physics has developed from classical physics in three steps each viewed as revolutionary:

1. Maxwell's equations 1867 describing all of electro-magnetics including light-as-wave.
2. Einstein's Special Theory of Relativity SR 1905 replacing Newton's mechanics without gravitation by relativistic mechanics.
3. Einstein's General Theory of Relativity GR 1916 replacing Newton's mechanics with gravitation by curved space-time geometry. 
4. Schrödinger's equation for atom physics.  

The present view is that a unified theory including all of mechanics + electro-magnetics + atom physics, is impossible because of severe incompatibilities between 1- 4, primarily because of SR and GR.  There is really no incompatibility between Newton, Maxwell and Schrödinger, if you do not seek incompatibility to boost your own favourite substitute.

Let us search the main reason why Newton was dismissed by Einstein and then all his followers. Newton's theory of gravitation as the main jewel of the infinitesimal Calculus created by Newton and Leibniz, describes the motion of all celestial bodies from the inverse-square law. But Newton's theory seemed to require instant-action-at-distance, which was exhibited by Newton's critics as a mystery/physical impossibility, even acknowledged by Newton himself, from a prevailing understanding that forces only can act by instant direct contact, with the history described in Newtonian Studies by Koyre. 

In GR the inverse-square-law is replaced by curved space-time geometry without instant-action-at-distance (from an ad hoc assumption that the speed of gravity is equal to the speed of light9,  thus circumventing the mystery, but at the price of an even more mysterious concept of curved space-time.

But there is a way to get around instant-action-at-distance even in Newton's theory, which is explored in posts on New View on Gravitation. The basic idea is that gravitational potential is primordial with mass secondary as the result of local instant differentiation. 

We may thus see a return of Newton, and let us then recall that Newton's world can be constructed starting with free fall of a small test particle of unit mass with velocity $v$ in a given gravitational field $\phi (x)$ depending on a Euclidean spatial coordinate $x$ described by:

•   $\dot v=-\nabla\phi$,                     (N0)

where $\dot v$ as the time derivative of $v$ is the acceleration of the test particle under the gravitational force $-\nabla \phi$. Accordingly the time of Newtonian success was named "the dot-age". 

(N0) is then generalised to small material bodies of mass $m$ as collections of $m$ particles of unit mass expressing that all small bodies fall freely the same way in a given gravitational field according to

• $m\dot v =-m\nabla\phi$.               (N1) 

We here assume the material body to be small so that $\nabla\phi$ is the same for all parts of the body. 

The next step is to transfer gravitational force $\nabla\phi$ to mechanical force $f=-\nabla\phi$ e g  hanging a unit mass in a unit linear spring and measuring its elongation under gravitation. This makes it possible to generalise (N1) to Newton's 2nd Law for a body of mass $m$ 

• $m\dot v= mf=F$                           (N2)                        

where $F$ can be gravitational or mechanical force. 

The final step is to express Newton's law of gravitation in the form given by the mathematician Laplace as the differential equation

• $\Delta\phi (x) =\rho (x)$ for all $x$,          (NG1)

where $\rho (x)$ is mass density and $\Delta$ is the Laplace differential operator, which thus connects to gravitational potential to mass density. 

The common view is that presence of a unit point mass at $\bar x$ generates a contribution $-\frac{4\Pi}{\vert x-\bar x\vert}$ to $\phi (x)$ as an apparent instant-action-at-distance. But it is possible to turn the connection around and view instead $\rho (x)$ as being generated as

• $\rho (x)=\Delta\phi (x)$       (NG2)

where the process of differentiation is local and as such can an be instant in the same way as a contact force.

It is thus not necessary to dismiss Newton's theory of gravitation as requiring mysterious instant-action-at-distance, and thus combine with Maxwell's and Schrödinger's equations into a unified model of the world as outlined in Many-Minds Relativity, Real Quantum Mechanics, Computational Thermodynamics, Computational Turbulent Incompressible Flow and Computational Black-Body Radiation.

This means that Newton, as the greatest physicist all times, is welcome come back again to constructively contribute to a unified model of the world without fundamental incompatibilities and so leave SR and GR without mission and open a way out of the current crisis of modern physics. This has been a main theme of this blog with more details to come.

In particular, with the gravitational potential as primordial with everywhere presence there is no vacuum or complete emptiness of mysterious nature.  Moreover, it opens to connect regions in space where the gravitational potential is smooth with derivatives of only moderate size, to dark matter. Further, (NG2) opens to negative mass being created subject to repulsion from positive mass as a possible source of dark energy,

After all, the world must be rational to exist at all and so must be possible to describe in rational mathematical terms like (N2) + (NG) without incompatibilities. It is impossible that the world is incompatible with itself. Only models of the world can be incompatible if incorrect is some way.