Newtonian Gravitation Does Not Require Instant Action at Distance

                                                 Instant local action.

Recent posts describe a resurrection of Newton's Theory of Gravitation NG as the prime jewel of classical physics, which in modern physics formally has been replaced by Einstein's Theory of Gravitation EG, although in practice NG still reigns. 

The main reason to throw away NG is a common understanding that NG requires instant action at distance for which physics appears to be missing. The argument is that gravitational potential $\phi (x,t)$ of NG is connected to primordial mass density $\rho (x,t)$ as the solution to the differential equation in the Laplacian $\Delta$:

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

with the same time coordinate $t$ on both sides of the equation, with the solution being represented by the integral formula:

• $\phi (x,t) =-\frac{1}{4\pi}\int\frac{\rho (y,t)}{\vert x-y\vert}dy$           (NG2)

which appears to require instant action at distance or fast global action, because the integration variable $y$ covers all of space at a given time $t$.

But it is possible to switch the roles in NG and view the gravitational potential $\phi (x,t)$ as primordial role from which mass density $\rho (x,t)$ is "created" by the local action of the Laplacian differential operator:

• $\rho (x,t) = \Delta \phi (x,t)$ for all $x$,         (NGnew1)

which can be assumed to act without time delay for all $t$ as local action. Mass is thus created locally for each $x$ by differentiation as an instant local operation acting at each time instant $t$, as considered under the tag New View on Gravitation. 

But (NGnew1) is not the full story because conservation of mass is described by the equation 

•  $\dot\rho +\nabla\cdot m =0$ 

where $m=\rho u$ is momentum with $u$ velocity, and the dot on top signifies differentiation with respect to time, which takes the following form with $\phi$ primordial:

• $\Delta\dot\phi +\nabla\cdot m =0$, 

 allowing $\dot\phi$ to be expressed by the integral formula 

$\dot\phi (x,t) =\frac{1}{4\pi}\int\frac{\nabla\cdot m (y,t)}{\vert x-y\vert}dy$.   (NGnew2)

Formally (NGnew2) appears to again require instant action at distance, like (NG2), but in a different setting with $\dot\phi$ as an integral over $\nabla\cdot m$, instead of $\phi$ as an integral over $\rho$, thus in terms of small changes instead of gross quantities.

With NGnew as (NGnew1) + (NGnew2) we can thus express NG with gravitational potential as primordial with instant local action for gross quantities in (NGnew1) and formally instant action at distance only for small changes of $\phi$ in (NGnew2), for which limitation to finite speed has little influence. 

The basic critique of NG takes the form: Suppose the Sun suddenly disappears. How long time will it take before the absence of the gravitational pull by Sun on the Earth will be noticed? Instantly? And if so how?

With (NGnew2) instead of (NG2) the formal appearance of instant action is reduced to small changes instead of gross quantities. In this setting the changes of the gravitational potential are slow because velocities are small and a sudden disappearance of the Sun is not possible. 

Summary: NGnew gives a new view of NG where instant action at distance for gross quantities is not required.  Is this enough to resurrect NG? In short, here is the story of a complete harmony in the spirit of Leibniz between gravitational potential and mass without any need of fast global action: 

• Gravitational potential gives mass to matter.  (fast local)
• Spatial change of momentum (mass x velocity), changes the gravitational potential. (slow global)
• The gravitational potential of the Earth/Sun/Galaxy…changes very slowly in a coordinate system fixed to the Earth/Sun/Galaxy…

Compare with a common popular description of EG as a "theory in curved space-time":

• Matter tells spacetime how to curve.        (fast global, speed of light?)
• Curved spacetime tells matter how to move.    (fast local?)

Your choice: NG for all normal physics or EG for non-physics? 

Everybody can understand NG. Nobody can really understand EG, only pretend to do so. 

PS But what about the precession of Mercury, as proof of supremacy of EG over NG? Is it clear that NG gives incorrect prediction when full input data to a NG computation is missing? How is it possible to claim that EG gives correct prediction when EG computation for the Solar system is impossible?