Instant Action by Local Differentiation vs Global Integration

                                     Local imbalance generating heat as instant local action.

Newton's Theory of Gravitation NTG is beautifully captured in the Poisson-Laplace differential equation

• $\Delta\phi (x,t) = \rho (x,t)$ for all $x$ in 3d space and  time $t$,          (G)

where $\phi (x,t)$ is gravitational potential, $\nabla\phi (x,t)$ is gravitational force and $\rho (x,t)$ mass density at $x$ and $t$.  

With $\rho (x,t)$ viewed as the source at $(x,t)$ and $\phi (y,t)$ for all $y$ as the effect, it appears that the presence of mass at $x$ at time $t$ without time delay creates gravitational potential at any other point $y$, as an expression of apparent instant action at distance. This was viewed as a mystery by Newton himself, but did not force him to give up his beautiful invention, which seemed to describe the whole world of gravitation in remarkably concise form as an ultimate expression of the power of mathematics.  

In search of modernity at the turn to the 20th century the mystery of apparent instant action at distance was chosen as target, which eventually led to Einstein's General Theory of Relativity as a geometric theory supposedly without instant action at distance, as an improved version of NGT albeit at the cost of being immensely more complicated. 

It is natural to ask if apparent action at distance poses a real problem to physics? If not, maybe NTG can be retained because it is so simple and effective, and GR can be reserved for Einstein's devotees in our time. 

Let us then compare with the heat equation describing conduction of heat in a heat conducting medium, which takes the form:

• $\dot u(x,t)-\Delta u(x,t) = f(x,t)$ for all $x$ in 3d space and  time $t$,         (H)

where $u(x,t)$ is temperature, $f(x,t)$ heat source and the dot represents differentiation with respect to time $t$. Also in this case is there is apparent action at distance in the sense that the presence of a heat source at $x$ at time $t$ appears to have instant influence on the temperature at all other points $y$, even if the influence decays exponentially with the distance from $x$. Again the actual physics of such instant action at distance is a bit mysterious, but it is not viewed to be so serious that the heat equation must be given up for some new maybe geometric complex theory of heat conduction. The rationale is that since the action decays exponentially with distance, in reality the action is not infinitely quick.  

But there is another way of interpreting (H), which does not involve instant action at distance, which is to view (H) and correctly so as balance between two sources of heat, an exterior represented by the right hand side $f(x,t)$ and an interior represented by the left hand side involving the time derivative $\dot u (x,t)$ and the spatial derivative $\Delta u(x,t)$ which both can be viewed to result from instant local action by differentiation. 

It is thus possible to give the heat equation (H) an interpretation as local instant action by differentiation: If  $\dot u(x,t)-\Delta u(x,t)$ does not vanish, heat is produced according to $f(x,t)=\dot u(x,t)-\Delta u(x,t)$. With this interpretation there is no instant action at distance. 

Instant action as local differentiation replaces instant action as global integration. In the setting of the heat equation this can quite naturally be understood as local imbalance of interior state generating heat locally. An explosion can be viewed this way as shown above.

In the same way we can turn around the view on (G) and thus avoid the mystery of apparent instant action at distance. I have discussed this in many posts on New View on Newtonian Gravitation.

Compare with the opening statement of Gravitation in the Twilight of Classical Physics: An Introduction from Vol 3 of Genesis of General Relativity (eds Renn and Schemmel) from 2007: 

• The history of treatments of gravitation in the nineteenth century reflects the transition from an era in which mechanics constituted the undisputed fundamental discipline of physics to an era in which mechanics became a subdiscipline alongside electrodynamics and thermodynamics.
• From the time of its inception, the action-at-a-distance conception of Newtonian gravitation theory was alien to the rest of mechanics, according to which interaction always involved contact. This explains the early occurrence of attempts to interpret the gravitational force by means of collisions, for instance, by invoking the umbrella model described above. During these early days the comparison of the gravitational force to electric and magnetic forces had already been suggested as well. 
• However, the analogy with electricity and magnetism became viable only after theories on these subjects had been sufficiently elaborated. There were even attempts at thermal theories of gravitation after thermodynamics had developed into an independent sub-discipline of physics. 
• Besides providing new foundational resources for approaching the problem of gravitation, the establishment of independent subdisciplines and the questioning of the primacy of mechanics that resulted from it affected the development of the theoretical treatment of gravitation in yet another way, namely through the emergence of revisionist formulations of mechanics. This heretical mechanics, as we shall call it, consisted in attempts to revise the traditional formulation given to mechanics by Newton, Euler and others, and often amounted to questioning its very foundations.