Physics Illusion 9: Instant Action at Distance Not Physical Nor Needed

                     Gravitational potential generating mass at the tip of the potential well.    

Viewing physics as a form of analog computation based on computational exchange of local information, suggests that instant action at distance is unphysical, because a large number of local computations would be required to transfer information globally, which would require time.

Newton's law of gravitation can be expressed through the relation $\Delta\phi (x,t) =\rho (x,t)$, where $\phi (x,t)$ is gravitational potential and $\rho (x,t)$ mass density with $x$ a Euclidean space coordinate. The potential $\phi $ may be computed by explicit time stepping to stationary state, each  step exchanging local information only, of the parabolic evolution equation (heat equation)

• $\dot\phi (x,t) - \Delta\phi (x,t)=\rho (x,t)$ for all $x$ and $t > 0$ with $\phi (0,x)$ given,

which would require time and thus cannot be accomplished instantly. Here a dot signifies differentiation with respect to time $t$.

We conclude that in the relation $\Delta\phi =\rho$ connecting $\phi$ and $\rho$, we cannot view $\rho$ as input or cause and $\phi$ as output or effect, but rather the other way around with the potential $\phi$ as primordial input/cause and $\rho$ as the output/effect. With this view $\rho (x,t)=\Delta\phi (x,t)$ is produced locally by the Laplacian differential operator $\Delta$ acting locally in space on the potential $\phi (x,t)$, thus without action at distance and thus in principle allowing instant action. 

We may compare with Faraday's law of induction in electromagnetics:

• $J = -\dot E + \nabla\times B$, 

where $J(x,t)$ is electrical current density, $E$ is electrical field and $B$ magnetic field, which can also be expressed by the wave equation

• $-\dot J =\ddot E -\Delta E \equiv\square E$, 

with a common notation. Here we see the current $J$ being generated by the electrical field $E$ by differentiation acting locally in space and time and we thus face a concrete physical example of a field by generating a flow of charge or current locally by differentiation. By analogy we may envision the gravitational field $\phi$ generating mass $\rho =\Delta\phi$ locally by differentiation.

We thus have the following two relations with the fields $\phi$ and $E$ acting as input and the output $\rho$ and $\dot J$ being produced locally by differentiation:

1. $\rho = \Delta\phi $,
2. $\dot J =- \square E$ or $J = -\dot E + \nabla\times B$.          

Here 2 expresses the familiar physics of induction, while 1 expresses unfamiliar generation of mass by gravitational potential. The question if we can learn to become as familiar with 1 as we already are with 2? Von Neumann reflected:

• In mathematics you don't understand things. You just get used to them.

Maybe it is the same with physics, and then it seems better to get used to constructive real physics which can be simulated by digital computation, than with illusionary physics, which can only be simulated by illusions of thinking without connection to the realities of physics. Such an illusion is instant action at distance as in the conventional view of gravitation with matter generating gravitational potential and force.