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,
- $J = -\dot E + \nabla\times B$,
- $-\dot J =\ddot E -\Delta E \equiv\square E$,
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:
- $\rho = \Delta\phi $,
- $\dot J =- \square E$ or $J = -\dot E + \nabla\times B$.
- In mathematics you don't understand things. You just get used to them.