# Action at Distance

• The real lover is the man who can thrill you by kissing your forehead or smiling into your eyes or just staring into space…I believe that everything happens for a reason. People change so that you can learn to let go, things go wrong so that you appreciate them when their right, you believe lies so you eventually learn to trust noone but yourself, and sometimes good things fall apart so better things can fall together. (Marilyn Monroe)

If local contact forces in a sense are easy to envision, action at distance by definition is mysterious. If we can see a chain connecting the source/cause to a distant effect, the action at distance can be explained as a form of chain reaction based on repeated local interaction like a row of domino bricks falling one after the other, with each domino brick knocking down the next.

The gravitational and electromagnetic forces are key examples of action at distance because the medium carrying the action seems to be a vaccum or nothingness.

Physicists like to believe that forces between elementary particles are transmitted through certain other particles carrying forces over distance. The gravitational force is conjectured to be transmitted by a hypothetical particle named \emph{graviton}, but nobody has been able to detect a particle like that.

The standard view of gravitation acting at distance is that the presence of a mass, like the Sun, creates a \emph{gravitational field} or gravitational potential, the variation of which gives rise to a gravitational force in the same way as a variation of pressure in the air can give rise to a pressure force.

We shall below present an alternative view only based on local interaction without any gravitons, where it is instead the gravitational field which creates the mass. This view is like a hen as gravitational field laying an egg as mass, while the standard view is an egg as mass generating a hen as gravitational potential.

We believe it is more difficult to explain how an egg can create a hen, than how a hen can lay can egg.

# Local vs Global in Digital Simulation

We shall meet the aspect of local interaction vs action at distance, or local vs global, in both Calculus of derivatives and integrals and in digital computation.

Computing a derivative is like digging where you stand.

Computation of a derivative of a function is a local operation involving comparison of function values at nearby points in space/time, while computation of an integral of a function is a global operation involving summation over many function values points in space/time which are not close.

Differentiation is like digging a whole where you stand, while integration is like a rumour spreading over distance by mouth-to-mouth communication.

In general ntegration requires more computational work than differentiation because information needs to spread. Differention is a local process, while intergration is global.

Computing an integral is like walking from one point to another, step by step.

Derivatives of analytical functions can be computed analytically/symbolically, while integrals in general cannot.

In digital computation the aspect of processing of local vs global information relates to how information is stored in a computers memory, and how fast it can be accessed. The memory storage pattern can reflect physics so that nearby points in space are stored nearby in the memory, but in digital computation action at distance is possible, by addressing any point in the memory. This is like sending information by emailinstead of by person-to-person mouth-to-mouth.

In computational digital solution of differential equations, information is processed on a computational mesh reflecting a physical structure. If physical flow of information between nearby material particles in space/time is reflected computatinally by communication only between nearby mesh-points, then the digital flow of information mimics the physical flow and thus can be termed as “physical”.

But in digital solution also communication between distant points is possible, which as we will see can speed up the computational process, like email communication can speed up communicationby surface snail-mail.

We will meet computational processes with direct meshpoint-to-meshpoint communication in the form of explicit methods (of time-stepping), while implicit methods will involve more or less global communication. Explicit methods are “physical” and “simple” but sometimes slow, while implicit methods are “artificial” and more “complex” but possibly much faster.

Person to person communication reflecting meshpoint-to-meshpoint communication in computational mesh