torsdag 30 januari 2014

From Spooky Action at Distance to Dig where You Are

                                               Spooky attractive action at distance.

Newton admitted in a letter to Bentley February 25 1692/3 that his universal law of gravitational attraction based on instantaneous action at distance, was a true mystery:    
  • It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact…
  • That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it. 
  • Gravity must be caused by an Agent acting constantly according to certain laws; but whether this Agent be material or immaterial, I have left to the Consideration of my readers.
Leibniz picked on Newton claiming that Newton's gravitational action at a distance is a perpetual miracle because it does not follow from the essence of matter.

Einstein was also critical to "spooky action at a distance" as it seemed to require infinite speed of propagation of the gravitational force in violation of the special theory of relativity. In his general theory of relativity Einstein turned the question around and replaced the non-local gravitational force acting at distance with the local property of "curvature of space-time" commonly interpreted:  "matter acts upon space-time geometry, deforming it, and space-time geometry acts upon matter".

The dominating idea of modern physics is that gravitational force is mediated by specific "force particles" named gravitons supposedly being exchanged between two bodies attracting each other gravitationally. But no evidence of the existence of gravitons has been found, neither the existence of "gravitational waves" as ripples of curved space-time.      

The nature of the gravitation force is thus remains as one of the deep mysteries of physics with no real progress since Newton. Newton's universal law of gravitation can be formulated as follows:
  • $\Delta \phi = \rho$   
where $\phi (x,t)$ is the gravitational potential at a point $x$ in space and $t$ in time associated with a mass or matter density $\rho (x,t)$, and $\Delta$ is Laplace operator in space. The gravitational force at $(x,t)$ is then given as $\nabla\phi (x,t)$, the space gradient of $\phi$. For two point masses this reduces to Newton's law of gravitation in its common form with the gravitational force along the line connecting the masses and scaling with the inverse of the distance squared.

The conventional way of viewing the connection between $\phi$ and $\rho$ connected by Poisson's equation $\Delta\phi =\rho$, is to view the matter distribution $\rho (x,t)$ to be given and the gravitational potential $\phi$ (and gravitational force $\nabla\phi$) to be "created" from $\rho$ with $\phi$ the solution to the equation $\Delta\phi =\rho$ "created" in some form of "solution process" in space for each given instant of time.  

Now, the "solution process" generating $\phi$ from $\rho$, is a non-local process and since it is supposed to be performed all over space at a shared given instant of time, it seems to involve infinite speed of propagation. 

But there is a another way of viewing the equation $\Delta\phi =\rho$ and that is to view $\phi$ as given and $\rho =\Delta\phi$ as the result. In this case the "solution process" is local since it involves locally determined space derivatives of $\phi (x,t)$ without action at distance, in a form of "dig where you are". This is somewhat more detailed in The Hen and the Egg of Gravitation with the Hen the gravitational potential and the matter distribution $\rho$ the Egg and in Many-Minds Relativity 20.3.

With Egg from the Hen there is no need of instantaneous action at distance which requires a universal time and infinite speed of propagation, both seeming to be miracles. 

The only thing to explain is how a Hen can lay an Egg, that is how a gravitational potential $\phi$ can generate a mass distribution $\rho =\Delta\phi$. In particular it opens to view local concentrations of $\phi$, with delta functions representing point masses, as visible matter and more smooth distributions as dark matter.

Connection to Incompressible Flow

The velocity $u$ and pressure $p$ in (nearly) incompressible flow are connected by an equation of the form

(1)  $\Delta p =\frac{\nabla\cdot u}{\delta}$,

where $\delta$ is a small positive parameter. We see that the pressure $p$ (with a minus sign) corresponds to the potential $\phi$ and $\frac{\nabla\cdot u}{\delta}$ to the mass density $\rho$. With $p$ given by $\frac{\nabla\cdot u}{\delta}$ as solution of a Poisson equation, incompressible flow appears to require infinite speed of propagation of pressure, which is unphysical.  With instead $p$ given, mass is generated/destructed from the equation $\nabla\cdot u=-\delta\Delta p$ with the result of maintaining incompressibility under motion of matter.

Note that (1) represents a certain form of regularization of the velocity-pressure coupling in incompressible flow. Adding a first order time derivatives of $p$ to the left hand side gives a parabolic equation (heat equation) with infinite speed of propagation, and with a second order derivative a wave equation with finite speed of propagation, in the perspective with $p$ the solution and $\frac{\nabla\cdot u}{\delta}$ given. But as we said, there is a different perspective...

Connection to the World as Computation

With Newton and Laplace the view of the world as a clock got momentum leading into a view in our computer age of the world as giant computational process. With this view we would look at Poisson's equation $\Delta\phi =\rho$ together with some computational process for solving the equation. With $\rho$ as data, this corresponds to a process of integration typically performed by iterative (repeated) local relaxation, which requires computing time and is restricted by finite speed of propagation in some medium carrying the propagated information in its physical realization. Asking for instantenous solution would lead to the absurdity identified by Newton.

The alternative would be to give up the idea of solving the equation $\Delta\phi =\rho$ with $\rho$ given and $\phi$ the solution, and instead view $\phi$ as given with $\rho$ locally computed by evaluating $\Delta\phi$, which can be done instantaneously because no iterative relaxation is needed. It may be that this is what makes the world go around...what else could it be?

Connection to Faraday

Michael Faraday expressed in a letter to R. Phillips in 1846:
  • You are aware of the speculation (2) which I some time since uttered respecting that view of the nature of matter which considers its ultimate atoms as centres of force, and not as so many little bodies surrounded by forces, the bodies being considered in the abstract as independent of the forces and capable of existing without them. 
  • In the latter view, these little particles have a definite form and a certain limited size; in the former view such is not the case, for that which represents size may be considered as extending to any distance to which the lines of force of the particle extend: the particle indeed is supposed to exist only by these forces, and where they are it is. 
  • The consideration of matter under this view gradually led me to look at the lines of force as being perhaps the seat of vibrations of radiant phenomena.         
Faraday here expresses that gravitational force (from gravitational potential) is primary and matter is secondary, in line with the above alternative view.

What You See Determines Your Model of the World

We can observe matter distribution $\rho$ (the position of the Sun, Moon, stars) since matter emits/reflects light, while the gravitational potential $\phi$ is "invisible". We are thus (mis)led to consider $\rho$ as given (we see what we have) and consider $\phi$ to be a miraculous effect  of $\rho$ from action at distance, which we cannot see. Our model of the World would be different if instead matter was invisible while we could observe the gravitational potential.

Complete Gravitational Model 

If now the mass distribution $\rho$ is given as $\rho =\Delta\phi$, the equation expressing mass conservation $\dot\rho + \nabla\cdot m$, takes the form:
  • $\Delta\dot\phi + \nabla\cdot m =0$   
where the dot signifies time derivative and $m=\rho u$ is momentum and $u$ velocity. This equation monitors the change of the gravitational potential in time through the change of momentum $m$ in space. The change of momentum $m$ in time is described by Newton's 2nd law in the form
  • $\dot m +\nabla\cdot (um) - \nabla\phi =0$, 
assuming only gravitational forces. For further details, see Many-Minds Relativity 20.3. A relaxation of the equation of mass conservation could possibly have the form
  • $ - \delta\ddot\phi + \Delta\dot\phi + \nabla\cdot m =0$,   
with a small positive parameter $\delta$. This could allow a finite speed of propagation of effects in the gravitational potential $\phi$, which possibly could be identified and measured, while the gravitational force $\nabla\phi$ still could appear to be determined by a miraculous instantaneous action at distance, which would however only be an illusion.

In other words, a planet would be geared locally and instantaneously by the gradient of the gravitational potential, and not by somehow feeling the position of the Sun through miraculous spooky instantaneous action at distance:

This connects to Einstein's general relativity with the planet locally geared by the "curvature of space-time" and not by spooky action at distance. But instead of a mysterious "curved space-time" we are here dealing with a gravitational potential in Euclidean "flat space-time" which may seem less mysterious.

A model of the World as a compressible gas interacting by gravitation, thus can take the form: Find $(\phi ,m,e)$ such that
  1. $\Delta\dot\phi + \nabla\cdot m =0$
  2. $\dot m +\nabla\cdot (um) +\nabla p + \nabla\phi =0$
  3. $\dot e +\nabla\cdot (ue) +p\nabla\cdot u =0$,       
where $\rho = \Delta \phi$ is matter density, $u=\frac{m}{\rho}$ is matter velocity, $p=(\gamma -1)e$ is pressure with $\gamma > 1$ a gas constant and $e$ is internal (heat) energy, see Many-Minds Relativity 20.3.

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