måndag 3 februari 2014

Gravitational Law as Perfect Harmony as Perfect Marriage

This is a continuation of From Spooky Action at Distance to Dig Where You Are.

We are used to saying that the Earth moves around the Sun geared by a gravitational force accelerating the Earth towards the present position of the Sun of magnitude proportional to the inverse of the square of momentary distance. This makes the Earth track an elliptical orbit around the Sun with the Sun in one of the foci, as proved mathematically by Newton in his Principia Mathematica and thereby rocketing him fame. Newton could thus describe the motions of the planets from one single law of physics, Newton's law of gravitation of the form:
  • $F = G \frac{M_eM_s}{R^2}e$, 
where at a given moment of time shared by the Earth and the Sun, $F$ is the force acting on the Earth directed towards the Sun, $G$ is a universal constant of gravitation, $M_s$ is the mass of the Sun, $M_e$ the mass of the Earth, $R$ is the distance from Earth to the Sun and $e$ is a unit vector pointing from the Earth towards the Sun.

Despite the fame his law of gravitation brought, Newton was not happy because his required instantaneous action at distance, which Newton admitted was an absurdity.

We have noted that Newton's law of gravitation can be formulated as follows (with suitable normalization)
  • $\Delta\phi (x,t)=\rho (x,t)$ for all points in space $x$ and moments of time $t$,
where $\Delta$ is the Laplace differential operator, $\phi (x,t)$ is the gravitational potential at $(x,t)$ with the gravitational force equal to the gradient $\nabla\phi$ and $\rho (x,t)$ density of matter or mass density with $(x,t)$ serving as space-time coordinates. This was the starting point in Laplace's monumental Mecanique Celeste in 5 volumes 1829 - 1839.

The conventional way of looking at the equation $\Delta\phi =\rho$ is to view the mass density $\rho$ to be given and $\phi$ to be created instantaneously by instantaneous solution of the differential equation $\Delta\phi =\rho$.

For example, if $\rho$ is a delta function at the origin, then $\phi = -\frac{1}{4\pi\vert x\vert}$ with $\nabla\phi$ giving Newton's law of gravitation.

Because mass density $\rho$ is visible by light emission/reflection while the gravitational potential $\phi$ is invisible and can only be felt as a force,  it was natural to Newton and still may appear natural to us today to view $\rho$ as primary given data (what we can see) and $\phi$ as secondary derived quantity (what we cannot see) somehow caused or generated by some physical "solution process" producing the solution of the equation $\Delta\phi = \rho$, instantaneously without time delay. This amounts to a cause-effect relation with mass the cause and gravitational potential the effect with the solution of the differential equation being a non-local operation of integration requiring transmission of information over distance and thus massive computational work, which is in conflict with instantaneity.

From this perspective major efforts have gone into uncovering this solution process as an exchange of some form of force carrying particles named gravitons, but without any form of success. Newton's law of gravitation with instantaneous action at distance, is to us today the same absurdity as it was to Newton!

We thus have to shift strategy, and it is then natural to reverse to the cause-effect and view the gravitational potential $\phi$ as given data and $\rho =\Delta\phi$ as solution as the reverse operation of differentiation which is local in space and time. With this perspective there is no absurd inexplicable instantaneous action at distance, but instead we face some form of "creation process" with matter emerging as an effect of the "curvature" $\Delta\phi$ as the cause. 

A third possibility is to give up the idea of cause-effect in the relation $\Delta\phi =\rho$ and in the spirit of both Leibnitz and Einstein to view this relation as a form of "perfect harmony" like a "perfect marriage" with each partner following individual wishes turning out to be exactly what the other partner would like to see, without any form of domination of one over the other. 

In any case, there seems to be some hitherto unexplored possibilities of getting around the spell of absurdity of Newton's law of gravitation as requiring instantaneous action at distance.  Let's get to work!  A starting point can be

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