Newton's Sphere Theorem: Review & Analysis

[MetaFrizzics]

Stability in unequally Spaced Configurations

Counterintuitively, the charges (if constrained to a sphere) could take odd positions, and find a minimal stable energy state, even though in the conventional way of viewing things, the charges are not 'equally' spaced on the surface, but are varying distances apart. What makes the arrangement stable is symmetry of repulsive forces over the surface of the sphere. Being confined to the surface puts additional constraints upon the motion of mobile charges and allows stable configurations not otherwise possible.

 

 

How can repulsive force allow a non uniform arrangement of repulsive particles over a sphere?

 

You will notice that the arrangements of successive rings of charges away from a given particle on the surface are quantized discrete sets of distances, and that the number of particles in each successive ring is odd-even-even-odd-odd. It is especially stable that the first ring is triangular and creates a strongly stable central saddle-point in the centre of the spherical cap on the surface.

 

 

This means for instance, the particle in the first ring (3 surrounding particles) is held in a stable position because of the triangular arrangement, combined with the physical curvature of the proposed constraining sphere (say an aluminium ball). The spherical cap forms a minimal potential energy well, literally! The saddle point is in the middle of the cap.

 

Similarly, the next larger ring of particles also pushes the particle into the centre of the polar cap. It is a physical minimax solution to the potential formed by other charges on the sphere. By simple extrapolation, (all particles on the sphere are geometrically identical) all the the particles are stable and so the whole lattice is stable.

 

Keep in mind that although this symmetrical pattern of charges is stable, this is *not* the general case with an arbitrary number of charges. It only works with certain numbers, like 4, 12, 20 etc. Other numbers will not spread equidistantly and retain correct symmetry.

[MetaFrizzics]

Quantifying the deviation from a Flat Field

 

We can model the quantization of mass by using the pure (unintegrated) form of the equations we have already derived. This will leave the mass discretely distributed at any level of coarseness we wish:

 

 

Of course once you get over a few hundred charges, the perturbations in the field are individually small.

 

However, since we are really discussing the flatness of field inside a hollow sphere it is interesting to note that even with a few thousand charges there is a significant distortion of the flatness resulting in all particles inside quickly accelerating and sticking to the inside surface. Here I show 2000, and 8000 charges:

 

 

One can see that the force near the inside surface of the sphere is as high as the force a few radii away outside, even with as many as 10,000 or 20,000 atoms or charges.

[MetaFrizzics]

Summary

 

Of course Newtonian Gravity, including the Sphere Theorem is well substantiated for large quantities of atoms, and is best and most accurately known at the Solar System range of distances. Also, at these distances, electrostatic forces are cancelled out on average quite quickly. In fact, using lasers and complex computations, NASA scientists claim an accuracy of many significant digits for the Inverse Square Law and the Gravitational Constant as we have calculated it. (I will discuss the merits of those claims in another thread.)

 

While the Sphere Theorem and Newtonian Gravitational theory generally can be challenged on any of the three axioms/assumptions previously mentioned above, only on the basis of discrete localization of mass can it be limited in the range of its fitness for physical applications. To challenge Newtonian gravitation on another basis would require formulation of alternate models.

 

QED.

[DQW]

Nice discussion !

 

I have just one question now :

 

Here I show 2000' date=' and 8000 charges:

 

[img']http://lentils.imagineis.com/rouge/2000balls.jpg[/img]

 

One can see that the force near the inside surface of the sphere is as high as the force a few radii away outside, even with as many as 10,000 or 20,000 atoms or charges.

Can you tell me, if there are [imath]\approx 10^{23} [/imath] charges in the shell, what distance (as a fraction of the radius) you must travel inwards before the field drops off to below, say 0.0001% of the field at the surface ?

[MetaFrizzics]

Can you tell me, if there are \\approx 10^{23} charges in the shell, what distance (as a fraction of the radius) you must travel inwards before the field drops off to below, say 0.0001% of the field at the surface ?

I'd be delighted to:

With such a large number of particles there is no real need to do any calculations. In the case of gravity the force is so weak that it is immeasurable at any distance significantly greater than a few thousand effective nuclear diameters (e.g. for a gold atom).

 

The engineer in me is tempted to frame the question in this way, but a much more productive and important approach is to view the entire field inside the sphere, and speak qualitatively about it as we consider a system evolving in time.

 

In the case of electromagnetic forces, not only is there hope of measurement of such effects, they become extremely important, as we use very small numbers of charges in controlled lab settings to measure electrostatic forces and set standards of units. Here the errors can be far from minor and it is important to set theoretical bases for interpretation of results very carefully, and physically construct our experiments with extreme accuracy.

 

I would take the view that the concept of interest is that no matter how small the imbalance in forces, as a system evolves, even particles placed at the very centre where the imbalance will be the absolute least will inevitably be forced to break symmetry sooner or later. This action for a single particle starting at the origin can be modeled as a 'biased' Brownian motion which finally is compelled to migrate away and accelerate toward the inner surface, its variations and uncertainties of velocity gradually converging to more and more efficiently directed path to the surface.

 

At first, imperceptible imbalances will be drowned out in 'Brownian noise' and so the process must at first rely upon a large enough random perturbation to allow the particle to be immersed for a significant amount of time in a given general direction of imbalance. Eventually however, the 'signal' will rise above the noise floor often enough to affect the future path of the particle.

 

From here the particle will begin a new 'life', now characterized by a 'leashed' type of Brownian motion of the most complex kind, in which its overall position is being constrained more and more along the 'normal' (perpendicular axis) to the sphere surface. In its wanderings, the particle will weave in and out of the sphere, but along with other particles will converge in a dust-like cloud enveloping both sides of the surface.

 

This second 'random walk' will be more like the particles in the Rings of Saturn: Most will remain trapped and the ring will narrow, while very rarely, some particular particle will be given 'escape velocity' through a random series of energy exchanges, and 'tunnel' out of the surface to either escape or begin again.

No Single Answer to the Question

By the way, there can be no single answer for such a question as the one you pose, with a simple mathematical result. If you thought the Three Body Problem was a tough nut to crack, this is many orders of magnitude greater. First even from a Classical point of view (simple point-masses and deterministic velocities) The field would have to be constantly recalculated using a perturbation method.

 

Once the mass is discretely localized in space, we can no longer talk about a radially symmetric field, but instead must talk about a 'scattering matrix' which reflects the exact locations of the point-particles making up the sphere.

 

Remember that all along, we used a test-mass and moved it in a straight line along the x-axis. This was possible because in a continuum-style radially symmetric shell of constant density and equal insignificant thickness, every trajectory through the Geometrical centre was identical, and our particle (in the Classical system) would naturally travel a straight line in Euclidean space.

 

Now with point-masses spread at random distances across the surface, every single trajectory of a particle will be unique and complex. No particle will be able to travel in a straight line. In the physical case (say with a sphere of gold atoms, or a charged aluminium sphere) we would have to fire millions of particles through the sphere and measure the results to build up a scattering pattern.

 

Each individual trajectory (even if theoretically deterministic) would be unique and have it's own history with an instantaneous force vector that was constantly changing in direction and magnitude.

 

Every point in space just inside the sphere would have an infinite number of actual vectors, depending upon the speed and direction of the particle passing through it. This is because the previous history of the particle in a deterministic system must also affect the lattice of charges/masses in the sphere according to Newton's Third Law. (Equal and Opposite Reactions)

 

Also of importance is the relative velocity of the particle compared to the force. For very fast or heavy particles, a small bending of trajectory would be expected most of the time, and the scattering matrix would look quite orderly and tractable. However, once we slowed the particle down and lightened it's load, trajectories would deteriorate into the Random 'Brownian Walk' we described earlier.

[DQW]

Can you give me your best guess (to within a few of orders of magnitude) then ?

 

Will the field be below this arbitrarily set threshold of 0.0001% E(surf) upon moving inwards a distance of 0.1R from the surface ? How about 0.001R ? And what about 10^{-6}R or 10^{-12}R ? Or might it get there even sooner than that ?

 

I'd just like to see a number, please !

 

In fact, could you also tell me what this number would be for smaller numbers of charges, like 10, 100, 1000 ?

[DQW]

At 10 charges, it looks like you might have to be essentially at the center (or within a very small fraction of it) for the field to be this low. At 127 charges, I suspect you may still have to travel inwards by over 0.9R for the field to be this low. At 2000 charges, it looks like you'd get low enough after about 0.5R and at 8000 charges this looks like it happens after about 0.2R (I'm just making very rough guesses from your pictures). Could you please look up the actual numbers and let me know ?

[MetaFrizzics]

I'll have a go at providing you with the information you need.

 

In the meantime I am posting this tantalizing graph of the gap between the equation for a solid sphere near the surface and Newton's 1/d^2 formula, for a quantization involving 4000 slices of a sphere (the number of particles would be approaching the millions).

 

 

(text from one of my articles analyzing various models of quantization of mass:)

 

A Closer Look at the Solid Sphere Formula

 

It is even more remarkable when we enhance the formula for accuracy and plot the results. Note the n-1 summation endpoint and +.5 centre-point:

This doubles the accuracy of the finite approximation. The convergence with Newton’s 1/d2 formula is actually tighter than current capabilities for measuring the gravitational constant. Nonetheless we cannot be pleased with the formula from a theoretical standpoint. The reason is that for the Sphere Theorem to be self-consistent, which is a mandatory requirement, it must hold for both hollow and solid spheres. It clearly does not. This means that some other mechanism must be sought to explain the apparent validity of the solid sphere portion.

 

(note: here the jiggling of the plotted lines reflect the rounding errors due to long floating point computational variations/rounding errors. Interestingly, this actually mimics the Brownian-like motion expected from small fluctuations of force. The error in the equation at 4000 slices is still way above the noise floor of the computer algorithm calculating the forces. )

[MetaFrizzics]

If people enjoyed this thread, I could continue with an analysis of the Centre of Mass, and tie them all together.

 

[MetaFrizzics]

I have taken the essential steps I expounded here in deriving the Sphere Theorem, and converted them to a .pdf format document with TOC.

This will match the (2nd) pdf document showing the integration of the formula.

I plan a 3rd document discussing the applications and limitations of the Sphere Theorem as a companion piece.

(1) Formulating the Sphere Theorem (pdf)

 

(2) Integrating the Sphere Theorem (pdf)