(D) Failure of Sphere Theorem at Molecular Level
CONTENTS
1. A Simple
Quantization of Mass
2. General
Method for Discrete Distributions of Mass
3. Discrete
Form of Hollow Sphere Equation
4. Formula and
Graph for Quantized Hollow Sphere
5. Sample
Hollow Spheres of Discrete Particles
6. Summary and
Conclusions with FAQs
1. A Simple Quantization of Mass
In a
very general and straightforward sense, Newtonian gravity and
electrostatics are already ‘quantized’ theories. That is, they
begin and end with ‘corpuscles’ or ‘atoms’, discrete packets
of finite, fixed mass and charge. These packets are viewed as
concentrated point-masses or particles separated by relatively large
amounts of empty space. This picture has turned out to be
surprisingly accurate.
A
salt crystal for instance is a regular cube-like pattern of atoms
held in place by electromagnetic forces. Although rigid, it is
actually mostly ‘space’, and its uniform density amounts to
simply an even spacing in close-up view. While ‘regular’, the
distribution of mass is not really and cannot ever be ‘uniform’.
Instead, the mass ( or charge ) is ‘quantized’ spatially. We
could say instead that the mass is uniformly distributed in discrete
packets, equally spaced either throughout a volume or across an area.
Similarly, we can construct ‘hollow spheres’ at near-atom sizes by careful arrangement of atoms to form regular polyhedrons of various sizes and shapes.
A simple hollow sphere can be constructed out of carbon atoms for instance, with the required uniform spacing of atoms across the surface area, by placing the atoms at the vertexes of a dodecahedron.1 Although the synthesis of these exotic molecules is not easy, there is in principle no limit to the size or complexity possible of such spheres of ‘discrete uniform distribution of mass’. Naturally, one wants to know what the actual gravitational field for these objects is, both inside and out, and it turns out there is a straightforward method of calculating those forces.
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1. See Nazaroo’s article, A
Fourth Isomer of Carbon
, 1972
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2. General Method for Discrete Distributions of Mass
Obviously for a handful of atoms one can simply calculate and sum the forces. For spheres larger than 20 times the size of an atom, where you have a hundred or more atoms distributed over a spherical surface, this is no longer practical. Yet we have not yet reached the size and quantity of particles required to treat the sphere as a smooth continuum of mass or charge spread over its surface.
Consider a sphere, of a size where about 20 atoms evenly spaced will reach around the circumference. As before, consider a test-mass, which we are going to move along the x-axis in a straight line through the centre and off again to infinity. At this size, the actual position of individual point-masses (charges) is not crucial, because our test-mass is not gettng that close to most of them.
Instead, we can construct an equivalent model of the sphere (as in the diagram) by placing rings equally spaced on the surface at about the same spacing as the original atoms. Along each ring we space equivalent atoms, again at the same spacing. The result will be an approximately even distribution of atoms at the desired resolution or spacing, which is all we really need. By inspection, we should be able to see that the overall sum of forces will not be affected by changing the actual latitudes of the individual rings, if we keep the spacing. We could have an even or odd number of rings, or slide the pattern across the surface like wallpaper, taking rings away at one end and adding them at the other.
At this point, the reader might ask, what if we can’t spread the atoms along a ring evenly at the required spacing? Suppose we need an extra half-atom to match up with our tail on some ring: No problem. From our previous analysis, we know that rings of evenly spread discrete points act just like continuous rings. That is, the y and z components cancel and we can simply use our ring formula. From the point of view of our test-mass, it makes no difference whether we have rings of discrete points or continuous rings. As long as we assign the correct total mass to each ring, we will calculate the correct force. We emphasize that this is does not compromise accuracy. It translates directly to the physical situation, at least as far as the force in the x direction is concerned.
In fact, using smooth rings will actually smooth out errors caused by our initial substitute model. It would have had some artificial ‘bumps’ since it isn’t possible to align rings with different numbers of point-masses properly. This isn’t the critical issue it appears, since plenty of polyhedral molecules might actually have such features.
It does matter that we assign the correct mass to each ring. The mass will naturally be based upon the number of particles. Since the particles are equally spaced all over the surface, each particle represents a fixed, equal amount of surface area, and hence particle number is also a measure of surface area. Since we have ‘unquantized’ the sphere in the y and z -direction by using smooth rings, we can now have non-integer values for ‘particle units’. These ‘particle units’ are interchangable with area units.
If this sounds familiar, it should. We could use exactly the same summation equation we developed previously for the smooth sphere. Only this time, miraculously, it is no longer an approximation increasing in accuracy as we increase N. Now it is an exact equation, in which ( 2N + 1 ) represents the actual number of rings, and hence the quantization of the mass (or charge) in the x - direction. An N of 5 for instance would correspond to our current example above, giving 11 rings.
An astute reader might raise the following objection: In the original equation we spaced the rings equally along the x-axis using Archimedes’ Theorem. Shouldn’t the rings in this case be spaced equally along the sphere surface? Yes they should, and this can easily be incorporated into our equation, as we show in the following pages.
But in the present case we wouldn’t even need to do this to get an immediate, qualitative and very accurate understanding of what quantization of any kind does to the force in the x-direction. After all, we would like to know about almost any hollow polyhedral molecule. For this purpose we could assign arbitrary masses to each ring, and space them in almost any rough fashion. This would correspond to replacing various atoms in a polyhedral molecule with different elements, or slightly changing the bond angles.
An important point is that we don’t need calculus ( integration ), and in fact it would defeat our purpose, since we want to keep the mass ( or charge ) quantized spatially, not smear the actual gradient of forces into a blurred average of the real picture.
3. Discrete Form of Hollow Sphere Equation
In order to space the rings equally along the surface, we divide the distance into equal parts. For ( 2N + 1) rings, the distance (and angle) in radians is
phi = ( i / N ) * ( pi / 2 ).
We continue our previous strategy of sweeping i from -N to +N. in whole integers. This again allows the sign to automatically handle ring distances, angle directions, force directions, and both small and negative values for the position of the test-mass on the x – axis.
Since there are a lot of variables, the flow chart will help clarify the chain of dependence.
Each variable is related to those below by a simple formula. This allows us to build an overall formula in terms of the loop counter i :
…And we are ready to assemble a summation formula for the quantized sphere.
4. Formula and Graph for Quantized Hollow Sphere
We have plotted a few sample values. ( N = 2 for instance corresponds to a dodecahedron. ) As we increase the # of rings, ( = 2N + 1 ), we can see the curve evolving from that of a single ring toward the shape for a continuum sphere. Even with as many as 41 rings ( # of particles over 2000! ) there remain significant forces inside the sphere. Thus for small numbers of particles ( mass or charge ) the Sphere Theorem is extremely inaccurate, and this is independent of sphere size, applying for instance to room-size objects with small quantities of charge.
5. Sample Hollow Spheres of Discrete Particles
6. Summary and Conclusions with FAQs
We have shown that when mass is 'quantized', that is, localized in particles or discrete packets in space rather than spread out as a continuum over the surface, the Sphere Theorem fails. Forces are not balanced or neutralized at every point inside a hollow sphere. And the net results of the imbalances cause the whole inside volume of the sphere to be unstable, with a net force attracting particles to the nearest surface. The field strength and attraction increases as the inner surface is approached.
Q: 'So what? Gravity is only an approximation at the atomic level. Big deal.'
A: But this is not the actual case at all. Our findings are not based upon absolute sizes or fixed scales in any way. They are simply a result of the clumping of mass, i.e., discrete packeting, or 'quantization' of distribution. While the Sphere Theorem certainly does fail at the molecular level, it also fails at any size where one is dealing with clumping or discreteness with the aspect at hand. This could be mass, or charge, or any other localized object or attribute which induces a force. While the actual 'size' of particles relative to their spacing may be unknown, scattering patterns clearly indicate that both mass and charge are indeed practically 'point-particles' in their effect. The collapse of the Sphere theorem is based not upon sizes, but 'point-like' behaviour.
The error is more closely related to numbers of particles than sizes. For instance, a three-meter aluminium sphere could carry a static charge of only a few hundred electrons. In this case, the Sphere Theorem would be grossly inaccurate, since the point charges would be spread out and the field would be as uneven as a golf-ball, inside and out! Expecting charged particles inside it to drift free is absurd. If anything, with fixed particles, errors will be amplified at larger sizes and distances.
As another example, being near the surface of the earth using the Sphere Theorem as a special case of the Center of Mass theorem would result in an inaccurate measure of the gravity field, and would mislead us as to the true value of the Gravitational Constant (or its universal component).
( That's right: We don't know the real Gravitational Constant at this time: The calculated value is currently only useful as a coordinator of mass to distance units in the vicinity of the earth's surface. There is no theoretical justification to treat it as a universal constant for all sizes and distances. )
Q: What happens when the test particle passes through the surface of the sphere?
A: Under the discrete particle model, the actual surface of the sphere doesn't really exist. It is only a geometrical surface upon which we locate the particles of interest. It could be 'real', in the sense of it being a thin layer of atoms or molecules, as in the three-meter aluminium sphere containing charges. Or we could simply be interested in the gravitational field of the atoms themselves. In either case, the particles of interest are still particles. The chances of actually having a collision or a near-collision with our test-particle would be extremely small, but determined by a typical scattering-matrix. In most instances, the particle would experience the 'softened' and unbalance field we have shown in the text.
Now and then, a particle passing too close to a concentrated point-mass or charge on the surface would experience extremely high forces. Note that even here, those forces would not in any way cause the test-particle to experience a 'balanced' potential field entering the sphere. Instead, the test-particle would be extremely deflected off it's path. It would not proceed straight ahead as predicted by Newton's Sphere theorem.
Our method of approximation is the most accurate and useful model, because we don't attempt to posit extremely rare head-on collisions (or near-collisions) between test-mass and point-mass on the surface itself. For most real trajectories, our equations will reasonably model the general forces experienced by a test particle passing through.
In passing, we have proposed a new and useful significance for discrete models and quantized summations which take them beyond mere approximations of Newton's Sphere Theorem and place them squarely in the practical category of reasonably accurate models for actual expected and observable effects.
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