Posted: Wed Jul 13, 2005 6:26 am Post subject: The Flaws in the Sphere Theorem
Newton's Sphere Theorem apparently remains a bit of a sacred cow to physicists, because it is a standard integration learned in school, and its results are claimed to be exact. The claim of exactness if of course crucial to the resulting claim of the theorem:
The Sphere Theorem
The gravitational forces from a hollow sphere of uniform density and negligible thickness are balanced everywhere within the sphere, and so any particle within experiences no force from the sphere. (Outside the sphere, particles experience the force of attraction as if the mass of the sphere were concentrated at the geometric centre.)
Posted: Wed Jul 13, 2005 8:56 am Post subject: detestable laziness
Rogue Physicist,
It's good that you question everything but you shouldn't be talking about mathematical limits as if it were a conspiracy theory. Do a computer simulation for n=10,000 or 100,000 point masses on a n-sided polygon of variable size and get a practical result for a cumulative inverse square law. Computer programs can do this remarkably fast and accurately. Vector forces add linearly so conveniently stack the slices with the required spacing between the layers for a final, glorious computation.
If you really have confidence in the outcome and if your suspicions are correct, you will be overturning a fundamental assumption in basic science and engineering and become incredibly famous. What's stopping you?
Please note these findings have nothing to do with absolute sizes.
They are strictly a function of the spacial quantization of mass or charge.
There is no stability even at the very centre, for structures at the 2000-10,000 particle fragmentation range.
The units of distance for these equations is in radii of the sphere.
To get right at the issue, since mass is in reality distributed in clumps, and is not a continuum, the Sphere Theorem is only a valid or useful approximation for very large uniform spheres at a macro-level, involving millions of atoms.
It is not in serious dispute that the bulk of the mass of atoms reside in the nucleus and this is demonstrated both mechanically and vis the gravitational field by the scattering matrix. Most scientists agree that this experiment has already been done to death.
(1) If we were to make a sphere out of a thin layer of gold atoms for instance, the actual gravitational potential field would not at all reflect the result of an integration of the continuum model proposed by Newton's Sphere Theorem, and so the integral is inapplicable to this problem. The *REAL* field would look more like a golf-ball, and there would be no flat field inside the sphere. All particles floating inside would accelerate outward toward the inside surface due to imbalance of forces, no matter how carefully the sphere was constructed.
(2) One might think, "So what? The Sphere Theorem fails at the molecular level. Big deal." But this is not the case at all. The failure is independant of size entirely. It is not tied to physical size, but to the coarseness of the quanization of the mass distribution. This would also be true for charge distribution as in both classical and relativistic electrostatics.
(3) You could also have a 3-meter diameter aluminium sphere which would for all intents and purposes would be a continuum. However, once the static charge on it dropped below a few thousand excess electrons or hole charges, even though these charges would spread out as evenly as possible due to repulsion, the electric field would be as lumpy as gravitational potential of the gold-leaf ball we referred to above.
This thought experiment is all that is required for any reasonable person to see that the Sphere Theorem is an approximation similar to the Center of Mass theorem, and it fails miserably in many physical situations. you don't have to be a rocket scientist to see this.
The Sphere Theorem is not accurate, practical or useful. QED.
The failure of the Sphere Theorem is provable mathematically on many levels, the simplest being known mathematical theorems of topology and tesselation of the plane and spherical surface.
For instance, of course you can tessellate the Euclidean plane evenly with equal sized spheres (which would represent equal sized/spaced charges). Recall a ball is surrounded by exactly six equal sized balls, and forms a 'honeycomb' pattern of touching balls on the plane. (straight rows oriented 30/60 degrees apart)
This is impossible on the surface of a sphere. Lets see why:
It would require either varying sizes of balls, or symmetry-breaking spaces which could not accomodate a 'whole' ball. The physical result in either case would be an uneven field extending into the hollow area of the sphere.
Lets try varying the ball size. This is not possible in the electrostatic case, because charges have fixed discrete values, but with mass-clumps we can vary the density or spread of a clump to allow different size radii of equal gravitational strength in the plane of the tangent to the sphere's surface.
The requirement that adjacent balls touch is just the equvalent of equal spacing since the ball surface represents a surface of equal gravitational (or electrostatic) force around a particle.
Now for each concentric circle of balls around a given starting point, we can shrink the ballsize for that ring, to squeeze up the previous inner ball/ring. Balls can remain touching, but now notice that one geometric feature, the hexagonal shape of each ring gradually becomes the shape of the spherical approximation. The rings cannot conform to circles and still maintain contact with adjacent balls. The two constraints, keeping balls touching and conforming to the sphere surface are mutually incompatible.
You can quickly demonstrate this for yourself by taking a billiard ball and spraying it with a layer of mounting glue. Now dip it in a bowl of small ball-bearings, and try to push around the balls to make a uniform surface.
No matter how many balls one tries to cover a sphere with (even to the tens of thousands) there would be symmetry breaking, which is all that is required to create an imbalance of forces inside the sphere.
Rogue, you don't really seem to understand the significance of the sphere theorem. When was the last time you saw a perfect, hollow spherical shell of enough mass to have a sizeable gravitational field...
The application of the spere theorem is its ability to treat spherical mass distributions as being pointlike. Also, on the inside you can simply ignore the massoutside of your radius. The gravitational field due to the amss inside your radius dominates over any inhomgeneities in the field due to the outer mass.
No, actually. It's you who don't seem to understand the physical significance of the Sphere Theorem.
No one disputes it as an acceptable approximation when dealing with masses the size of the earth, and Solar System distances.
The whole point of questioning the theoretical foundations of current models of gravitation is not to produce better approximations to problems that have already been solved, but rather to move on to the next gravitational theory by searching for the next logical step in any theoretical advance.
The only interest anyone should have in Newtonian Gravitational theory is to probe deeper into the mysteries of physical causality. It's fine for launching satellites. It is of obvious interest as a starting point, since it is the single most commonly used set of physical laws and methods today.
General Relativity, after nearly a hundred years has done very little to change mechanical engineering methods. Only three things come to mind: the perihelion of Mercury, the bending of light rays, and the possible detection of gravitational waves. While the mathematical apparatus of GRT is elegant and sexy, it is almost completely useless for building bridges, designing jets, and navigating spacecraft. Even Einstein, when seeking a new formulation of gravitation started with Newtonian theory and had his own GRT reduce to Newtonian results at slow speeds and 'normal' mass systems, like the Solar System.
It seems reasonable to start with Newton as well as GRT when laying any groundwork for a new theory or modification of current formalism.
Everybody knows Newtonian gravity is only an approximation. You don't need to somehow prove it wrong because no one is claiming it is correct.
GR has no engineering applications??? Ummm, have you ever heard of, oh, I don't know, GPS?? It would quickly become inaccurate if you did not employ GR. I don't know if it's specifically used for navigating planes at all, but it certainly could be. Obviously, is has a hell of a lot of engineering uses.
I still don't understand your dispute.
Is there a personal problem?
If you think it is not worthwhile to discuss or analyze scientific theories, why post here?
Have you ever heard of, say, LOSA? It's the List Of Scientific Acronyms. It currently stands at about 30,000. If you want to single out some meaningful topic, don't use acronyms.
I am not surprised to find out I am not a very knowledgable man at all.
I only recently learned I was a Neo-Nazi high-school dropout:
Oz-bored wrote:
Guess you probably don't have any college training at the very least, which is what I expected. ...OK Mr. On Topic Nazi,...
I am enrolling in night school to learn about the holocaust.
Thank God I found this out after accepting my $100,000 /yr post,
or I might have had to turn it down out of honesty. Now however,
since I signed a contract, I am committed to cashing the cheques.
I have decided to embrace the regret of Liberace,
who, when accused of being a homosexual by a music critic in the newspaper,
responded in an interview,
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