Flaws with the Sphere Theorem
by Rogue Physicist (Nazaroo, (c) 2001)
A. Introduction *
B. Newton’s Proof for Hollow Spheres *
2. The Law of Projection, Mass and Force *
3. Opposing Disks Balance the Pull *
4. Parallel Planes versus Spheres *
5. Objections and Questions *
2. Most of Newton’s Cones Cut Sphere on an Angle *
3. Tilted Disks Pull Off-Centre *
4. The Center of Mass Theorem *
5. How the Center of Mass Theorem Fails *
6. Watching the Direction of Force change *
7. Tilted Disk Pairs on Sphere Don’t Balance Out *
8. No Special Tricks available *
2. Even the Single Untilted Cone Pair Fails to Balance *
3. Expecting Polar Caps to Balance leads to Absurdity *
4. Balancing Residual Forces between Components: The Last Hope *
5. An Alternate Graph of Forces inside a Sphere *
This paper should be a fun read. We have kept the math simple and added lots of diagrams to make it as easy as possible for people to understand.
What is this paper about?
This paper disproves the Sphere Theorem ( ST ). Briefly, ST is a key part of Newton’s theory of gravity. It says that a uniform sphere acts as if it were a simple point-mass of the same mass, located at the geometric centre of the sphere. It is a special case of the Centre of Mass Theorem ( CMT ).
Aren’t both CMT and ST just handy approximations?
Not really. While CMT is generally admitted to be an approximation in some sense, there is no real clarification as to its accuracy in the current textbooks. The assumption is that is essentially an accurate approximation. The case is qualitatively different for the Sphere Theorem. Here the current belief seems to be that it is an exact claim. Uniform spheres act just like point-masses. Mathematical proofs are often given for this claim without reservations.
Didn’t Einstein’s General Relativity (GR) replace Newton’s Theory?
No. Although preferred as a theoretical foundation, GR is impractical for most problems, including ordinary and celestial mechanics. While GR may be the only ‘approved’ gravity theory, Newton’s methods are the only ones we have for most ordinary situations.
Isn’t General Relativity more accurate than Newton?
Not really. For ordinary low speeds, and smaller masses under moderate gravity, GR reduces down to Newton’s formula anyway, just as it was designed to do. GR was only meant to correct for huge masses and near light-speeds, where relativistic effects are believed to occur. While current measurements are interpreted as supporting GR, it is not at all clear that a Newtonian theory properly developed could not predict the very same results. (see below.) In any case, GR also has its share of problems as well, and may be modified or replaced in the future. There seems to be no clear way to distinguish between the two theories regarding the static cases we will examine below.
Why is the Sphere Theorem important?
The Sphere Theorem is used in a wide range of disciplines and in other theories, not just Newtonian gravity. For instance, ST appears without modification or reservation in Electrostatics, as well as indirectly in General Relativity. It should be understood that any problems with ST will not be restricted to or contained by Newtonian mechanics.
How can disproving the Sphere Theorem affect General Relativity?
It should be understood that GR is also a theory of gravity, based upon the effects of mass (or mass-energy). Although Spacetime is curved by the presence of mass, the basic inverse-square law is presumed to hold on some level in ordinary circumstances. By designing GR to reduce to Newton in normal cases, Einstein left his theory as vulnerable to the same potential flaws and contradictions with experimental data as Newton’s theory is vulnerable to.
In what way is the Sphere Theorem disproved?
Specifically we show that the Sphere Theorem is false for hollow spheres. If however ST is false for hollow spheres, then Newton’s argument for the gravitational field inside a solid sphere is also false. This would be bad enough, but it will be apparent from the nature of the case that the hollow and solid instances of the Sphere Theorem stand or fall together. Current mathematical analyses of the two are mutually contradictory: Therefore the Sphere Theorem as currently understood is self-contradictory. This leaves us with several options, none of which vindicate Newton, Electrostatics, or General Relativity, except partially.
What is the end result of the analysis?
In essence, Newton’s general theory has to be revised. As a consequence, Electrostatics also must be modified. Finally, the consequences for General Relativity must also be carefully considered also.
Can any of the theoretical options be tested empirically?
Yes. We believe that in spite of the difficulty in testing gravitational formulas themselves beyond 3 significant digits or so, the nature of the findings in this paper indicate that experiments in Electrostatics can establish one way or another the physical situation and assist in choosing our theoretical options. We first have to know what we are looking for, in order to make those tests.
|The Sphere Theorem says that there is no gravity inside a hollow sphere.|
Picture a deep well, with sloping sides and a flat floor. The steepness of the sides reflects the strength of the force outside the sphere, and the flat floor represents the lack of force inside. As a final touch to our model, we might add a small 45o reflector at the floor’s edge to deflect falling balls across the bottom. A marble will roll toward the well and speed up. Inside, a resting ball can move anywhere but has no reason to, due to the lack of any slope, or potential energy difference.
Why does the Sphere Theorem matter? Newton used his Theorem for a Hollow Sphere to explain the gravity field inside a solid sphere. He did this by imagining a series of hollow shells, one inside the other, building up a solid sphere out of them. The Sphere Theorem is also important because it is a special case of Newton’s Center of Mass Theorem. We will also examine that too.
How did Newton go about proving his view of the inside of a sphere? He creatively combined several basic ideas: the Law of Projection and cones, uniform density, the relation of area to mass for flat surfaces of constant thickness, and his own self-invented calculus. We will follow Newton’s argument and explain his steps in simple terms.
2. The Law of Projection, Mass and Force
By this law, the area of a shadow increases as the square of the distance. Here we use a real square as an easy-to-see example, but any convex shape would work fine. Notice the straight lines that trace the rays of light that cast the shadow’s edge:
Similarly, the area of a cone’s base increases as the square of the cone length, if we keep the cone-angle constant. Lets place a test-mass at the vertex and point the cone at a flat sheet of uniform density and thickness. Moving the sheet various distances away, we can use the cone to mark out a disk on it. The mass of the disk will increase as the distance squared, just as the area does.
Moving a fixed sized disk away would have decreased the force with distance, but since the size and mass of the projected disk increases proportionately, the force stays constant regardless of distance. The cone acts as a guide as to how big a disk is needed at any given distance to keep the force upon the test-mass at a fixed value.
3. Opposing Disks Balance the Pull
We can take two opposing cones with equal cone-angles, along with some appropriate disks, and place our test-mass in the middle at the vertex, balancing the forces pulling in opposite directions: The smaller disk pulls on the test-mass just the right amount to balance the force pulling it the other way, because it is closer.
Suppose we extend opposing cones from a test-mass floating between two parallel planes. The disks marked by the cones exert identical fixed forces upon the test-mass. The disk forces are constant and equal regardless of where the test-mass is placed. We can also move the parallel planes independently as well, and the force still stays balanced.
It seems reasonable to assume we can always match any part of one plane with a piece from the opposing plane, so that forces balance, and particles float freely between the two (infinite) sheets.
4. Parallel Planes versus Spheres
Now we place our test-mass and cones inside a sphere: In a similar manner, the spherical caps marked by the intersection of our cones should balance too.
Newton’s idea was that opposing parts of the sphere could be mapped off by opposing cones of equal angle, packed next to each other. Every part of the sphere could be accounted for, and matched to a corresponding part on the opposite side, balancing the force.
Although spherical caps aren’t exactly the same as flat disks, Newton argued simply that smaller disks were flatter. By making the disks as small as we like, we can also make the spherical caps as flat as we wish, removing any inaccuracy. So by covering the sphere with very many small disks, we can account for the whole surface, and correct for the distortion that would result from larger, more curved disks.
Although there would be spaces between the cones if they were equal sized no matter how tightly packed, you could simply fill those spaces with smaller cones and so on, covering the surface of the sphere as completely as desired.
5. Objections and Questions
a) Spherical Caps Aren’t Disks: The first objection, namely that the sphere-pieces are really spherical caps, not disks, is apparently not a problem: We can make the cones narrow enough so that the caps become as flat as we wish. The sample size becomes infinitesimally small in the limit. The approximation to the actual surface area becomes virtually absolute the same way a circle is more closely approximated by inscribed polygons with more sides.
b) Can you really accurately cover the surface of the sphere? This is a fair question: Even tiling a flat plane using circles is not trivial: each smaller size sphere added then requires another smaller size to fill the new gaps. You really need an infinite number of sizes and disks. Still a procedure seems definable, and while tedious, any level of accuracy for our approximation appears achievable. Although the surface isn’t really flat, you can achieve ‘local flatness’ in the same way that General Relativity does by using a local ‘Special Relativity’ Euclidean Spacetime in the small picture, and spreading out curvature.
In fact, one could probably relax the constraint that the cones be circular. The Law of Projection works for almost any shape, including triangles and squares. That is what the basic cone argument rests upon for calculating mass, assuming uniform density and thickness of surfaces. In fact, globes are often divided into quadrants using great circles and latitude circles. The result is simple rectangles and triangles, and if an even number of great circles is used and an odd number of ‘latitude’ lines, you automatically have equal areas and matching opposing shapes.
Since the test-mass could be anywhere inside the sphere, it follows forces are balanced everywhere inside it. Any misgivings we may have must simply be our intuition misleading us, as often happens in science and mathematics.
Now that we have the word from the very founder of modern physics and co-inventor of the calculus, what can we possibly add? Rightfully, we must acknowledge the brilliance and originality of Newton. Nonetheless, the proof needs closer examination.
Firstly, all points inside the shell are not equal. There are two distinct sets of points inside a sphere. Set 1 contains only one point, the origin, or geometric centre (GC), equidistant from all points on the surface. Set 2 contains all the other points. All the other key geometric features follow from this.
Let P be a point anywhere inside a sphere, but not at the centre. Only lines passing through a sphere's centre will pierce the surface perpendicularly. All other lines pierce the surface at some other angle. So a line passing through P must also pass through the centre to be perpendicular at the surface. Only one line passes through both P and the centre, and is perpendicular at the surface.
2. Most of Newton’s Cones Cut Sphere on an Angle
Newton’s scheme is to cover the surface using cone-pairs. Each pair must have a different axis, so only one pair can be perpendicular at the surface if P is not at the centre. All the other cone-pairs pierce the surface on an angle, and make caps or disks which are tilted relative to P.
3. Tilted Disks Pull Off-Centre
A point-mass is only pulled directly toward the centre of a uniform disk when the point-mass lies in the same plane as the disk or when the point-mass lies on the axis of the disk. That is, whenever a disk has any other tilt relative to the point-mass, there is a residual force toward the nearest disk edge, pulling the point-mass off-course from the disk centre.
Does the force really change as a disk revolves around its own centre? Physics students will immediately ask how this statement can be reconciled with the Centre of Mass theorem. Every rigid body, whatever its shape, has a ‘centre of mass’, the weighted average position of all its parts. This fixed point acts as a point of balance. If you want to push an object straight without causing it to spin, you apply force along a line through this point. According to Newton, every object behaves as if all its mass were concentrated at this point.
However, we can’t appeal to the Centre of Mass theorem here, since it is actually only an approximation. It breaks down with larger objects at close range. The Sphere Theorem is just a special case of the Centre of Mass theorem, which is what we are trying to prove (or disprove). To see why the general case is only an approximation, let’s review it in more detail, so we can understand why it fails.
4. The Center of Mass Theorem
Every rigid body, whatever its shape, has a ‘centre of mass’, the weighted average position of all its parts. If you mark this point on an object and throw it, the point will trace out a parabola just as if the whole object were a point-mass located there.
If the object is isolated and drifting through space, this point will move in a straight line, even while the object is spinning. The centre of mass is the very point about which an object will naturally spin.
At a glance, we know the centre of mass and the geometric centre are identical due to symmetry. And speaking of that, the disk is radially symmetric on its axis, and bilaterally symmetrical around every plane containing the axis. In the light of the Centre of Mass theorem, it seems especially ludicrous to insist that the gravitational force changes as we turn a uniform disk right on its centre. Yet this is precisely the case, as we shall see. For edge-over-edge spin, the force changes in both strength and direction.
5. How the Center of Mass Theorem Fails
The balance of forces from all parts of the disk is only strictly maintained along the axis of rotation. Only from a spot along this axis does the distance to moving parts stay constant.
For test-masses facing the spin, the movement can be roughly represented simply as a pair of point-masses rotating around the GC in the same plane.
Although changes in distance seem balanced, the forces acting on the off-center test-mass are not balanced at all, since they follow the inverse-square law.
To show this without mathematics, you can simulate the gravitational field using a compass and a ball of unmagnetized iron in the same horizontal plane. If the force between ball and south pole, and between ball and north pole were equal regardless of compass swing (corresponding to tilt above), the needle would sit wherever you placed it, instead of turning toward the ball. Magnetism also decreases as the square of the distance, and gives us a feel for gravity. Gravity is simply too weak between small objects to measure at home, without extreme cleverness.
6. Watching the Direction of Force change
We can easily show the change by using the parallelogram rule. The final vector will always point toward the true location for an equivalent point-mass ( EPM ), and this is not usually at the GC. During a tilt, the closer part (a) will always be a greater angle from the test-mass, since both (a) and (b) are always an equal vertical distance from the GC.
For the final vector to point along the GC line, the furthest part would have to give the greatest force, to counteract the smaller angle. This is clearly impossible.
This is independent of the inverse-square law, and is true for any force which decreases with distance.
7. Tilted Disk Pairs on Sphere Don’t Balance Out
Could disks balance their forces in spite of tilt? Yes, but only between two (infinite) uniform parallel planes, where the tilts cancel. Then masses between the planes can indeed experience zero net force. Newton's claim for hollow spheres actually turns out to be true (in theory) for parallel planes.
Opposing disks on a sphere have the wrong orientation when they tilt:
We can understand intuitively that even though the direction of force is off-center because of tilt, we can exactly counter that with an equal and opposite tilt on the opposing disk, without having to correct the direction change from the tilt. But on the sphere this is impossible: The disks actually double the error!
8. No Special Tricks available
Except for the one (untilted) pair, disk pairs pull off-centre, exerting a net force upon the point-mass toward the nearest inside surface. The net pull is always along the axis through P and GC of the sphere.
If individual pairs of cones/disks leave a net pull, can we construct pairs of pairs in some way, so that the residue again balances out? Could the net force of one pair of disks be balanced by another pair of disks? (i.e. groups of four disks, or some larger number?)
It seems not: To continue supporting Newton’s argument and maintain consistency, we have to keep constructing more disk pairs based upon opposing cones. Only this way can we continue to tile the surface completely.
Since all the nearer pairs that can be formed on the near side of the sphere will pull in the same direction, we must look to the far half of the sphere for counter-forces. But even here, cone pairs and disk pairs leave a residue which pulls essentially in the same direction: toward the nearest surface. Every disk pair we can envision just adds fuel to the fire, pulling the particle away from the center and toward the inside surface of the sphere.
Newton sought to avoid questions of error in approximation by taking the cones and disks down to infinitesimal limits. But Newton’s argument can’t just be extended down to points, as he basically did here.
Of course tilt will be insignificant if the disk is virtually a point-mass relative to its distance from the test-mass. The effect of tilt really does vanish in the limit. Wait! Didn’t we say the tilt can’t be ignored? Yes! Or more accurately, the information contained in the tilt. The answer lies in what the tilt is really doing for us in the finite case. Whatever the disk size, the tilt and position is approximating the curvature of the spherical surface, which is very real, reflecting the actual distribution of mass.
Nothing is gained for Newton’s argument if we reduce the ‘tilt’ effect of each disk by resizing downward. The curvature and mass distribution information has merely changed form. Curvature becomes expressed less and less by tilt and more by the individual locations of the tiny ––––disks making up the surface.
You can divide a circle into pieces which are locally virtually straight, but this has no real effect on curvature. Putting the pieces together puts us where we started. When curves are chopped into microscopic bits, the ‘local curvature’ of the segments or even tilt per se, may not matter, but their mass and absolute location in space does. In the big picture the curvature information is gradually passed over from tilt to position. In fact, if we chop a circle into small enough pieces, those pieces can be rotated or bent without significantly affecting the curvature of the circle, or the distribution of mass and its gravitational field.
We must take into account tilt (or rather the actual distribution of mass) for accuracy. It follows that if the appearance of balance is achieved by ignoring tilt, the ‘balance’ is actually a convenient but naive illusion, or else two hidden errors are cancelling each other out.
The idea of shrinking the disks down to a size where tilt is insignificant is so compelling that it’s failure seems anti-intuitive. So let’s look carefully at why it does indeed fail. Start with large (approximate) disks: These can be treated by slicing them into strips of constant width and thickness, as finely as needed to minimize edge effects. Now let’s examine one such strip, from which we can extrapolate results for all. This curved line segment is just the two-dimensional case of the approximation process.
In the first picture we can clearly see that simply tilting the line would better approximate the distribution of mass along the original curve. Instead let’s follow Newton, dividing the line and approximating with smaller perpendicular lines (ignoring the tilt of each segment). Newton’s method does indeed give a better and better approximation of the original curve, although individual segments will always have less than true tilt, in some cases actually increasing the error. Only by using infinitesimals will the method actually converge to the correct answer.
Has the tilt really been ignored? No. Tilt is real: It approximates the actual distribution of mass, which is fixed and remains unbalanced. In the process of breaking up the line into smaller segments, the pieces have been re-oriented and moved in space to better conform to the actual distribution of mass. All that has happened, is that the information (and the unbalancing force) originally contained in the tilt has changed into a form more directly representing the distribution of mass.
|The point is this: Newton’s method may very well converge to the right answer. But his argument, which depended upon balanced forces between opposing disks is utterly false. If the forces inside a sphere do balance after all, they must do so in a different manner and it must be for other reasons than the ones Newton gave.|
2. Even the Single Untilted Cone Pair Fails to Balance
|If we use Newton’s own trick of breaking up the surface into smaller disk pairs, it is clear that all the pairs will have tilt.|
The tilt will simply be the tangent of the sphere’s surface, regardless of disk size, which will always be perpendicular to the sphere’s origin (but not to the test-mass!). It is important to grasp this: The tilt does not vanish if we break up disks into smaller disks. Instead the tilt has a value, namely, the surface tangent. A plane that cuts a sphere will always be parallel to the tangent at the centre of the cap created, regardless of size. This is because the curvature is constant.
While there may be slight errors in approximation of tilt due to the small directional difference between the geometric center of an approximating disk and the surface center-point of a spherical cap, the overwhelming majority of disks will retain a basic tilt in the same direction, with a cumulative net effect pulling the test-mass toward the nearest surface.
These tilts will always add a residual upward pull, and the vertical components cannot cancel out. Only the horizontal components will balance, due to symmetry. Even the one untilted pair is really made of many smaller tilted pairs. The polar caps don’t balance (unless they vanish entirely), and so there is nothing left of Newton’s original argument.
This is a reasonable expectation, because the fixed uniform curvature of the polar caps should not be compatible with a gravity law that is non-linear (inversely proportional to the distance squared between points), combined with arbitrary positions for the test-mass. That is, unless it happens in some kind of special circumstance by symmetry or via a plain compensating mechanism.
3. Expecting Polar Caps to Balance leads to Absurdity
Breaking up the surface into zillions of tiny cone-pairs is a flop. In fact, we have to go the other way to salvage the theory: Instead of creating many tilted cones, we need to look to the one untilted cone. Suppose We accept Newton’s claim that perpendicular polar caps will balance. This actually imposes a much stronger constraint than first suspected: Since we can arbitrarily make one polar cap as flat as we wish, not only must the opposing forces balance each other, but they must also exactly equal their flat disk counterparts!
That is, any untilted piece of a sphere cut by the cone must have the same pull as a flat (untilted) disk cut by the same cone! Somehow the curvature must ensure that the area (and mass distribution) increases just enough to keep the force constant, as the surface stretches away from the test particle.
The cone argument requires flat disks to exactly equal convex polar caps. Put another way, any arbitrary uniform curvature with no tilt will balance at the vertex! Otherwise, extending the argument from disks to polar caps is invalid, even for the untilted cone pair. But we know this is false from using Newton's own argument. (see previous page: 2. )
4. Balancing Residual Forces between Components: The Last Hope
All is not lost yet however, because we can relax our requirements: The force does not need to be balanced at the vertex of the cones, because we still have the rest of the sphere to account for. We could still save the Sphere Theorem for Newton if we could show that any net force produced by the polar caps was counter-balanced by forces from the ‘pumpkin’ shaped remainder of the spherical surface:
In the diagram at left, the polar caps are exerting a net pull to the right on the test-mass. To compensate for this, we must propose an equal and opposite counter-force pulling to the left, from the remainder of the sphere, as in the diagram below.
The curves below each picture show the strength and direction of the force from the system as the test-mass is moved back and forth along the horizontal axis. Upward on the graph means a pull to the right, and downward means a pull to the left. We know what the basic forces will look like between the caps. The position will be unstable: pull will increase as the test-mass approaches either end, and will accelerate toward the nearest pole. To counter this, the forces inside the midsection must do the opposite and possess a stable equilibrium point.
Again, due to symmetry we assume only horizontal net forces are possible. But now we have moved from Newton’s incredible idea that polar caps balance, and hence are equal to flat disks, to a new idea that can only be described as fantastic:
The changes in force on the test-mass from any arbitrary pair of polar caps are exactly matched by the changes in counter-force from the remaining mid-section. This is the hidden demand of the Sphere Theorem! That there could be a finite construction that could satisfy this demand is incredible. That it should be a radially symmetrical sphere seems miraculous and fortunate. It goes without saying that a proof for this solution is hardly trivial.
5. An Alternate Graph of Forces inside a Sphere
Forces are balanced (=zero) only at the exact centre of the hollow sphere, and net force increases outward in the direction of the nearest part of the sphere surface. The true gravitational field inside a hollow sphere is not zero, but a spherically symmetrical gradient increasing with distance from the centre to the edge, set by the density of the surface.
Our new two-dimensional model of the force appears below. Immediately one can see that even the one point where the forces are balanced is unstable. All particles in a hollow sphere will fall into and stick to the inside surface.
A. No. Newton did not actually develop a mature method of multiple integration to tackle such problems, and so did not actually carry out his program or test his prediction. But suppose Newton had actually done a series of finite approximations using smaller and smaller disks: He would have found that the forces, instead of converging to zero everywhere, converge to a value dependent upon distance from the centre. As the sphere is more closely approximated, so is the gravitational field. It’s hard to fault one of the co-inventors of calculus for attempting an ‘infinitesimal’ style argument to reason out the gravity inside a sphere. This had been one of the most fruitful approaches to geometrical problems for many centuries.
Q. Could Newton be right anyway?
A. It should be noted that any credible ‘proof’ for Newton’s claim concerning hollow spheres must now rationally and with certainty account for the contradictory results of our analysis above.
One way to try to salvage Newton’s result would be to carry out a complete calculation by some reliable method to see if indeed the forces balance in the case of a uniform hollow sphere. Especially important will be the hidden assumptions and axioms entailed by both the method and the interpretation of the results.
Another way to try to save Newton’s result, would be to postulate a hidden mechanism or governing adjustment: We could suppose there was a hidden solution or mechanism which could acount for the results in spite of the fact that the opposing parts of a sphere don’t balance in a simple way.
There is one final way to hang onto the Hollow Sphere claim: by abandoning the primal formula for point-masses itself, specifically, the ‘inverse-square’ portion in the denominator. A new formula probably wouldn’t look anything like gravity as we know it, if it were even possible.
Q. What caused Newton to make such a fumble?
A. The answer is probably his own centre of gravity theorem. Newton must have had some understanding why his theorem was so remarkably accurate for small objects near the earth’s surface. He surely related it to the relatively huge radius of the earth, and the resulting virtually flat gravitational field. He was probably thinking along these lines when he constrained his cone-angles to infinitesimally small values. He was so confident of his method, that he gave insufficient thought to his argument, and failed to describe the real mechanism, if any, by which the field inside a hollow sphere is null.
Q. What has been shown here?
A. It is important to note that we have not simply disproved Newton's proof. The disproof goes much further, clearly showing that Newton's description of the gravity field inside a hollow sphere is wrong. This has important implications for any further 'proofs' or notions about both hollow and solid spheres built upon Newton's premise. Any claims about spheres based upon false descriptions of the gravitational fields or their causes need re-examining. All bets are off. So many things are interdependent in Newtonian theory that any discovered weaknesses affect a great deal of theorems and claims.
Q. What about modern proofs?
A. We have found that available modern ‘proofs’ all suffer from similar flaws in method and argument. Any truly modern proof must also account for all the objections uncovered above, with an as yet undiscovered mechanism or description of forces and balances. We will examine modern ‘proofs’ in the next section.