Science a GoGo's Home Page Forums General Discussion General Science Discussion Forum Disproving Newton's Hollow Sphere Theorem

In his Principia Newton claimed that a hollow sphere exerts no force on a particle floating inside (Propositions 70 to 74 of Book I). His argument went as follows:

(1) Imagine a test-particle at the vertex of a cone of fixed acuteness. You can picture the cone as a flashlight beam coming from the particle. The cone makes a circle on a flat sheet of uniform density and thickness. Moving the sheet further away makes a larger disk on the sheet. The size of the disk increases as the square of the distance (d^2) and so does the mass, while the gravitational force of the disk on the test-particle decreases as the square of the distance (d^2), cancelling out any change. That is, at any given distance, the cone traces the size disk needed to keep the attractive force constant.

(images can be provided if I am allowed)

(2) Now imagine the particle is between two parallel sheets: You can picture two equal opposing cones beaming out from your particle like a lighthouse beam. (Equal cones can be made by rotating one line passing through the particle around another line-axis through it.) Either sheet or the particle can be moved independantly, and the force from the disks will remain equal and balanced, as long as the sheets stay parallel.

(3) Similarly, opposing cones will cut out opposing spherical caps on the surface of a hollow sphere if we put our particle and its cones inside. The arbitrary position of the particle and the varying distance of the caps is compensated by their size and mass. Newton proposed constructing more cone pairs, covering the entire surface and showing that the forces inside balance.

(4) Someone might object that the spherical caps marked by the cones aren't really flat disks. Newton responds by shrinking the cones to very small (infinitesimals) so that we can make the caps as flat as we wish.

(5) Another might ask how you can cover the sphere surface completely with circles. Again, Newton argues that spaces between disks can be filled with smaller circles and so on. We can iterate the procedure as many times as we wish to closely cover the surface area.

Your task is to disprove Newton's argument in a few simple steps using high-school level mathematics, in a way that anyone can understand.

It's easier to prove that 1 + 1 = 732

If ten of you give up, I'll post what I have in mind. You can decide if it is easy enough for a high-school student with a reasonable grasp of ordinary geometry and basic mechanics.

I accept Newton's conclusion. The sphere shell must be well shaped with a coradial hollow and rigid enough to not flow to any slight initially more massive parts and revert to a solid sphere. The flowing shell will increasingly attract the originally floating inside objects to the most massive part.

Aloha, Charlie

What I am getting at here is that the Sphere Theorem, which claims that there is no gravitational field at all inside a hollow sphere, is false.

I read the cited part of Newton's Principia to be a claim that universial gravitation is present but balanced so that it dosen't exert force on floating particles.

Aloha, Charlie

Yes. Newton is in this passage claiming that particles inside a sphere float freely, because the gravitational forces balance wherever the particle goes.

Now for the mathematics:

(1) Not all points in the sphere are equal.

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

Similarly, since a cone-axis is a line, only one cone-axis passing through P is perpendicular at the surface. Only cones formed on this axis will make perpendicular disks or spherical caps. Cones on some other cone-axis will cut tilted ellipses or spherical caps.

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. This is because the Centre of Mass theorem fails in close proximity or for significant spreads in distribution of mass, since it is only an approximation.

(4) 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

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. This is why Newton's thought experiment works with parallel planes, but not with spheres.

Similar arguments apply to electric fields which are strong enough for mid easy experimentation. There is stuff light enough to float in air filled hollow spheres to offset gravitational influence. If I use spherical infintismals the tilt is lost but is this fair? I'll think some more.

Aloha, Charlie

This is a higher order correction which is irrelevant when you consider infinitesimal disks to prove the theorem. Newton invented calculus to deal with these sorts of problems and his arguments based on geometry are a bit intuitive.


It is plainly stupid to replace the infitesimal disks by finite ones and then say that the argument fails.


If you have a problem with infinitesimal disks then use modern methods of analyses you can find in your maths textbook.

Proving that a thin spherical shell exterts no net gravitation in its interior is a trivial exercise in calculus. Your incompetence in math and physics is no qualification for your abuse of this forum.

I have no problem doing the integration. My claim is quite different. The integration technique and its result are inapplicable to the physical situation in many instances.

Insults and accusations are unscientific,
that is where the abuse is found. so stop.

Actually, it is not stupid at all to use finite methods here. Mass is *not* a continuum, but localized in discrete packets widely spaced. Only finite methods will be accurate in this case. Likewise for charge distributions:

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.

I agree with Newton and Gauss. Gauss's theorem works for the same reason, because electromagnetic forces, like gravitic forces, vary as the square of the distance. I also agree with you. The sphere as a continuum is an approximation, but more so it is a simplifying assumption. When Newton was talking about spheres he was talking about idealized spheres, not real ones. (And so was Gauss.)

One use would be a microgravity cavity in the center of the moon. For the price of one ~1,700 km entrance tunnel or a network of tunnels, lunar material can be fed from the innermost receeding upwards radius to microgravity processes. Masses of material within the cavity still have gravitational attraction for each other where they can be kept separate by gentle orbits easiest in a high vacuum.

Aloha, Charlie

How deep a radial tunnel as a fraction is needed to cut gravity at its bottom face to 1/4 a planet's surface gravity? Please first simplify in the first round with an assumption of consistant density even though planets tend to be denser in their center. Correction for greater central density is a more advanced exersize.

How much pressure can a carbon nanotub reinforced tunnel wall take?

Aloha, Charlie

Edited to add notice of the advanced exersize.

Rouge, I don't understand what your point is. The multipole expansion is part of the curriculum of first year physics students. It is even mentioned in high school.


And if you use the multipole expansion to calculate the gravitational potential inside the sphere, you'll see that you are wrong about your claims.

Rogue is trying to prove that he, with apparently minimal education, is deserving of a Nobel Prize in physics.

What he has proven so far is that moderators are extremely moderate in their moderation of worthless content: And little else.

Let's have a comprehensive collective discussion on sciece / math / technology points on the web so people will correct educational blind spots. On the web knowlege can be subjected to blind testing meant as the opposite of blind obedience.

There was a section here I need help with in that I can not imagine center of mass being differnt than center of gravity, was there tongue in cheek there or a real issue? What shapes have the maximum distinction between COG and COM if there ever is a distinction?

In psycology there is the concept of a non critical very concrete subconscience that is a very powerful problem solver. If it can solve almost impossible problems than it has done a great service. The subconscience dosen't get the hint in stretching arguments to absurdity easily because it tries to solve any problem it is presented with. I try to be extremely clear to make sure my subconscious is fed well. I enjoy irony when I read it so it has a role. My sense of its abuse then is if it is used to reinforce my group right or wrong feelings which is rare in the books I read.

Aloha, Charlie

There are four different concepts here that are commonly confused, because they are not taught properly in high-school and 1st-2nd year elementary physics courses. There simply isn't time, and later in specialized courses there isn't room:

(1) The Centre of Mass. This concept is used most often in classical mechanics problems. It has a clear unambiguous mathematical definition, but has no physical counterpart or concrete meaning, aside from being a convenient way of managing a system of particles. It is the arithmetic mean of position for each particle in a system weighted by each particle's mass. (for charges weighting is not necessary, so there is no corresponding concept in electrostatics.)

(2) The Geometric Centre. A similar idea, mostly applicable to regular shapes having some form of symmetry, and meant to represent the mean position of a volume in space.

(3) The Centre of Gravity. A point with behavioral properties that help to describe the conditions of balance of a rigid body under the influence of a uniform (or zero) gravitational field. Essentially the opposing axis through a balancing point on a rigid object must pass through this internal point.

(4) The Equivalent Point-Mass. The position in space one must place a point-mass of the exact same mass as the system to be replaced in order to exert the exact same force in magnitude and direction upon another external test-mass.

A brief discussion of the similarities and differences is in order.

(a) For regular shapes of uniform density, or symmetric arrangements of discrete components in a system, the Centre of Mass often coincides with the Geometric Centre, making it easy to locate without having to average all the particles in a system.

(b) The Centre of Mass is used as the common or best estimate of the location of the rest frame for rigid bodies.

(c) The Centre of Mass is useful for studying rotation, since a system of particles (rigid or not) which is free of external forces or in a uniform field will rotate about this point if it contains any stored rotational energy. This is due to the Conservation of momentum and energy.


(d)The Centre of Mass is expected to follow trajectories as if all the mass were concentrated at that point in a rigid body (ignoring things like wind resistance etc.) For example, a spinning hammer may appear to move irregularly, but if the Centre of Mass is followed, this point will move in a straight line through space or trace a parabola if thrown near the earth's surface, obeying Newton's laws of motion for point-masses.

(e) The Centre of Mass is acknowledged to be only an approximation when dealing with gravitational and electrostatic fields. This is because systems that don't have uniform geometric symmetry don't have distributions of mass that exert uniform forces on arbitrarily located external bodies, even when they have uniform densities.

(f) A Key point is that the Equivalent Point Mass location (EPM)is only definable from the viewpoint of a test-mass with a fixed and defined location. If the test-mass is moved, the EPM also moves, when the object under consideration is not uniform in density or radially symmetric in all directions.

(g) An important concept to come away with here is that an irregular object looks different to all other parties and locations in space, even if those objects are point-masses, and so the EPM is in a different location for each observer under the influence of the gravitational force from the object. There is no unique EPM for an object.

I have no problem doing the integration. My claim is quite different. The integration technique and its result are inapplicable to the physical situation in many instances.

Gauss' Theorem fails for the same reason: You are trying to apply an elegant theorem to a physical situation which is inappropriate.


Rogue Physicist does some Integration...

Here is an example of the force experienced by a test-particle moving along the X-axis through the sphere when the mass distribution is discrete points (127 particles forming a hollow sphere):