Tuesday, July 31, 2012

Newton's Gravity Waves (Part I) Dumping Center of Mass

 Many are under the impression that 'Gravity Waves'  are a prediction of Einstein's General Relativity (GR), and their discovery would be a proof of GR against Newton's Gravity Theory (NGT).

This is actually completely false.  A quick review shows this is not the case at all.

Newton vs. Einstein

On the one hand, the confusion was mainly caused by Newton's state of knowledge at the time.  Newton had no suspicions concerning an absolute limit to speed, and so conceived of gravity as an 'instantaneous' force, which took no time to propagate between locations.
Newton had no need of an elaborate mechanism such as a 'field', a wave, or a spray of particles to transmit gravitational force between objects.  It was simply an axiom that gravity was 'instantaneous'.  This force was treated as a 'static' or unchanging, unmoving force, based on distance and mass alone, independent of time.

Forces between masses

Einstein on the other hand, had already interpreted time geometrically as a 'special' 4th dimension.  This went beyond a mere charting or diagramming technique, but became fundamental, with the suggestion of a 4-dimensional geometric 'manifold' known as Spacetime, combining both space and time.

Minkowski Space

Einstein then created "Special Relativity", a theory about electromagnetic waves, which posed that light was made of massless particles (photons) that always traveled at "lightspeed".  This theory seemed to suggest that no object with mass could travel as fast or faster than light.

Light Speed Barrier

Einstein now incorporated "Special Relativity" into his new Gravity theory, his "General Relativity".   If gravity was to be treated as a 'field', like other forces, it would also be limited in its speed of propagation, and might be mediated by 'particles', now called "gravitons".  One could expect to detect 'gravity waves' traveling through Spacetime at the Speed of Light (or slower).

Basic  Newtonian Gravity

Newton's Gravity theory however also predicts Gravity waves, when stripped of its naive baggage.  To do this we have to keep some notions, and reject others.

Fundamental to Newton, are the following Axioms:
(1)  Space is three-dimensional and Euclidean.  This is one of the features that distinguishes Newton from Einstein, and modern formulations of Newton use a Euclidean Spacetime Manifold.

(2)   Gravity is fundamentally a static force, based on mass and distance alone, and independent of time.  This is how Newton formulates it at a fundamental level.  All dynamic effects of Gravity via motion, are based on treating Gravity as an instantaneous force, and applying it using Newton's equation, in combination with Newton's other laws of motion.

(3)  The Gravitational force falls off according to the Inverse-Square Law.   We can accept this as fundamental,  because its apparent failure at galaxy-distances may be accounted for in a variety of ways.   The same Inverse-Square Law in regard to electric charges, and appears to hold universally:
"...experimental results reveal that the validity of its inverse square nature can be unassailable almost to a
certainty at the macroscopic level, the length scale of which
has been shown to be of the order of 10^13 cm by laboratory and geophysical tests reviewed above. As for the microcosmic scale, the well-known Rutherford experiments on the scattering of alpha particles [indicates] that
Coulomb’s Law would be valid at least down to distances
of about 10^−11 cm, which is roughly the size of the nucleus.  Modern high energy experiments on the scattering of electrons and protons proved that Coulomb’s inverse square law was successful even down to the fermi range. Thus, the evidence from experimental results reveals that the inverse square Coulomb’s Law is valid not only over the classical range, but deep into the quantum domain also, a total length scale spanning 26 orders of magnitude or more: this range is impressive but still finite.
- Experimental tests of Coulomb’s Law,
Liang-Cheng Tu and Jun Luo (2004)
 Thus scientists tend to prefer to keep the Inverse Square Law as a fundamental concept for such forces, look for other explanations for anomalies in star-orbits around the galaxy.

Rejecting the Fixed Center of Mass Concept

Next, we remind the reader that there is a key problem with Newton: The Center of Mass concept (CM) contradicts the Sphere Theorem (ST).

This is not an experimental issue, but a problem of self-contradiction between two ideas that have been naively combined in popular expositions of Newtonian gravity. 

Newton's Sphere Theorem (ST)

Newton claimed that a Uniform Solid Sphere (uniform in mass density) acts just as a point-mass of the same mass, located at its center.  Thus the force between two spheres (e.g. celestial bodies) can be calculated by simply using the distance between their geometric centers:
 (There is a minor qualifier, namely that the distance 'd' is greater than that given by adding up the radius of each object, ensuring that the objects don't overlap, i.e., share mass.  Rigid objects should be able to approach to touching without a failure of the formula).

In Newtonian Gravitational theory this is not considered a mere approximation (i.e., true for distant objects only), but a fundamental theorem.   Any anomaly caused by say, equatorial bulge (from spin) or shearing (from proximity effects) would be accounted for by deformation of the spheres and displacement of mass, not any kind of failure of the concept or of the formula's predictive power.

The Sphere Theorem is "proved" by extrapolating from calculations for hollow spheres and combining the results.  Thus comes the importance of Newton's claim and 'proof' of the force exerted by a Hollow Uniform Sphere of Negligible Thickness (HST).   Newton's original 'proof' depends upon a result of infinitesimal Calculus, (the so-called mathematical "Sphere Theorem").  We will look again at this in a moment.

Gauss' Law for Gravity

Meanwhile, Newton's original proof was neglected, in favour of a mathematical result by Gauss, applied also to the Electric Field.  Gauss' Law is built from a different approach, that of classical 'field theory'.

 The Center of Mass Concept (CM)

 The Center of Mass concept (CM) builds upon and generalizes Newton's original idea with the Sphere (ST).    The idea behind a Center of Mass is intuitively attractive.  In a static case (no motion), every massive object must exert a fixed force upon another nearby object.

The CM for a Half-Sphere for instance, is defined and calculated as being fixed at 1/3 the distance along the 'radius' from the flat side. 

The Equivalent Point-Mass (EPM)

 The force an object exerts on a test-mass will be calculated from Newton's equation, by treating it as a point-mass, with distance and direction measured from the CM of the object to the CM of the test-mass.
This force will be a fixed size and direction, and so becomes a vector.  Any such force-vector can be represented by an equivalent point-mass (EPM) at a specific location.  Note that this mass could be anywhere along the direction-axis until we specify the actual mass of the replacement.  By definition, the EPM is formally given the same mass as the object it is to replace, fixing its location. 

 Using this definition and usage, the EPM is both a hypothetical object (a point mass located so that it exerts the same force as an object under discussion), and also a local position in space, relative to the geometry of the object being replaced.

With a sphere, the EPM is placed at the Geometric Center (GC) by fiat, and should in any case be placed somewhere along the axis between the two objects, due to symmetry.  The direction of the force vector (along this axis) is not in dispute. 

It is also assumed by Newton that the distribution of the mass (near and far halves) in a uniform sphere balances out as well, and results in nailing the EPM at the GC. 
Note that this second result doesn't simply follow from the symmetry of the sphere, since the viewpoint of the test-mass is from outside the sphere, not from the center, and the only symmetry from that viewpoint is radial  around the center-to-center axis.   By inspection, we can see the following embarrassing facts:
(1)  If the Center of Mass (CM) of each half-sphere is fixed relative to the half-sphere's geometry, (e.g., 1/3 along its axis of symmetry), then the CM of each half is an equal distance from the CM of the whole sphere (by symmetry and inspection).

(2)  The force exerted from each half cannot be equal.  The nearer half exerts a greater force.   Suppose equivalent point-masses (EPMs) are placed at each location (each half the mass), to replace the half-spheres. 

(3)  Suppose each EPM is moved toward the CG of the original sphere:  The far EPM does not increase in force the same amount as the near one decreases, because each is a different absolute distance, and the force must follow the Inverse Square Law, meaning it is non-linear with distance.  Equal adjustments in distance at different locations cannot have the same value. That is, the two EPMs for the half-spheres cannot equal in force the EPM of the whole sphere.
 Moreover, calculating total force by addition of vectors using the CM gives contradictory results, because the force is less if we divide the sphere into halves through the axis, and more if we divide through the plane perpendicular to the axis through the sphere and test-mass. 

 For a fuller treatment of the definition and method of calculating the Center of Mass and EPM, see our Article on the CM here.

 Since the Center of Mass concept is in fact self-contradictory  with extended objects and close distance, we must reject it as only an approximation, which fails when the radius/extension is in the same order of magnitude as the distance.

Thus physicists naturally reject the CM in favour of the Sphere Theorem. 

But if the naive and oversimplified method of calculating the CM and EPM is dangerously false, what is really going on?  The force must be deterministic, and the EPM must be defined somehow.  Far worse, we've really proved too much.  If the CM concept has problems, what kind of confidence can we have in the Sphere Theorem? 

The problem is more slippery than we might think.   For instance, suppose that Newton's claim about spherical objects is also an "approximation"?   How could we test it?  A theorist might suggest that the choice to use the Geometrical Center of a sphere is possibly arbitrary, meaning that the correction might only mean an adjustment of the Gravitational Constant.  Or perhaps the error is 'self-correcting' in that the error for one sphere-size balances the other.  The pragmatic experimentalist might simply say, "Measure the Gravitational Constant, and use it.  Don't worry about esoterical questions."

Sunday, July 29, 2012

Newton's Black Hole

Yes, it may surprise many, that the concept of "Black Hole" goes back, not to Hawking, Schwartzchild, Einstein, or even Maxwell, but Newton.

Newton's attempt to put the intuitive and useful gravity equation on a secure footing, required many unusual concepts, the most remarkable being the Uniform Hollow Sphere of negligible thickness.  This indeed, was the first "Black Hole", once Newton decided to assign it the most remarkable property: it is the only Newtonian object that can create a space inside itself which exerts zero gravitational force upon any massive test-object, and vise versa, regardless of the size/mass of either the sphere or the object, and irregardless of the test-object's shape.

Newton had decided (probably through clever self-deception,) that the gravitational pull would be perfectly balanced inside such a sphere, and the net force would be zero.

Outside its surface however, the sphere would exert the same force as a point-mass of the same mass, located at its Geometrical Center (GC).

Hollow Sphere Exerts Gravity Outside its surface only.
We are justified then, in portraying Newton's Hollow Sphere as a kind of "Black Hole", in which the forces are represented as light colors of increasing strength approaching the surface, but having no color inside (representing zero gravitational force).

Newton's Gravitational "Black Hole" (shading indicates field strength)

Outside its surface, objects flying by would travel in elliptical orbits, or follow the same path just as if they were circling a point-mass of the same mass, according to Newton.  Below we trace the orbit of an approaching test-mass as if it were trapped in an orbit and otherwise 'unpowered' and simply free-floating in space.  It is assumed to have had an already existing (constant) velocity before becoming entrapped:

Orbital Path of a Body trapped in gravitational Pull of Hollow Sphere

We mark the path around the Center of Mass and Geometrical Center, just as the test-mass would travel if the sphere were very small, but of the same mass.   Here the CM/GC is marked as a small dot.

Note however, that once the object pierces the surface of the sphere, it would no longer feel the pull, and would continue in a straight line (in Euclidean Space) due to inertia, and without any friction or force, would maintain the same velocity (speed and direction). It would continue coasting straight, until it exited the sphere at the opposite side.

At that time, the test-mass would again take up an elliptical orbit, although offset, and starting up rather sooner than expected (after a short time inside).

This new elliptical orbit would not have had the benefit of acceleration (and balancing deceleration) while inside.  Thus, it exits at the same speed it would have exited, but at a different than expected point, and at a different time.

One can see that depending upon the angle of entry, one can have almost any pattern of precession of the orbit, with accompanying arbitrary changes in phase or timing.

Orbit Displacement of a particle Passing through Sphere
Given that we could also be talking about electrical charges as easily as gravity, one can see the potential for all kinds of phenomena in regard to orbital patterns, including quantized and stable orbitals with synchronizing 'beats'. 

It doesn't take much to picture molecular arrangements which would stabilize or enhance synchronization in interesting and unusual ways.

Newtonian theory then, has plenty of potential for mechanisms of quantization without resort to "unknown" causes.

Saturday, July 28, 2012

Disproving both Newtonian Gravity and General Relativity in Five Minutes

Newtonian Gravity and Einstein's Gravity attempt to describe/explain the apparent 'force' between massive bodies.

 Both theories offer similar shortcut equations to calculate these forces between any two masses:

 Einstein's General Relativity Field Equations conveniently reduce to the same formula as Newton's equation as a very close approximation, where the relative curvature of Spacetime can be ignored (locally) because of the scale of the bodies and the distances involved. Newton makes no claims at all about the possibility of non-Euclidean effects due to a distorted Spacetime manifold.

Both Newtonian Gravitational theory and Einstein's theory of Relativity reduce to a simple but purportedly accurate algebraic equation, used to calculate the 'force' of attraction between two massive objects in space at appropriate distances, such as stars.

The equation is:

 The Gravitational Constant in this equation, "G", was meant by Newton simply for the purpose of scaling or 'renormalizing' the chosen units of mass and distance.  In other words, to match them, so as to produce correct units of 'force'.

Einstein did not like this constant, since it gave the impression that it might be a 'real' physical universal constant rather than just being a units-matching adjustment.   Others since Einstein have exploited this "Constant", by actualy inserting various variables in an attempt to force the equations to better approximate real-world measurements, and improve gravitational theory.

The Empirical Failure of Both Theories

Both theories however, make predictions as to the speed of orbits in space as a result of the mass and distance of an object circling around another (heavier) object.   This is precisely why the equation, and why both physical theories, Newton's and Einstein's, are held to be physical theories with predictive power, not just mere mathematical formulas.

Given the apparent mass of the earth and Sun, and the speed of earth's orbit in relation to the background of stars, we can choose an appropriate value for G.  (This is done alongside other experiments and measurements using Newton's Laws of Force etc.)
This lets us calculate the expected speeds of other planets and their masses, from their observed distances and motion around the Sun.  We can extend this to stars circling the galaxy, and this works quite well out to distances approaching 2/3 the radius of our galaxy, the Milky Way.

(1)  Failure at Large Distances

Both theories however, fail in their predictive power at distances greater than this:
Speed of Stars at Edge of Galaxy
Rotation curve of a typical spiral galaxy: predicted (A) and observed (B). The discrepancy between the curves is attributed to dark matter.
The invention of "dark matter" however, is a form and distribution of matter which would interact gravitationally but not electromagnetically.  This sounds plausible, until we note that  "not interacting electromagnetically" is just a euphemism for being invisible!

Even this "fix" doesn't save the two theories:  It doesn't account for observed "extra-energetic photons" which also seem to indicate that gravity falls off slower than inverse-squared rates at certain distance scales.

In all, there are some half-dozen observed anomalies which contradict both theories of gravity, and which to this date have not been adequately explained by any theory at all.


Thursday, July 26, 2012

The So-Called "Proof" of the Shell Theorem

Here is the typical "proof" for the Shell Theorem (the Hollow Sphere Theorem), as presented by university math faculties (this one was from saddleback.edu).

It was presented by a reader as "proof" that Newton was right about the force inside a uniform hollow sphere.  Of course the 'proof' does nothing of the kind.

As another reader posted, a "theorem" in mathematical parlance is a mathematical statement considered to be "proven true" only in the sense that the mathematical result follows logically from the starting premises and the application of appropriate algebraic and rules and conventions.  Even this idea of 'proven mathematically' isn't always clear, and theorems are constantly modified, limited, and expanded in the field of mathematics.

And a mathematical "proof" is not a in any sense a "physics" proof, which is based upon how well a given piece of mathematics actually applies to a physical situation, and its predictive power, when the mathematical variables and other elements are assigned to physical entities and measurements.

The so-called "proof" given begins by implicitly claiming to model a vague physical situation.  None of the items below are properly defined, and none of the physical theory relating the idealized abstract model and its elements and diagram to a physical reality is ever presented or articulated.  All of that remains not only unproven, but actually unstated.  The premises and assumptions remain unexposed.  Its appropriateness and accuracy of the theorem in encompassing a physical situation is left completely unexamined.

Essentially, the "proof" begins by ignoring the main physical problem.  The author might have happily begun by explicitly making the admission:
"This 'proof' does not expound any physical theory.  Nor does it examine the strength of the connection of its elements to physical realities.  For that see a physics textbook.  We are here only outlining a mathematical theorem:

 Unlike our own diagram, nothing here is properly explained or labelled.  None of the mathematical entities are described, and even key mathematical axioms and premises are left unacknowledged.  For instance, the application of Sin and Cos functions have no meaning, even in mathematical proof, without the assumption of a Euclidean Spacetime manifold. But this is one of the very things to be proved in a gravity theory, and something that requires explicit discussion.  We can't fault Newton for his ignorance of spacetime manifold options, but we can certainly fault a modern proponent of the "Shell Theorem" for failing to qualify his 'proofs' in the modern context.

 Remarkably, a form of Newton's classic gravity equation simply appears in the third line here without any qualification or explanation.  None of steps of the algebraic manipulation is shown, and none of the theorems or axioms required for these steps is given.  No derivation or justification for any of the steps that are show are explained.
We now have an equation, an integral, to which someone has assigned a physical meaning, by identifying the elements in the equation with elements in the original physical problem.   While the substitution of variables and elements may be acceptable, the student/reader is left abandoned to the "God-like" presentation of the 'teacher'.  The only hope is memorization of the forms acceptable on the exam, when the question is posed.

Now comes the hilarious part of the "proof":

 "From the Tables:..."  yes, that's right, the four difficult pages of real hardcore integration, and their theoretical underpinnings are skipped entirely, unlike our own appendix on the integration of this difficult integral (previous step).  Naturally the 1st year student is not expected to know how to actually integrate a function like this.  Like a trained monkey he must look it up in an advanced volume/table of Integrals, and simply jump to the next part of the problem.

This is not a formal 'proof' in fact of the Shell theorem or any other theorem.  Its a "memorize" the entry in a table method, and don't ask any questions style of teaching suitable only for dummies.

Now, magically, the Force Vector F just appears.  An algebraic equation has been derived (skipping all critical steps and techniques) for the force exerted on the object (not the sphere).  No explanation for how the mathematics has been connected to a real physical vector is given.  Essentially, all that has happened is that an equation from a book has been assigned a task. 

A similar phoney implicit step has been performed for the Force vector in the separate case when the point-object is inside the shell/sphere.  No notice of the failure of Newton's equation in the case where the particle is actually at the border of the shell is given.  No explanation for what happens when a particle pierces the shell, exits/enters,  is attempted. 

The student is left with two equations, one of which reduces to a constant (0), but he has no idea of why there is no 'force' inside a 'uniform shell of insignificant thickness with its mass evenly distributed on its surface', or whether it is even true.

This is a classic example of the academic "bluster".

A bunch of algebra and some calculus references are waved in front of the inquirer, and they are just left feeling stupid, because in fact it is impossible to derive any proof of the Shell Theorem from this terse, highly condensed and deficient presentation of a complex operation which invokes literally hundreds of mathematical and physical assumptions. 

The student is told to "go read the textbook discussion" on the Shell method of integration, or 'Newtonian Gravity'.    He does the only thing possible in such a ridiculous situation:  He memorizes the steps approved by the lecturer, gets his grade, and has no clue what is really transpiring in gravitational theory.

Wednesday, July 25, 2012

Disproving Newton Part 3: Center of Mass

Disproving Newton
Part 3
The Death of the Center of Mass Theorem

by  Nazaroo

Newton’s Centre of Mass Theorem is shown to be self-contradictory
(With a New Introduction)
Last Updated:  June 19, 2005


This paper examines the Centre of Mass Theorem ( CMT ), originated by Newton, as it is currently formulated and presented in physics textbooks around the world  today.
Specifically, after examining the details of the theorem, we show:
1) That  CMT  is incapable of properly handling external forces exerted upon a system, because its method discards crucial information regarding those forces.
2) That  CMT cannot be patched, modified or extended simply and easily, because its flaws are deeply entrenched and are systemic to its derivation and formulation.
3) That  CMT  is self-contradictory and mathematically incoherent.  Its theoretical foundations are misguided and confused.
4) No sensible or useful reformulation of  CMT  is really possible, and its only value is historical.  It cannot be retained as an integral part of modern Newtonian mechanics.
5) That the few valid components contained in  CMT  should be left to stand alone under restricted circumstances, such as the Conservation of Linear Momentum ( CLM ), or else be absorbed into other valid  theorems and methods, such as those covering Angular Momentum and Equilibrium of forces.
6) That  CMT  should be abandoned as a teaching tool, since it misleads people regarding both its meaning and value for Newtonian mechanics.
7) That  CMT  cannot  be used either as a theoretical foundation or even an auxiliary support for the Sphere Theorem ( ST )  in regard to Newtonian gravitation.  
In summary, the Centre of Mass Theorem is essentially incorrect, and flawed as a theoretical basis for understanding the physical phenomena it is expected to account for.  Teachers and students are better off avoiding this theorem entirely and waiting for a better theoretical overview or presentation to be offered.


Table of Contents

1.  Quick Review:  Newtonian Gravity    
2.  Gravity  made Practical    
3.  The Background and Purpose of the Centre of Mass Theorem    
 4.  The Centre of Mass Concept  ( CM )    
5.  Special Properties of the Centre of Mass    
6.  The Centre of Mass and the Geometric Centre  
7.  The Generic Nature of the Centre of Mass  
8.  Practical Methods of Calculating the CM   
9.  Derivation of the Centre of Mass    

The Basic Theorem of the Centre of Mass 
( CMT )   
1.  A Typical Formulation
2.  The Scope and Claims of  CMT   
3.  Interpretation of  CMT     
4.  Counterexamples contradicting  CMT   
5.  Realistic Restrictions for  CMT    
6.  Re-interpretation of  CMT    ( ‘no free lunch’ theorem )  

Analysis and Criticism of  CMT   

1.  Flaw A:   Failure to distinguish between types of systems   
2.  Flaw B:    Non Uniqueness of the CM    
3.  CMT :  Blow by Blow Critique   

Demonstration of the Failure of CMT  

1.  Analyzing Newton’s Balls    
2.  CMT contradicts NTF     
3.  CMT contradicts  Itself    
4.  The  'Success'  of CMT:  A Cosmological Coincidence   


1.  Quick Review:  Newtonian Gravity

In brief, Newton viewed gravity as a force of attraction between bodies.   Matter is made of basic particles, such as protons and electrons.  Each particle attracts every other, and so there is a small force between every pair of particles.

 These forces are independent, in the sense that they do not interfere with each other as sound waves or light waves can.  That is, unlike waves that vary over time, gravity is a static force that only varies with distance.  Gravitational forces pass through one another in a ghostlike fashion, only acting on actual particles.
Gravity apparently reaches infinite distances and penetrates everything.  Interposing a third mass between two objects does not diminish or deflect the force.  In fact it has no effect at all, except to add two more paired forces.  The gravitational force between each pair of objects is dependent upon their own mass and distance alone.  The total net force is just the superimposing and adding up of all these forces between pairs.

Every object exerts the same total  force 100 light-years away as it does next door (!):
Although gravity weakens as distance squared, this really just reflects a fixed amount of force spreading out over an increasing spherical surface area.  It can be represented by a fixed number of lines of force spreading out like spokes of a wheel.  This essentially lossless spread could even be called the Conservation of Gravity.

Even more remarkably, gravity seems able to draw upon infinite amounts of energy to apply itself.  One object exerts the same force upon one other object, or an infinite number of them, as necessary.  Gravity doesn’t run down like a clock, or drain like a battery.  The force doesn’t become defocused, diluted, or randomized as it stretches across space.  These basic discoveries are mystifying.  Nor is there any obvious mechanism or medium for the actual transmission of the force. This led Newton to view gravity as a spooky, instantaneous action at a distance, and led Einstein to interpret it as a geometric property of space-time itself.

2.  Gravity  made Practical

Although mysterious in terms of ordinary notions of cause and effect, at least gravitational forces are easy to handle in principle.  We don’t need to worry about extra interactions or side-effects.  For any given pair of particles, Newton’s basic formula could be used to calculate the force.  With more particles we just add all the gravitational forces together like we do other forces as vectors to get the total net force on any one particle or object.
Easy in principle however becomes impractical with billions of particles per handful of matter.  Useful formulas must apply to large groups of particles at a time.  Newton addressed this problem with two claims:
(1) The Sphere Theorem:  Uniform spheres act as if they were point-masses, and so the simple formula for point-masses can be used ‘as is’ with celestial bodies.  One assumes the mass is concentrated at the geometric centre (GC), and distances are measured centre-to-centre between spheres.
(2) The Centre Of Mass Theorem:  Similarly, any object’s centre of mass  ( a fixed point of balance inside it1) also moves according to the laws of motion, as if all the mass were concentrated there.  External forces can be summed and applied to the centre of mass (CM)  to predict its new position etc.2
These two key theorems are the basis for practical calculations and predictions for the motion of large masses under Newtonian gravity.
1. An object’s centre of mass ( the weighted average of position of all particles ) can be found by measuring weight, volume, composition, density & shape, without descending to the atomic level. The geometric centre ( GC ) of an object is determined by its shape, i.e., distribution of volume and extension in space. The centre of mass is determined by the distribution of mass. If the system is symmetrical in both shape and distribution of mass, its centre of mass coincides with its geometric centre. These two are not the same as the centre of gravity, which is only relevant in a uniform field. Nor are they the same as an equivalent point-mass position, which is what we have to calculate.

2. That is, the centre of mass traces out a parabola or moves in a straight line, even if the object appears to move eccentrically.

3.  The Background and Purpose of the Centre of Mass Theorem

To assess the impact of a flaw in the Centre of Mass Theorem ( CMT ),  we need to review its function.  Newton’s theoretical foundation rests upon atoms exerting mutual forces of attraction upon one another.  That is, his gravity formula actually only applies to particles.   This would be hopelessly impractical unless we could economically deal with large groups of particles at once.
The whole purpose of  CMT  is to enable us to do so. CMT then is a key part of Newton’s complete theory.  Without  CMT or some other practical technique any gravity theory is just a curiosity.  Newton’s  spooky notion of invisible, instantaneous action-at-a-distance would probably have been branded as mere magical thinking long ago, if not for the spectacular success bought for it by  CMT. 3
 That is, people accepted Newton’s theory, because his formulas and methods actually worked, or seemed to.  Even those today who reject Newton’s theory presumably don’t object to engineers continuing to use Newton’s techniques  ( i.e. CMT ) to build bridges and to calculate rocket trajectories.
It is hard to deny that  CMT is still presented as a currently valid theorem, and that most scientists assume the consistency and validity of its derivation and formulation, at least within the sphere of Newtonian mechanics.4   Now and then CMT is called an ‘approximation’, although no quantification is offered.  The basic position given on  CMT  is that it is accurate. 5
3. Even with the success of Newton’s techniques, many brilliant minds questioned his theoretical foundations. Newton’s success regarding celestial bodies relied upon the Sphere Theorem ( ST ) but this appears to be just a special case of CMT.
4. Angular Momentum in particular has been modified to accommodate Relativity, but this is really only relevant for particle physics. For ordinary objects Newtonian versions of angular momentum are expected to be valid and applicable to gravitational problems also.
For instance see Elements of Newtonian Mechanics Knudsen/Hjorth, Springer, 2000 pg. 195


The Current State of Affairs as to Gravitational Theories

The only viable alternative is General Relativity ( GR ).  But  GR is still in its infancy and in its current form is simply impractical.  For most purposes Newton’s  CMT is all  we have.  Even those who prefer  GR  as a theoretical foundation still use  CMT  for ordinary calculations. This is no surprise, since Einstein designed  GR  to mimic or reduce to  CMT in ordinary situations.  For these cases GR then offers no increased accuracy or practical advantage, nor does it make any alternate predictions.  Apparently  CMT (with a few corrections for relativistic effects when necessary) is all we really need. So Newton’s method ( CMT ) is used for accuracy and convenience, while Einstein’s theory  ( GR ) is preferred as a theoretical foundation for other reasons, such as compatibility with Special Relativity.

Other Related Theorems and Methods

To complete the picture presented so far, we must mention the Angular Momentum and Equilibrium theorems.  These handle rotation and the application of forces to rigid bodies, and they also handle balance under uniform gravitational fields. Both theorems make use of the Centre of Mass concept, ( CM )  but do not directly rely upon the Centre of Mass Theorem. ( CMT ).  One might argue that most engineers rely upon these other theorems more heavily than they rely upon  CMT .

Perceived Agenda for Gravitational Theories and Techniques

Naturally,  Newton’s theoretical foundation ( NTF )  is expected to fade away because it can’t explain new discoveries.  It wasn’t written to explain them, and would need ongoing revision to do so.  In hindsight,  NTF  appears not to be a real explanation at all, but simply the ad hoc concept  Newton needed to justify  CMT , in the same way he used ‘Absolute Space’ to justify inertial frames and simplify the laws of motion.  That is, NTF really depends upon  CMT  for its existence, not vise versa.  Both  NTF and Absolute Space were necessary for Newton, but are no longer so for us.  If this is so, we can discard  NTF  and keep  CMT as one removes the scaffold once the house is built.  If  CMT  doesn’t  require  NTF, or they were found incompatible in some unforeseen way, we could even now choose  CMT  over  NTF  on more than empirical grounds.
In a happy world, so it would be.   Newton needs  CMT , and we need  CMT, but we don’t need Newton’s  NTF.   It could be left as a learning tool, or retired to a history shelf, and  CMT  would be retained in the honourable service of  GR.
Probable Outcome of a New Analysis

This however appears not to be the future of the Centre of Mass theorem.  The reader should discover as we did that  CMT has fatal flaws, if they can find the patience to step carefully through the arguments below.  Is this an exaggeration?  Surely people will go on using  CMT  as they always have, in the same manner as they did after  GR.  Not exactly.
GR was clearly revolutionary, overturning theoretical foundations.  Yet, for ordinary mechanics, life went on pretty much undisturbed:  In this case, the theoretical modifications are obviously far more modest.  Yet something very different has happened qualitatively.  The function of  CMT  has inevitably changed, because it can no longer be trusted.  Its results, even its successes, require careful reinterpretation.  CMT  must now be applied more cautiously, keeping an eye out for much more than just relativistic effects.  Many results and theorems will need reformulating in cases where it is clear that errors may become significant.

 4.  The Centre of Mass Concept  ( CM )

The centre of mass ( CM ) is not a vague concept in Newtonian mechanics.  It is a unique point related to a system, and is located within the volume of space that encompasses the system.  A system in this context is a group of particles, each having a definite mass and position at some instant in time.6  A system need not be a rigid body, and systems can overlap and contain one another.  However, the instantaneous position of the  CM  for a system is always defined in principle, because a reliable method is given for calculating it.  The  CM  is called the ‘weighted average of position’7  of the particles, and each coordinate is defined by a calculation averaging the coordinates and masses of the particles in the system:

This is simply an arithmetic mean of position, weighted by mass. 

This procedure always gives an unambiguous location, which is fixed relative to the rest frame of the system it belongs to.  This relation can be called  an ‘internal position’ to the system.

7'   e.g. University Physics 7th Ed. Sears/Zemansky/Young, Addison Wesley 1987 pg. 197


5.  Special Properties of the Centre of Mass

The  CM  cannot change without a change in the distribution of mass.
So the  CM  is always a fixed position for a rigid object, which is defined as a system with a fixed relative distribution of mass.  This allows us to relax the simultaneity restriction for rigid objects, and we can track the motion of the  CM  as easily as other parts of a system without having to constantly recalculate it.
It is trivially true and desirable that a change of coordinate system, or scaling of all the masses and/or distances by some factor (i.e., a change of units) has no effect on the internal position of the  CM.  For instance, since the  CM  is only a point, altering the orientation of a system has no effect on the  CM.
For the  CM  to hold its position relative to other systems however,  the system must be rotated only on an axis through the  CM  itself.   It is also true that changing the orientation, distance, mass or size of a system relative to a second system, or changing the second  system’s orientation to the first, also has no effect on the internal location of the CM.  But this latter case is not a mere change of coordinates, scales or units.
We expect the  CM  to be unaffected by trivial transforms like rotations, scaling and reflections.  However, a system can actually be squashed flat, exploded, or stretched symmetrically in any number of dimensions without affecting the CM. One can even add or subtract mass, particles, combinations, whole systems, provided the effect is symmetrical.
Conversely however, even extreme changes in the distribution of mass don’t necessarily affect the position of the  CM.  
Any number of systems could average out to the same CM.  Although dependent upon distribution, the  CM  is a one-way trapdoor, preserving no information whatsoever as to the actual distribution of mass.  Neither its location nor its motion can really tell us anything more than the general point of balance within a system.

6.  The Centre of Mass and the Geometric Centre

The  CM  is closely related to the geometric centre ( GC ), because for systems of point-masses,  the GC  can be similarly defined as the ‘unweighted’ average of position, which simply means the ‘equally  weighted’ average.
 When dealing with systems of point-masses, the CM and GC will be the same as long as all the points have equal mass.
Where the mass can be considered continuous because of the quantity of point-masses, all that is really required is uniform density for the CM and GC to coincide.
If the mass is distributed symmetrically around either the CM or GC, then it is so around both, and they are identical.  For many symmetrical objects, such as spheres, disks, cylinders, and regular polygons, the  CM  and GC are again the same, as long as the objects have merely a radially symmetrical gradient.
Because of this, the CM and GC can share both topological and geometric properties.  For instance, if the system’s point-masses sit on a straight line, the  CM  is also on that line.  If they lie on a plane, the CM  is also on the plane.  Like the  CM, the  GC  is also indifferent to orientation, is a fixed internal point in a rigid object, and lacks unique association to a specific point-mass distribution.

7.  The Generic Nature of the Centre of Mass

Above, we referred to different systems having the same  CM. This is only meaningful with an independent way of defining the position of a system, which  the  GC  provides.  The  GC gives an alternate means of defining position, and helps to locate the CM,  but does little else for us. The  CM  is the more relevant concept for gravitational forces, and for rotation as well.
Using the  CM  and the  GC  alone only allows us to distinguish two kinds of systems topologically, those in which they coincide, and those in which they don’t.  For example, take any two non-symmetrical systems, each of which has non-coinciding CM and GC.  Superimposing them with their GC at same location, we can rotate and scale one of them until their  CM s  also coincide, making them equivalent systems according to  CMT.  Clearly,  even when the  GC  and  CM  do not coincide there is little to distinguish between systems, applying these concepts alone.
If the current claims found in  CMT  actually hold in the physical world, we would expect diverse systems to behave in an identical manner, namely the manner predicted by  CMT.

8.  Practical Methods of Calculating the CM

Obviously we can’t know the instantaneous position and mass of billions of particles, let alone calculate a weighted average from the data.  The whole interest in the  CM  is to develop practical methods to measure and predict motion of large systems, without  having to handle individual point-masses.

Ultimately, people rarely use the formal method of calculating the  CM , except perhaps for a system with only a handful of particles or celestial bodies.  Otherwise, matter is treated as though its mass were continuously distributed, and cases are limited to rigid symmetrical shapes with simple density patterns.  However, the CM  is invisible, often sits in empty space between masses in a  system, and sometimes can only be found through the GC.  If they don’t coincide, the orientation of the system is also needed as well as the GC, in order to find the  CM.  A combination of geometry and calculus is used along with assumptions about the behaviour of the  CM  found in explanations of  CMT  stated below.
In a locally uniform gravitational field, such as near the earth with small objects, a simple method for finding the  CM  is to exploit symmetry (it must be on an axis of symmetry in an object with a known symmetrical distribution of mass).  One can also hang an object from some point two times, and in each case extend the vertical line through the object or along its surface, and mark their intersection.

9.  Derivation of the Centre of Mass

Modern treatments derive the  CM by first applying Newton’s 2nd law (F = mA) to each particle in a system.   Position is substituted for A  as a derivative (acceleration is the 2nd derivative of position).  Mass is allowed to slip inside the derivative, since it is assumed to be a constant for each particle: 8

Next, an important declaration is made:
‘The total force on the system 9  is the vector sum of all the forces on all the particles.’ 10   
This is such a remarkable theoretical step that it deserves a comment: The idea is that just as we get the net force on a particle by summing the forces, so we should calculate the final net force upon a system by summing the forces applied to each particle in it, regardless of where  in the system they are actually applied.
Thus, the total force of the system is written as the vector sum of the individual forces on each particle.  Using the sum of derivatives = derivative of the sum rule, the summation is also slipped inside:

Finally,  multiplying by  M/M  ( M = total  mass ) and slipping  the 1/M portion inside the derivative as well, puts it all back into the form of Newton’s 2nd law,             ( F = MA ) only this time  A  is clearly seen to be the 2nd derivative of the formula given earlier for the  CM,  namely the weighted arithmetic mean average:

From Newton’s third law it is noted that all the internal forces between particles within the system come in equal and opposite pairs, and so they cancel leaving zero net force.  Only forces from outside the system are unpaired and unbalanced and so represent a leftover net force.

That is, while   Ftotal  = Fexternal  +  Finternal ,       Finternal   = 0   and so    Ftotal  =  Fexternal

Thus the final equation  Fexternal  =  Mtotal * AccelerationCM   seems to imply that the  CM  actually obeys Newton’s 2nd law as long as  Fexternal  is understood to be the vector sum of all the external forces, as we originally defined it in the derivation.

8.  This is a minor fudge. Mass varies slightly as energy is stored or released in chemical bonding or atomic decay.
9.  Physics 3rd ed. Wolfson/Pasachoff, Addison Wesley, 1999 pg. 239
10.  University Physics 7th ed. Sears/Zemansky/Young, Addison Wesley, 1987 pg. 198 etc.
The Basic Theorem    ( CMT )  

We are now ready to state clearly what  CMT  essentially asserts:  Every system of particles having mass acts as a single point-mass concentrated at its CM, and external forces act upon the system as if they were all acting upon the CM.

1.  A Typical Formulation

1)  Suppose we have a suitable means of measuring the position and time of point-masses in any direction and at any distance.
2)We establish a suitable rest-frame by measuring the motion of a sufficiently isolated point-mass and assuming Newton’s first law of motion.
3)We define the  CM of a system as the weighted average of position (as above.)
4)The displacement of the CM is then the weighted average of the displacements.
5)The velocity of the CM is then the weighted average of all the velocities.
6)Momentum is defined for particles as  mv,  and the vector sum of all the particle momenta in an isolated system is shown to be conserved.                                            m1v1 + m2v2 + …  = ( a constant vector )
7)This vector sum is identified as a momentum, that of the system.                            (MVsystem ) =  m1v1 + m2v2 + …  and we get a law of Conservation of Momentum .
8)The total mass  M  is defined as the arithmetic sum of all the masses.
9)The velocity of the system is defined and isolated by dividing both sides of the momentum equation by M.
10)The velocity of the system then turns out to be the velocity of the CM. (!)
11)The total mass M is then assigned to the CM.
12)The momentum of the CM is then the momentum of the system.
13)The law of conservation of Momentum is applied to the CM.
14)The CM appears to obey the first law of motion when the system is isolated, and it has uniform motion, even if parts of the system have complex motion.  For instance, if the system is rotating, it rotates around the CM.
15)The system appears to behave as though it were a single point-mass located at the CM, with all its mass concentrated there.  This assumption is extended to non-isolated systems, under acceleration from the effects of external forces.
16)Newton’s second law  (F = mA)  is applied to the system as a whole (as above).
17)Just as the net force upon an individual point-mass is the vector sum of all the forces,  the net total force on the system/CM  is assumed to be the simple vector sum of all the external forces.  (In this case, the forces are those upon the individual particles at different locations however.)
18)The acceleration of the CM is calculated by dividing the total force on the system by the total mass ( by rearranging the equation to  A = F / M ).
19)It turns out that with force so defined, the CM is the position found on the right  side of the equation following Newton’s 2nd law (on paper), and vise versa.
20)Therefore, the  CM  acts as a point-mass of the same mass as the system, and all external forces can be simply applied there (as a vector sum).  We can either predict the  CM  motion from known forces, or the forces from the CM  motion.

2.  The Scope and Claims of  CMT

    The scope and generality of  CMT  is as follows:   It applies to all systems of point-masses, rigid or not, isolated or not.  It is not dependent upon ‘uniform fields’ or special cases.  This is often explicitly stated in physics texts, for instance:

    ‘ ( in contrast to the centre of gravity  concept, ) …the centre of mass, conversely, is defined independently of any gravitational effect.’ 11

At this point the following claims are typically made:
‘(The CM ) does obey Newton’s law…We have defined the  CM  so that we can apply  the 2nd law to the entire system rather than each … particle.12 …As far as its overall motion is concerned, a complex system acts as though all its mass were concentrated at the CM…we defined the CM so that its motion obey(s)…F = MA. ’ 13
‘When a (system) of particles is acted on by external forces, the  CM  moves just as though all the mass were concentrated at that point and it were acted on by a resultant force equal to the (vector) sum of the external forces on the system’14
‘The ( CM ) moves as if the entire mass were concentrated in that point, and all external forces act there.’15
‘The ( CM ) of the system moves as if all the mass of the system were concentrated at that point. … the system moves as if the resultant external force were applied to a single particle of (equal) mass … located at the centre of mass.’16
Physical support for  CMT  is usually offered in the form of examples such as non-uniformly weighted hammers or wrenches.  When rotating, they revolve around their  CM.    The CM can be marked on the object and observed to move in a straight line as the object spins through space or slides across a table.

3.  Interpretation of  CMT

One can search through the most recent physics texts, yet not find any qualification on the claims for CMT.  The word  ‘approximation’ does not appear there, although liberally used in other contexts.  The one time we have found it used in relation to  CMT is as follows:
‘Newton’s second law guides the motion of the CM  just as it guides the motion of a single mass point.  This is the justification for the often used approximation of an extended body as a mass-point.’ 17
This is hardly a statement that  CMT is an ‘approximation’  in any significant or quantifiable sense.  If anything,  CMT is once again asserted to be accurate.  There is no hint of when  CMT  might prove inaccurate, or by how much.  There is no mechanism, no example offered, no acknowledgment of any failure of  CMT.

4.  Counterexamples contradicting  CMT

Ordinary physics students can be forgiven for thinking that here at last is some promise of a general method for dealing with systems and forces that is practical, reliable, and easy to use.  They will not stay misled for long however, once they experience problems involving rotation and torque.
 Note that if one applies equal and opposite force vectors sideways to each end of a simple rod, it spins (about its CM in fact! ).  But if we were to follow  CMT  as presented,  and simply add them vectorally and apply the resultant zero vector  to the CM of the rod, we would never predict rotation or discover the rules of torque.

This isn’t a new discovery, obviously.  But we would do well to discover the nature of the failure of  CMT  in cases such as this, so that we can modify its scope and its interpretation to reflect reality.  One simple way to do this would be to just include theories of torque and equilibrium within  CMT as a basic modification.

This would be unsatisfactory for several reasons.  For one thing, it assumes the validity of  CMT  in other cases not involving torque.  But we have already seen that  CMT  has been poorly defined as to scope and interpretation, and there may be other flaws in  CMT. 

5.  Realistic Restrictions for  CMT

CMT only deals with linear (translational) motion.
How did  CMT  fail in the previous example?  Because in fact it has no mechanism at all to deal with rotation and associated forces.   It cannot, because by nature a point-mass has  no orientation, nor any mechanism to connect to, store, or even measure relative rotation.  Rotation can be thought of as balanced acceleration.  CMT ‘handles’ balanced acceleration by canceling it out and ignoring it.  One way to modify  CMT  to handle rotation would be to convert  angular momentum into a kind of mass vector.  In this case however, mass becomes a variable, and some of the mathematical steps in  CMT obviously become invalid.
CMT only deals with vectors passing through a CM  axis
Rigid bodies can be thought of as machines able to convert linear motion into rotational motion.  As such, they are unsuitable for a method using a simple summation of vectors regardless of application point.  Arbitrary forces often produce a mixture of translational and rotational motion.  One solution would limit external forces under consideration to those applied along an axis passing through the CM.  Then at least one point-mass in the system needs to be on this axis to receive the force.  We could also handle cases having a secondary mechanism that distributes and delivers force in a parallel direction, such as a flat plunger.
In other special cases,  CMT could handle balanced  forces acting on parts of a system.  For instance, one could simultaneously apply two vectors having appropriate directions and proportionate magnitude relative to a CM axis.   This would work like a uniform gravitational field, where forces trivially move the system as if  the sum of forces were applied to the CM. 
Otherwise,  CMT can only handle forces (or fields) acting symmetrically upon the system, around the axis of net direction through the CM.
These restrictions limit forces and combinations of forces to those  that produce pure linear (translational) motion without rotation.  The careful expansion beyond uniform fields allows for the kind of spatial distortion of a system that doesn’t affect the relative location of the CM on any axis of symmetry.

6.  Re-interpretation of  CMT   
( ‘no free lunch’ theorem )

With a sober assessment of the restraints upon  CMT , we can now carefully reinterpret its claims.
‘The ( CM ) moves as if the entire mass were concentrated in that point, and all external forces act there.’18    And,
‘… the system moves as if the resultant external force  were applied to a single particle of (equal) mass … located at the centre of mass.’19
These statements are true (and only true) if by ‘resultant external force’  we mean the result after  we filter all those external forces through a technique which calculates and discards all torque and rotation effects.  This technique would leave us with a net linear force related the system as a whole through the CM.
External forces  act on the CM  indirectly since they are altered by mechanisms unique to each particular system, and their effects rely upon things like rigidity, interconnection of parts, and distribution of mass.  There is no universal or simple method to convert real external forces into linear motion of the system or CM.

‘the  CM  moves just as though …it  were acted on by a resultant force equal to the (vector) sum of the external forces  on the system’20
This statement is blatantly false.  In fact, if a force is applied to a point on a  rigid body so that its directional axis does not pierce the CM, it produces a torque, making a variable  amount of its magnitude available for application to the overall linear motion of the system.  There is no way of knowing the net contribution of the force to linear (translational) motion without calculating and subtracting the torque.

The example above is even more damning for  CMT than the last.  The ratio of translational to rotational motion which vector force  F gives to the rod depends upon the distance R from the CM.  The vector itself gives no indication of linear motion, and a sum of such vectors can’t either.21


Analysis and Criticism of  CMT

We will skip general criticisms  of Newtonian mechanics regarding relativistic effects.  Our concern is whether  CMT  is consistent with the rest of Newton or even self-consistent.  Granted that it appears to give good results in many instances, we want to know why, and if it fails or can fail, where and why it fails.

1.  Flaw A:   Failure to distinguish between types of systems

When  CMT leaves the definition of ‘system’ open, allowing both arbitrary groups of point-masses in space, and also rigid bodies, it clearly fails:
The vector sum approach of dealing with external forces is only valid If we have a system of truly independent point-masses.  That is, if we have a purely classical system of elastic particles having no significant gravitational, electromagnetic or other forces connecting them.  In this case, each particle is truly isolated and has a definite uniform velocity, which cannot change without a collision and an exchange of energy, preserving the law of conservation of momentum.
External forces then act upon the system by direct application of a force vector to an individual point-mass, say through a collision.   In this case, the definition of the velocity of the system as the weighted average of the individual velocities will naturally and correctly reflect the application of a force vector to any particle of the system.  This force vector, although only physically affecting one particle, will be correctly assimilated into the average position, velocity, and momentum of the system.
But what systems conform to this classical billiard-ball style model?  A bag of billiard balls released into deep space.  Each exerts negligible gravitational and electrostatic force upon the others.  We can even correctly handle uniform gravitational fields, since each ball receives a force proportional to its mass independent of position, and the force vectors can be simply summed.  Adding the individual masses gives a total mass independent of position as well, but that is fine.  Thus, if the particles are spread apart far enough so that they act independently, we can use  CMT  safely, and the CM will reflect the system’s velocity and momentum.
But this is precisely not  the type of system we want to apply  CMT  to!  All large masses of interest consist of closely grouped particles  which are bound together by strong electromagnetic and gravitational forces.  These lumps of actual matter form a class of rigid or ‘semi-rigid’ bodies in which the particles interact strongly through fields, and are capable of converting linear forces into rotation and storing energy through angular momentum.
Thus a group of (non rotating) billiard balls is a legitimate ‘system’ for  CMT, but an individual  billiard ball is not.  Real billiard balls and other objects can rotate.
Since a raw vector sum of external forces on the system cannot distinguish between rotational or translational results, it cannot reliably represent the translational motion of any rigid or semi-rigid system of closely interacting particles.  The vector sum cannot properly reflect the motion of the system or  the CM, and is a meaningless value.  To work, any version of CMT  would require pre-treatment of external forces via torque handling techniques, and a consideration of the effects of rotation on the inertia of a system.
 An easy fix is not possible for the current  version of  CMT  however, because the vector sum is already deeply imbedded in its structure and formulation!  The current version of  CMT  is clearly and fatally flawed.  The next question is where exactly, and can it be fixed?

2.  Flaw B:    Non Uniqueness of the CM

Suppose we try to extend  CMT to non-rotating rigid bodies.  In this case,  we would still need the  CM  concept in order to apply force without causing rotation.   Thus the  CM  itself is retained in the theory of torque.  We would also have to restrict external forces to those essentially applied along an axis of direction through the CM,  but this generalizes the concept from a point to an axis.
The body would then move according to the laws of motion.  However, now so does every particle in the rigid body, and nothing distinguishes the  CM.  In fact, part of the interpretation in the Centre of Gravity Theorem ( CGT ) predicts that all the particles in a system have the same velocity and parabolic trajectory, even when separated in the air, excluding any extra motion imparted in separation.  The function of the  CM  is then reduced to an aid in applying force, but has no physical uniqueness in a rigid system , and no other physical meaning.
What is now left in our hands, after all these exceptions, constraints and limits?   We had to abandon the summation of external forces, except for ‘systems’ of free non-interacting particles.  We had to abandon rotating and rotatable systems to methods devised to handle torque.  We had to abandon applying forces except uniformly or on a special axis, according to laws of equilibrium and balance.  We had to abandon uniqueness and other overstated claims about the  CM.  The only substantive item left is a basic conservation of momentum law.

    The theorem of Conservation of (linear) Momentum:
 ‘When the external force is zero, the total momentum P (the vector sum of the individual momenta) remains constant.’22
And,  ‘the velocity of the  CM  is the same before and after (an interaction) in which the total linear momentum is conserved.’23
An understated, (and possibly unintended) sober assessment of  these results is found in Knudsen:
‘The two theorems, the conservation of (linear) momentum for a closed (isolated) system, and ( CMT ) have the same physical content.’24
Note particularly the necessary word ‘closed’,  i.e., completely isolated with no external forces.  What is implied for systems with  external forces?  Not much.

Where does this leave us?

It leaves us with the perhaps daunting task of explaining the successes of CMT, removing the contradictions in it, and reformulating it in a manner that saves the phenomena, gives us a more solid theoretical foundation.  Also desirable is increased accuracy and reliability, some practical methods, and some new and interesting predictions regarding Newtonian Kinematics and Dynamics.  CMT itself now lies in shreds before us.  The consequences for the Sphere Theorem ( ST ) which is a special case of  CMT  will be considered later.



3.  CMT :  Blow by Blow Critique

Referring to the boxed and numbered steps in the formulation:
On  3)  :  The choice of weighted arithmetic mean could be challenged.    It is the ‘first moment’ or centroid.  This comes out of applications involving balance in a uniform field or equilibrium under rotation.  Because of the physical properties of levers, torque is proportional to  m x d.  But field  forces are proportional to m / d2  or q / d2   and other ways of weighting particle position could easily be proposed.
Take a non-uniform hollow sphere:  At one point inside the gravitational field is zero ( ZG ) .  This is not the  GC or the CM.  For obvious reasons, one may prefer this as the fundamental location of the sphere.  The theoretical justification for the CM  therefore is weak for both rigid and non-rigid systems. 
Any supplementary argument drawing on rotation or equilibrium is irrelevant  if we restrict CMT to systems only having linear motion.  That the CM can actually sit in space between particles should remind us that it is not a real physical entity.
On  5) :  Having the velocity  etc. of the system dependent on the same definition gives consistency and beauty.  But the potential circularity from interdependence throughout is far more important  and worrisome.  A solid physical basis for every observable, not relying upon the properties of numbers, vectors and algebraic systems, is desirable.
On  6) & 7) :  Momentum is a powerful and deep concept, and total momentum of a system is also clearly a useful number, but are they fundamental physical entities?   Does adding bound vectors together give us another bound vector, or an unbound abstraction?  We don’t need the vector sum of individual momenta to be a ‘real’ vector that can be decomposed into ‘Mass’ and ‘Velocity’, in order to have Conservation of Momentum.  We could just as well have Conservation of a Sum of Momenta.
On 8) :  The total mass may be the most questionable entity of all, although on the surface it looks unassailable.  Yet can we just add the masses together, when they are really spread all over space?  This is actually one of the very things that  CMT  is supposed to demonstrate, not assume.  
On  10) - 13):     velocity CM = momentumsystem / Mtotal   Is this miraculous confirmation of a theory? … or just  an illusion predetermined by bad definitions and pulled off by taking abstractions too literally and allowing nonsensical operations?
On  14) & 15):   The characterization of the CM seems extreme and overextended.25  But does the behaviour of the CM at rest or in uniform motion really tell us anything about how the system  will accelerate under external forces?  Should a baboon observe a sleeping lion to determine its temper before poking it?
On  16) - 19):   This result is probably the fatal flaw in  CMT and seems to have been completely misunderstood. Force applied from outside to systems as simple as a rigid stick  have variable effects on linear motion, and the effects are not reflected in vector sums. We know that just adding the forces gives ludicrous results.   The fact that combining such a sum with another suspect entity, total mass , results in a ‘solution’ for Newton’s 2nd law ought to make us jump out of our skins.  - Not in the direction of making magical claims for the CM, but in the direction of questioning the applicability of  the 2nd Law to abstract non-physical entities, and toward abandoning the current formulation of  CMT.
Generally, force vectors are a restricted type of vector.  They are not ‘free vectors’, such as displacement vectors.  They are linked to a location and time, or to a particle, and so they are called ‘bound vectors’.  Normally, they can only be added vectorally (i.e., by parallelogram law) if they are bound to the same location and time, or the same particle.  The directional sense of a vector often only has physical meaning when assigned to point to or from a location or source of the force.
The resultant vector is also bound to a particle or location in the same way and it also can only act upon the system through actual physical mechanisms.
25.  Theorists seem to have stopped just short of giving it intelligence and a will of its own! Not bad for an abstract point-mass whose location in real space is actually empty.

Demonstrating CMT Failure 

1.  Analyzing Newton’s Balls

First, let's take a pair of equal point-masses joined by a rod of negligible mass (a barbell), and call it  System A. The centre of mass ( CM ) for system A  is at the geometrical centre (GC).  For clarity we draw our point-masses as a small balls. To get the total force on another test-mass, (System  B), aligned on the same x-axis we just add the forces from each end of the barbell using Newton’s formula.  We don't even need the Sphere Theorem or vectors with the point-masses are all in a row in a single dimension.

For clarity we normalize the equations by making the masses of each system 1 unit, and choose distance units to make the gravitational constant G = 1 as well.  This lets us drop the Gmamb part of Newton's formula.  Now the formulas simply become 1/ d2  for each part, the d in this case is the distance between the test-mass (system B) and each each end of the barbell.  The new equation for the force between the systems is restated as

 In this form we can keep the system to system distance constant while spreading our balls.  We could normalize d as well, ( i.e., d = 1 )  but we leave it in to show that the formula reduces to the inverse square law.  We construct a simple triangle in the diagram to identify  Theta  in the equation.  By inspection the angle Theta  in the diagram is always less than 45o because the  right triangle is formed using the distance  d ,  the distance between the geometrical centre ( GC ) of system  A  and the test-mass.
The force depends upon the spread in system A and becomes Newton’s formula when the two point-masses merge as   r = 0   and  Theta  = 0 o.
We use this formula for the force of system  A  to calculate the position of a singular equivalent point-mass  ( EPM ).  That is, we wish to replace system  A with a single point-mass having exactly the same mass as system  A.  We need to place it so that it exerts the very same force on  B.  By inspection we know that this location will be somewhere along the axis of alignment.  We only need to find the distance from test-mass  B.
Since we want a new distance ( dnew ), giving the same force,  and the EPM is just a point-mass, we simply rearrange Newton’s point-mass formula ( NGF ) to solve for distance, and substitute our new equation for force into it.  The Gravitational Constant and the masses cancel and drop out, leaving a general formula for the distance from the test-mass, based on Theta alone.  Remember, Theta  is just a convenient way to express the spread of the balls in system  A:

Keeping the centre-to-centre distance  d  constant,  we spread our balls. Naturally, the  EPM distance ( dnew  )  decreases as the spread and the force increases.   The position of the  EPM  drifts closer to test-mass  B , catching up with the inner point-mass by the time it reaches test-mass  B.
This is precisely as it should be.  Gravity is an inverse-square law. It is non-linear (exponential) by nature.  The increase in force as one point-mass approaches test-mass  B is not  balanced by a complimentary decrease from the retreating point-mass, because the rate of change slows down farther away.  This would be true of any non-linear distance law, and is not remarkable.

…Except that the system does not behave ‘as though all its mass  was concentrated at the CM’  at all!  At least as to magnitude of force, it behaves as though its mass were located at an entirely different position.  Rotating system A  will change the direction of force as well.  This will be generally true of any object with a  mass distributed in space in a non-spherically symmetric manner.

 It is important to grasp the quality of the variation here:  It is not a small error of approximation, or even adjustable with a simple factor.  The force can fluctuate almost infinitely in any direction < 90o.  It is virtually unpredictable without exact information as to both the distribution of mass within systems and the distances between the systems.

2.  CMT contradicts NTF

It is important at this point to carefully explain what actually went wrong here.  We calculated the correct location for an equivalent point-mass the hard way, by actually computing and adding the (two) individual forces acting on the system from outside. It is not the location predicted by  CMT.  That is,  CMT  contradicts results using Newton’s Gravity Formula for point-masses  ( NGF ), which is the core of  NTF.  But  CMT was supposed to be a shortcut for  NGF, to save us having to use it.
Is  NGF  at fault, or  CMT?  In fact, the fault is clearly with  CMT.  It simply provides no mechanism  to accommodate the vast fluctuations in force possible with even the simplest gravitational objects.  The failure of  CMT is built in.  Creating a weighted average of position just throws away critical information as to distribution of mass right from the start.  This self-mutilation fatally cripples  CMT.  It simply can’t do what is being asked of it, and so  CMT  invariably produces force vectors which are incorrect in magnitude and/or direction.  It is no surprise that  CMT is internally inconsistent.   As a result,  CMT  not only contradicts  NGF, but also  ST , as we will see below.  Since  ST is just a special case of  CMT, it simply contradicts itself!
Although we’d prefer  CMT  to  NGF, we can’t.  Abandoning  NGF  wouldn’t save  CMT  in any case.  There is no way to relax the absolute requirement that force vectors sum in a coherent and self-consistent way.  Couldn’t   CMT  be valid for some other unknown reason?  No.  CMT itself is an invalid method for calculating external gravitational forces and predicting motion, because it wipes out critical information.

3.  CMT contradicts  Itself

A simple example shows that   CMT  and  ST  contradict one another.  Take a solid uniform half-sphere, and calculate its centre of mass ( CM ).  For our purposes, it is irrelevant where it actually is, but by inspection it must be somewhere on the axis of symmetry.  We can mark it with a dot .  The important thing is that the CM  is definitely a fixed  point inside the object, since it is the weighted average of all the atoms making it up, and its a rigid solid.

Placing two half-spheres together to make a whole sphere puts the  CM  for each half at a fixed, equal opposing distance from the GC of the whole sphere. By symmetry, the CM  for the whole sphere is at the GC  itself.

Now add a test-mass.  This is a precise parallel to the case above with the balls.  The forces upon each half  should add up to the force upon the whole sphere.  Remember, this is just a static case:  there is no motion involved. CMT should enable us to replace the halves with point-masses at their respective CM s , and add the vector forces.  If  CMT is self-consistent,  an EPM for the halves should be at the same location as the CM of the whole sphere.  But we already know this is not the case.  Each half contributes unequally to the force, because of its position.  The resultant force gives an EPM location that is closer to the test-mass than the CM of the sphere.  Clearly  CMT can’t deliver the goods.

Similarly, One could split up the sphere horizontally: Again treating the halves separately and using the Centre of Mass Theorem diverges from the result predicted by the Sphere Theorem:

According to  CMT, any half-sphere, spinning or not, and regardless of orientation, will follow the laws of motion as if its mass were concentrated at that point.  This must entail applying vectors to the CM and using actual masses for practical calculation.  For such purposes, one is to treat the half-sphere as a point-mass of exactly the same mass located at the CM. That is, if  CMT  has any value, it must mean that we can use it along with  NGF  to calculate the external force between objects, and predict the resulting motion.  This is precisely how Newton uses it to calculate and predict projectile motion under earth gravity.  There, the gravitational field is virtually uniform for small objects near the surface.
This is not a trivial point, nor can one get around it by insisting that the centre of mass concept is only meaningful as a centre of gravity and ‘only applicable in a uniform field’.  The gravity field in the isolation of deep space would presumably be the most uniform field of all. There, the mass of the universe (the distant backdrop of stars) is randomly distributed but essentially spread evenly in all directions.
If you can use  CMT out in space, and also next to a huge mass like the earth, you must be able to use it pretty well everywhere.  But if we can use it almost anywhere, what are the real restrictions as to when and how we can use it?  CMT itself offers no help or explanation for such internal contradictions as plainly exist.

4.  The  'Success'  of CMT:  A Cosmological Coincidence

As it turns out, both Newton’s and our good fortune has more to do with circumstantial factors than universal physical principles.  The apparent success of  CMT is just that:   an appearance created by special circumstance.  The vast distances of space dwarf the actual size of stars so much that those spherical bodies are  essentially point-masses, and  ST while trivially true is also untestable as such.  On the other hand, gravity is so weak a force that it is virtually immeasurable between even adjacent objects at the normal human scale of experience.  Earth gravity originates at an average distance roughly the radius of the earth, again dwarfing the size of ordinary objects.
Thus  the mechanism for  CMT ‘s apparent success is really just a set of special circumstances and cases.  The root of the matter is simply that for most objects considered, the mass is sufficiently concentrated and point-like for any errors in  CMT to pass undetected under the umbrella of imprecision of measurement.  

CMT  and   GR 

But does this sufficiently explain the success and accuracy of  CMT ?  In fact, no it does not.  There remains another aspect of  CMT  not yet discussed, and a hidden mechanism to explain it.   Basing   CMT on  NTF  alone is a failure, because any simple version of  CMT fails a compatibility test with  NGF, the heart of NTF, as well as failing the basic self-consistency test.
Looking back at examples given to justify CMT, one notices that rotation is always involved, although not necessarily gravity.  This indicates there is real physical content behind the mistaken current formulation.  If our only real knowledge of the CM comes through rotation or interaction with a uniform gravitational field, we should look more closely here for a better foundation than CMT  can offer for both phenomena in relation to the CM.
But a new formulation of CMT is necessary and desirable, to account for rotation, moments of inertia and angular momentum.  That is, paradoxically, while rotation effects are the downfall of  CMT as currently formulated, and must be excluded from it, rotation is probably the only justification for any continued use of the CM concept!   Basing some new version of  CMT on  GR or a revised version of Newtonian Mechanics offers promise.

Hidden Penalties and Dangers

Our good practical good fortune has its flip side in our theoretical misfortune and the potential dangers in assuming the validity of the concepts packaged in CMT.  It seems reasonable to address these issues.