A paradox inside Newtonian world

[Tomaz Kristan]

There is nothing paradoxical about this, merely misplaced amazement due to ignorance of simple maths.

Then please explain, why the star drifts to the left, while the construction stays put.

Or explain, that it is not so.

Or admit, that it is you, who is ignorant.

 

[arildno]

Since you evidently don't understand that these paradoxes you present are wholly independent of the actual physical model we make, but merely one allowing the construction of a conditionally convergent series, I really don't see the point of discussing this any further.
Learn a bit of maths before you think you have discovered anything interesting about physical models.
Conditionally convergent series are not interesting.

 

[Tomaz Kristan]

the limit of a conditionally convergent series will depend on the order you sum the terms together

Which conditional convergence do you have in mind? I see no conditional convergence here at all. Divergence, perhaps.

 

[arildno]

Add your infinite set of equations together, and all force terms cancel pairwise, in accordance with Newton's 3.law.

 

[Tomaz Kristan]

Please explain how, arildno.

 

[Tomaz Kristan]

There is no conditional convergence when we have only a finite number of negative (or positive) elements.

To have a conditional convergence, there must be an infinite number of positive elements AND an infinite number of negative elements.

Every body knows this.

 

[ObsessiveMathsFreak]

There is nothing paradoxical about this, merely misplaced amazement due to ignorance of simple maths.

Woah man. Relax. There is nothing simple about this mathematics.

Now, the star slowly drifts toward this structure as a result of gravity. At the same time, every ball here is frozen by the gravity of those on its left side. Behind every ball, there is another.

If you compute this, you will find that every ball, in addition to being pulled to the left by gravity, is also being pushed from the left and the right by normal reaction forces. The normal reaction force on the left of each ball would be greater than the normal reaction force on its right. Furthermore, the reaction force on the right, would be equal to the reaction force on the left side of the previous ball(to the right). The lefthand(right pointing) reaction forces must therefore grow not only to account for the increased gravity, but also all the gravity terms that came before.

Again, as with the "point mass" case, these resulting forces increase without bound, and the resulting sum is no longer a number. As a result the motion of the conjoined ball mass is not defined, and so is the motion of its center of mass. Infinity is a double edged sword.

 

[Tomaz Kristan]

Again, as with the "point mass" case, these resulting forces increase without bound, and the resulting sum is no longer a number.

True. But in the ball construction case, every force is finite. Every (resulting) force can be represented by a finite number.

 

[Galileo]

Please explain how, arildno.

Well allow me, since I wrote down that equation, I know what the terms stand for. I`ll rewrite the last sum to make the argument clearer.

F_T=G\sum_{i=0}^{\infty}(25)^i\left(\sum_{k=1}^i \frac{2^k}{(10^k-1)^2}-\sum_{k=-1}^{-\infty} \frac{2^{k}}{(10^{k}-1)^2}\right)

Evaluate this sum by pairing the term i=N, k=N-M (This is the force mass M exerts on mass N)), with i=M, k=M-N.
Without loss of generality we can take N>M, then if we take i=N, k=N-M the term is:
G(25)^N\frac{2^{N-M}}{(10^{N-M}-1)^2}=G(50)^N\frac{2^{-M}}{(10^{N-M}-1)^2}
and the term with i=M, k=M-N is:
-G(25)^M\frac{2^{M-N}}{(10^{M-N}-1)^2}
multiply top and bottom by 10^{2(N-M)} and you get:
-G(25)^M\frac{2^{M-N}100^{N-M}}{(10^{N-M}-1)^2}=-G(50)^N\frac{2^{-M}}{(10^{N-M}-1)^2}
So the terms are equal and opposite. If you arrange the sum that way, you'll get zero.

 

[ObsessiveMathsFreak]

True. But in the ball construction case, every force is finite. Every (resulting) force can be represented by a finite number.

Yes but the magnitude of the forces is increasing without bound as you progress to the left, and the forces on each mass again increase without bound and the accelleration of the center of mass will become infinite.

How are you solving for the reactive forces? As I look at the problem, attempting to solve for the reactive forces gives and infinite recursion of linear equations, and that's if you assume that all particles have the same accelleration. The recursion stops when one of the reactive forces is set to zero, i.e. you have a final particle. But if this particle does not exist, then I do not think you can in fact solve for the forces here.

 

[Tomaz Kristan]

Galileo,

The sum of all forces inside the system is meaningless. All what counts is the net force to every point in the system. Since every point moves, as this net force wants to.

And to every point there is a finite number (ore even none, for the kilo mass point) of forces pointing to the right and infinite number of negative forces.

They can't be paired for the kilo mass, not for any smaller. It is always a net negative force to EVERY mass point.

What only counts.

 

[Tomaz Kristan]

The recursion stops when one of the reactive forces is set to zero, i.e. you have a final particle. But if this particle does not exist, then I do not think you can in fact solve for the forces here.

You have an infinite number of ever smaller forces, which sums up to a certain finite value, for every point. As I said to Galileo. We have a well defined force to every point every moment and it is a finite one.

And every point moves accordingly - to the left side.

 

[Galileo]

Galileo,

The sum of all forces inside the system is meaningless. All what counts is the net force to every point in the system. Since every point moves, as this net force wants to.

The sum of all forces inside is FAR from meaningless. The fact that these cancel out due to Newton's 3rd law leads to momentum conservation. The model of using point particles is a simplification justified by the use of this very rule. Extended bodies can be modeled by point particles situated at their center of mass because all the internal forces cancel, always. What you have shown is merely a somewhat amusing result you can get using a totally unrealistic mass distribution.

It not a paradox inherent in the Newtonian world. You can make something like us up in any theory using a weird enough fractal distribution of source terms. But even in a Newtonian world there are no real point particles which you can space infinitely close together, or an infinite number of particles.

 

[Tomaz Kristan]

But even in a Newtonian world there are no real point particles which you can space infinitely close together, or an infinite number of particles.

Of course you can. Inside mathematic formalism of Newton, you have every chance to do just that. Any distribution permitted in R^3 set, you can freely use.

Still, everything should work just perfectly for every imaginable construction.

If it doesn't ... then we have a problem.

 

[ObsessiveMathsFreak]

You have an infinite number of ever smaller forces, which sums up to a certain finite value, for every point. As I said to Galileo. We have a well defined force to every point every moment and it is a finite one.

The forces in question will become infinitly larger, not smaller. In any case, how are we to calculate them? The infinite amount of particles appears to set up an infinite system of linear equations wherein to solve for the reactive forces on each ball we must refer to the ball to its left. The reaction forces do not appear to be well defined, i.e. like the previous case, we cannot say what the motion of the mass is.

And every point moves accordingly - to the left side.

That depends somewhat on your definition of "move" and "left", I'm not kidding.

 

[vanesch]

In as much as it helps, I have a small mathematica notebook which confirms what the OP said, numerically. I was too lazy to work out things analytically, from which I'm certain one can prove that the force on every individual mass point does point to the left, but numerically already this seems to be clear.

However, I think the simple reason is that we are working with a set of forces which constitutes a "conditionally convergent series".
It is well known that such series can be re-arranged to yield just any result.

Consider the total set of forces, F_ij (from mass i on mass j).
If we sum them pairwise (F_ij + F_ji) then they yield 0 for each pair.
If we sum them (as we do here) first over all i, and then over all j, then we obtain an infinite force to the left.

You can do the same thing with the series:
1-1+1-1+1-1...

If you sum them pairwise, you find 0.

If you first sum the n available +1's and then the n-1 available -1 first, then you get a sum of +1's, hence infinity.

 

[arildno]

There is no conditional convergence when we have only a finite number of negative (or positive) elements.

To have a conditional convergence, there must be an infinite number of positive elements AND an infinite number of negative elements.

Every body knows this.

You DO have an infinite number of positive and negative forces, considering the total set of forces.
It is wholly unproblematic to regroup the summation of a conditionally convergent series to diverge in one direction.

Again, the sole features of your physical model adequate to create this situation is:
1. It is permissible to form a conditionally convergent series in the model.
2. There exists permissible permutations within the model which makes the given series diverge.

There is nothing "Newtonian" about this, and hence, the "paradox" is in principle independent from Newtonian physics (you might create it elsewhere).

Now, you could say that a proper model of reality should at the very least disallow 2. in all cases, but that means you'll have to formulate a physical-permutations principle, which is unobvious to see what should be a priori.
Nor does it seem likely that such a principle is deducible from present physical evidence.

Perhaps the most viable way to avoid this sort of "paradox" is to deny the physical existence of "infinities".

 

[ObsessiveMathsFreak]

Perhaps the most viable way to avoid this sort of "paradox" is to deny the physical existence of "infinities".

That in itself may cause problems later on with theories that rely on infinite arguments, for example continum mechanics.

If we assume infinities as in the problems posed here, we find that we are, in one way or another, unable to solve for the accellerations required. We have posed a mathematical problem to which we cannot find the answer. There is nothing very unusual about this. However, it is important to note that no conclusions can be drawn from these posed problems as no valid solution exists for them.

 

[arildno]

That in itself may cause problems later on with theories that rely on infinite arguments, for example continum mechanics.

Not really, mathematical models dealing with infinities of this form will just be interpreted as "intelligent simplifications/approximations".

 

[vanesch]

There is nothing "Newtonian" about this, and hence, the "paradox" is in principle independent from Newtonian physics (you might create it elsewhere).

Well, although I agree with your explanation, I also think that this situation is a problem for the "toy universe" of Newtonian mechanics. After all, what is disturbing in this example, is that the amount of mass is finite (although distributed over an infinity of point masses).
So maybe we should restrict Newtonian universes to those which have only a finite number of point masses.

In any case, what breaks down in the given example (for exactly the reason that you give, and that I also indicated, namely the resummability of conditionally convergent series), is the dynamical prescription of Newtonian mechanics. The dynamical prescription simply says that we should use superposition of the individual interacting forces, and in the given example, it turns out that this superposition, being a conditionally convergent series, is ill-defined. In other words, in this case, the prescription of Newtonian mechanics fails to give us a well-defined dynamics.

Nevertheless, the configuration, apart from the discussable fact that we have an infinity of point masses (but with a finite total mass), doesn't seem to be ill-defined, except for one thing: the mass density is infinite in x=0:

If we take the mass density averaged over an interval [0,x_k] with length 1/10^k as rho_k, then rho_k ~ 2^(k-1)/(1/10^k) = 10^k 2^(k-1), which increases without bound for k-> infinity.

Is this the reason ?

 

[arildno]

Well, although I agree with your explanation, I also think that this situation is a problem for the "toy universe" of Newtonian mechanics. After all, what is disturbing in this example, is that the amount of mass is finite (although distributed over an infinity of point masses).
So maybe we should restrict Newtonian universes to those which have only a finite number of point masses.

In any case, what breaks down in the given example (for exactly the reason that you give, and that I also indicated, namely the resummability of conditionally convergent series), is the dynamical prescription of Newtonian mechanics. The dynamical prescription simply says that we should use superposition of the individual interacting forces, and in the given example, it turns out that this superposition, being a conditionally convergent series, is ill-defined. In other words, in this case, the prescription of Newtonian mechanics fails to give us a well-defined dynamics.

What I'm saying, is that it isn't any particularly Newtonian cast over this.

Furthermore, that divergences may appear in a physical model is in itself not a death judgment over the model (the doom of Netwonian physics has been proclaimed much better by, for example, reality..).
For example, in QM, divergences appear that can be handled by a careful cut-off procedure.

 

[vanesch]

What I'm saying, is that it isn't any particularly Newtonian cast over this.

Well, the difficulty is mathematical of course, and I guess that what you are trying to say is that one can think of other kinds of toy universes in which it is possible to construct conditionally convergent series of the kind.

But that's exactly the point: given that within the axioms of the theory, it is possible to arrive at such series, means that there is a problem with the axiom set, in that it bumps into this mathematical difficulty.

Of course, our universe not being a Newtonian universe, we don't care. We're talking here - I presume - about a toy universe in which Newtonian mechanics is strictly valid, and in which there are genuine point masses. The surprise is here that it is possible to construct a totally acceptable (caveat infinite number of particles and infinite mass density, but nevertheless finite total mass) configuration for which Newtonian mechanics doesn't prescribe a well-defined dynamics.

This is a bit as having a force law F ~1/Sqrt[distance - 5.0meters]

Clearly this force law breaks down when the distance is smaller than 5 meters. Now, you could object that this is simply a problem because the square root of a negative number is imaginary, which cannot be a compoent of a tangent vector to a real 3-dimensional space, and hence that this is not any particular problem for this force law, it being a mathematical problem. The point is, the mathematical problem occurs because it has been prescribed by your force law !

So the appearance of the conditionally convergent series does have certain mathematical difficulties, but the very fact that this series appeared is due to the formulation of Newton's laws in the first place.

It is not so much its divergence, as its undefined-ness which is the problem.
The force is *undefined* according to Newtonian dynamics.

 

[arildno]

Well, you could equally well say that the divergences appearing in QM is due to that the QM model is dealing with a toy universe, and not the real one.
There's not any particular reason to be concerned about that, either, since QM works nonetheless.

 

[vanesch]

Well, you could equally well say that the divergences appearing in QM is due to that the QM model is dealing with a toy universe, and not the real one.
There's not any particular reason to be concerned about that, either, since QM works nonetheless.

Sure, I don't think that that was the point. As long as you say that the theory at hand is an effective theory, and not a genuinly fundamental model, then you can get away with pathological cases, of which you can say that they fall outside of the scope of applicability of the theory. But it does show that the theory as set up, cannot be the axiomatic basis for an entirely consistent fundamental theory ; unless you adapt the axiomatic basis such that you or resolve or eliminate these potential pathological cases, making sure, in the last case, that they cannot arise spontaneously.

 

[Tomaz Kristan]

I pretty much agree with vanesch. Newton's laws should be modified such a way, that this construction would become impossible.

Now it is a very possible one. Even the mass at 0 is 0. Everywhere else is finite or 0.

On one hand I also agree with arildno and Galileo, that those forces may be paired out. The "beauty" of a system with paradox inside is, that at the same time, this statement is clearly false. Every statement is true and false inside an axiom system with a contradiction. Anything can be proved. So, on the other hand I don't agree with them.

As I've said - "the beauty" of a paradoxical system!

 

[vanesch]

I pretty much agree with vanesch. Newton's laws should be modified such a way, that this construction would become impossible.
Now it is a very possible one. Even the mass at 0 is 0. Everywhere else is finite or 0.

Well, I suggested 2 difficulties. First, we could postulate that a Newtonian world only contains a finite number of mass points. But that's rather limiting, especially if we want to do things like continuum mechanics.

On the other hand, I pointed out that the mass density in your system goes to infinity at 0 (the ratio of the total mass in an interval divided by the interval's length). This is because your reduction in distances has a bigger factor (10) than your reduction in mass (2).

Now, the funny thing is that if you make these two EQUAL, so that you get a finite mass density on the line, then the total force doesn't diverge anymore, but goes to a non-zero constant. Nevertheless, it is still not 0, and now we have a finite line density, which still corresponds to an infinite volume density.
This can be seen in notebook massesonline2

If, however, you make the ratio of the masses strictly bigger than the ratio of the distances, so that the line density goes to 0, then the total force DOES converge to 0 (so that the CoG remains in place). An example in massesonline3.

So maybe the requirement we should impose is that the mass density in volume remains finite. The question is: how are we going to avoid DYNAMICALLY that this situation occurs ! It is pretty difficult to impose arbitrary constraints on configuration space: you have to make sure that your dynamical flow doesn't lead you there from within an allowed region.

 

[Galileo]

All this discussion...
Look, Newton's laws are physical principles, it's not mathematics. As such, you have to use a certain intelligence when applying them to physical problems. First of all, your model of the problem should be realistic. The given problem uses point particles, not as some approximation to real physical objects, but really as abstract mathematical points in R^3. That is just crappy physics.
Newton's laws do not speak of point particles. It is a useful approximation to physical objects in some cases, when used intelligently, which use is justified by the third law.
Newton's laws have not been shown to be inconsistent because of this and nor should they be modified because of this. If your model predicts something that can't doesn't happen in real life, then either the theory is wrong or your model is screwy. I easily vote for the latter on this one. It's an amusing observation, but let's not dwell on it.

 

[Tomaz Kristan]

I think, I should clarify those structures:

- the first (in linked pdf) is a structure of unsupported mass points

- the second has rods with negligible mass to keep the distance

- the third is of balls. Every ball 1000 times smaller in volume and 2 times in mass from the preceding sphere on its right. They all touch, except the big star far on the right side.

For the third case, there is no singularity anywhere. Small balls supports each other and drag the star closer.

The simplest and the strongest case!

For the singularity in the Newtonian world - every mass point is a singularity by definition.

 

[Tomaz Kristan]

Look, Newton's laws are physical principles, it's not mathematics.

Oh no, it is an axiomatic abstract theory based on the mathematics invented by Newton. About 100+ years ago, almost everybody asserted the infinite divisibility of the space, time and mass. That Newtonian world was generally regarded as the perfect model for our real world.

Today we know, it is not. We can't make arbitrary small and dense balls, since we soon produce a black hole. Which also evaporates, can't be squeezed and so on.

I do not assert, that this is a real life paradox. It isn't. But it is a paradox of the Newtonian theoretical space. We know for the last 100 years, that facts do not mirror Newton's theory. Now I think it is clear, it is not even coherent.

 

[Galileo]

Oh no, it is an axiomatic abstract theory based on the mathematics invented by Newton. About 100+ years ago, almost everybody asserted the infinite divisibility of the space, time and mass. That Newtonian world was generally regarded as the perfect model for our real world.

Newton paved the way for using a mathematical description of Nature. A method by which we could analyze and study forces in a systematic way. He didn't assume it was the ultimate or perfect model. There were lots of things unknown which he acknowlegded, but instead of making hypotheses he provided a means of discover more about the world so that we someday might learn more. About the indivisibility of mass, Newton believed everything (even light) was made of tiny indivisible particles.

Point is: A physical theory provides a framework to work with. To give insight in Nature and to predict what we can know. We use Euclidean R^3 to describe problems, it does not mean we live in R^3. Physics happens in the real world (or the laboratory), not in Hilbert space or something.