A paradox inside Newtonian world

[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.

 

[Tomaz Kristan]

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.

We know this today. Not so long ago, it was the same thing.

But it is not the question of the real world. It is a problem of a theory, which has a hole.

The so called Naive set theory also had a hole, named Russell's paradox. So they fixed it. At least they hope so, it has been fixed.

If I am right, the Newtonian world is also broken "a little" and needs a fix.

 

[ObsessiveMathsFreak]

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

Woah there cowboy. You still haven't proved that yet.

 

[Tomaz Kristan]

Woah there cowboy. You still haven't proved that yet.

Well, so let we do that, again!

You have the next configuration of spheres on the line, in the Newtonian ideal world.

At 1 mm we have centered a ball of 1 mm perimeter and 1 tone of mass. Next to it, on the left side, we have 0.1 mm perimeter ball, 1/2 tone of mass.

And so on. For every natural another 10 times smaller ball by its perimeter, 1000 times by volume, 2 times smaller by mass and 500 denser, then the right side neighbor.

The whole cascade is 10/9 mm long, weights 2 tons, does not moving anywhere for now.

Far away, on the right side, 10 light years away, on the same line, we have Jupiter, also just resting there.

Every ball is now under the net force of every other ball. Small balls are quite coined by gravity to each other. The gravity from the right side is smaller than from the left, for every ball. The reaction force of the surface prevent them to collapse, of course. So they stay put for now.

While Jupiter feels a tiny gravity toward those balls and has no choice but to go there very slowly.

Before the Jupiter's tidal forces will result a disruption inside this structure, we will see it coming closer and closer. The spheres inside the structure will not move anywhere.

What do you need to drag Jupiter? Enough balls. Infinite number of them, arranged as above. And with a very modest mass.

Of course, NOT in a real world. In Newtonian abstract world, only.

How could you not agree?

 

[Galileo]

We know this today. Not so long ago, it was the same thing.

If you are referring to relativity, that's not what I meant. Mathematics and physics are both able to stands on their own, since they are in essence separated disciplines in the sense, for example, relativity (curved spacetime) has not disproved Euclidean geometry (you can't disprove axioms, certainly not with physics, it's meaningless). Nor can a mathematically abstract construction (like the one you proposed) falsify a logically consistent physical theory, only experiment can do that.

So treat a physical theory from a physical viewpoint. Functions that are continuous everywhere, but nowhere differentiable are nice, but don't use it to describe as the trajectory of a particle (without due thought anyway).
I got the feeling you got a more mathematical background and in a much less extend physics.

But it is not the question of the real world. It is a problem of a theory, which has a hole.

I can find similar 'problems' in any other physical theory.

The so called Naive set theory also had a hole, named Russell's paradox. So they fixed it. At least they hope so, it has been fixed.

That's mathematics, not physics which I believe to be independent of the assumptions of set theory, choice axioms or continuum hypotheses.

If I am right, the Newtonian world is also broken "a little" and needs a fix.

I'm not surprised if Newton knew of similar constructions, but disregarded them on physical grounds.

 

[Tomaz Kristan]

I can find similar 'problems' in any other physical theory.

I don't see them that many. A few, dealing with the infinite mass.

That's mathematics, not physics which I believe to be independses.

An axiomatic theory is under fire. Doesn't matter the real world here, a bit!

I'm not surprised if Newton knew of similar constructions, but disregarded them on physical grounds.

Who knows? He forgot to tell us about this one, for sure.

 

[ObsessiveMathsFreak]

The whole cascade is 10/9 mm long, weights 2 tons, does not moving anywhere for now.

You still haven't proved that. You haven't shown how the accelleration of the mass is derived. You certainly haven't included a far off mass to the right in your calculations yet.

Far away, on the right side, 10 light years away, on the same line, we have Jupiter, also just resting there.

And each and every mass(an infinite amount of them), will be pulled to the right by Jupiter. There could be a problem here.

How could you not agree?

I'm stubborn that way. But you'll find an equation or two usually helps to move me. That is, if they can be solved.

 

[vanesch]

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.

Well, we have different points of view here. Normally, a physical theory is in fact a mathematical theory, with an axiomatic setup, which describes a certain toy world. This theory can now face 2 kinds of difficulties: it can be confronted with intrinsic difficulties as a mathematical theory, and it can be confronted with disagreement with experiment.
If it is in disagreement with experiment (that means, things happen in the toy world of the theory which happen observably differently in the real world, in the lab), then it is a theory which has been empirically falsified. it could have been correct, but it just isn't our world.
However, if it has an internal problem, then it isn't even a universal theory. It can then be at most a kind of effective model that applies more or less in certain "real-world" situations, but there is no consistent, well-defined toy world that goes with it, unless we can add axioms which eliminate the cases that make offense (and hence, in fact change the physical theory).
What the OP is claiming (in my opinion, correctly), is that up to now it was assumed that Newtonian mechanics DID have a consistent structure, and hence DID have a toy world associated with it, only, the real world wasn't completely like that. Now, it seems that it doesn't even have a toy world in the first place. That means, it is not conceivable that there could have been a universe in which Newton's laws are absolutely valid.

 

[vanesch]

If you are referring to relativity, that's not what I meant. Mathematics and physics are both able to stands on their own, since they are in essence separated disciplines in the sense, for example, relativity (curved spacetime) has not disproved Euclidean geometry (you can't disprove axioms, certainly not with physics, it's meaningless). Nor can a mathematically abstract construction (like the one you proposed) falsify a logically consistent physical theory, only experiment can do that.

The point is in fact that an inconsistency has been derived.
You have an axiomatically non-forbidden setup for which two contradictory answers to its solution (motion) can be constructed, and which all respect all axioms of the theory.

 

[vanesch]

What do you need to drag Jupiter? Enough balls. Infinite number of them, arranged as above. And with a very modest mass.

Of course, NOT in a real world. In Newtonian abstract world, only.

How could you not agree?

Yes, but again, you have a point of infinite volume mass density.
I'm starting to think that that is the problem.

 

[Gelsamel Epsilon]

I don't get it... Wouldn't the balls move towards Jupiter as well?

 

[Tomaz Kristan]

Yes, but again, you have a point of infinite volume mass density.
I'm starting to think that that is the problem.

Every standard Newtonian mass point has an infinite density. Here, in the ball case we have an infinite density out of the structure. Nowhere inside the complex.

 

[Tomaz Kristan]

I don't get it... Wouldn't the balls move towards Jupiter as well?

It would, if there were a finite number of balls. This way, every ball has a left companions, which easily overweights the Jupiter's gravity. They are much smaller, but very much closer, too.

 

[ObsessiveMathsFreak]

Every standard Newtonian mass point has an infinite density. Here, in the ball case we have an infinite density out of the structure. Nowhere inside the complex.

Well, the density is becoming infinite as you progress to the left. The density is M/V which is 2^-n/10^-n = 5^n, so things are becoming infinitely dense. You could reduced the ball mass, but what would that do to the gravity calculation?

 

[vanesch]

Every standard Newtonian mass point has an infinite density. Here, in the ball case we have an infinite density out of the structure. Nowhere inside the complex.

This is correct, but in standard Newtonian physics (with gravity forces only), you can replace every point mass by a sphere with finite density, as long as no trajectory comes closer than the radius of the sphere. So with a point mass distribution which has a finite lower boundary on inter-point distances, you could in principle replace them by spheres with radius = half of the lower boundary. Note that this doesn't even violate Poincare's recurrence theorem, because you could even introduce *shrinking* spheres of which the constant radius, at a certain time, is always half of the "distance of closest encounter". As such, the dynamics is not perturbed, and at any finite time, the volume density is finite with a global upper boundary.

In your construction, you don't have that, because arbitrary high densities are present from the start.

 

[Tomaz Kristan]

Well, the density is becoming infinite as you progress to the left. The density is M/V which is 2^-n/10^-n = 5^n, so things are becoming infinitely dense. You could reduced the ball mass, but what would that do to the gravity calculation?

I don't see, in which section of space, we have an infinite density here. It goes over every fixed number, but still remains finite everywhere, except in the point "the rightest point of the biggest ball minus 10/9", outside the structure.

Another peculiarity.

 

[Tomaz Kristan]

In your construction, you don't have that, because arbitrary high densities are present from the start.

No, no. I have! A few post higher I gave exactly this case.

 

[ObsessiveMathsFreak]

I don't see, in which section of space, we have an infinite density here. It goes over every fixed number, but still remains finite everywhere, except in the point "the rightest point of the biggest ball minus 10/9", outside the structure.

Yes, every ball does have a finite density. But, which density is the largest?

 

[Tomaz Kristan]

Yes, every ball does have a finite density. But, which density is the largest?

No ball has the largest density. No ball is the leftmost. No natural is the biggest.

That's the "beauty" of the infinty.

 

[Gelsamel Epsilon]

It would, if there were a finite number of balls. This way, every ball has a left companions, which easily overweights the Jupiter's gravity. They are much smaller, but very much closer, too.

Despite there being infinite balls inbetween 1mm and 10/9mm there is a FIRST at 1mm. The first ball would be feeling no gravity from his left.

Unless I'm still misunderstanding?

Assuming you have an infinite line of balls forever extending so that one always does have a bigger one to the left then my reply would be that we don't know how to properly define infinity. You can prove all sorts of things if infinity was a tangible value. But it's not. You can't have 'infinity' balls.

 

[vanesch]

No, no. I have! A few post higher I gave exactly this case.

I don't think so. There is, at a given time, no upper bound to the volume density in all of space. For each given value of volume density, you will find a ball which has a higher density.
This means that I can find a sequence of points in a compact space, x_n, such that rho(x_n) > N for any N ; in other words, I have a sampling of the function rho which diverges and hence rho itself diverges.
In my description, at each given t, there is an upper density in all of space (that is, the entire density function in all of space is bounded). That bound can increase as a function of t, but for a given t-value, there can be an upper bound.
For each given t-value, you cannot find a divergent sampling of rho.
rho(x) < Max(t).

 

[vanesch]

No ball has the largest density.

That's exactly what it means, for the density to diverge.

 

[Tomaz Kristan]

Yes. Density goes over every rho, just as natural numbers go over every N.

Still, every natural is finite, as every density in a finite volume is finite in this structure. You can't show me a finite volume with an infinite density inside the structure.

Nothing illegal here, at all! At least not for the Newtonian world.

In real physics, that is a different matter, of course. You can't get arbitrary high density in the real world.

 

[ObsessiveMathsFreak]

Yes. Density goes over every rho, just as natural numbers go over every N.

Still, every natural is finite, as every density in a finite volume is finite in this structure. You can't show me a finite volume with an infinite density inside the structure.

The density is diverging, it is becoming infinite, as are all the forces on every particle. There are an infinite number of particles so to find the rate of change of the center of mass, we need to sum the rates of change of every mass. But all the numbers involved here are divergent so we no longer have the ability to find the value of the infinite sum.

In order to say that the center of mass do not move, or moves to the left, you must in fact compute this movement, or the accelleration of it instead. It's not enough to make an argument without any mathematics at all, or with only half the mathematics present. You must show your calculations if you wish to make a conclusion about the accelleration. If you do, you will find that you obtain a divergent sums, something that isn't mathematically well defined.

 

[Tomaz Kristan]

I do not care for divergent densities. Why should I?

Ever smaller mass, ever closer distance, ever greater forces - but so what?

It is by no way something illegal.

The calculations? What calculations? They are simple, almost trivial, when we talk about this "touching balls" example.

Case pretty much closed, I reckon.

 

[Galileo]

If it is in disagreement with experiment (that means, things happen in the toy world of the theory which happen observably differently in the real world, in the lab), then it is a theory which has been empirically falsified. it could have been correct, but it just isn't our world.

You're right. So I`ll just wait for experimental confirmation of the prediction that the center of mass in this construction will go to the left. Then the theory can be adjusted.

Even an axiomatic physical theory is not pure mathematics. It is not as strict. As I said: A certain intelligence has to be used when applying the theory to problems.

 

[ObsessiveMathsFreak]

The calculations? What calculations? They are simple, almost trivial, when we talk about this "touching balls" example.

There's quite a few of them though. The calculations are as you say, relatively straightforward. There may be a slight issue with the answers though.

Case pretty much closed, I reckon.

Before you go, if you could just throw down those equations, for the sake of completeness if nothing else. I'd just like to say I saw them. You'll forgive an old man his idosyncracies.

 

[vanesch]

Even an axiomatic physical theory is not pure mathematics. It is not as strict. As I said: A certain intelligence has to be used when applying the theory to problems.

Ah ? To me an axiomatic physical theory is a mathematical theory + a rule to link the mathematical objects in it to observations (which, in the case of Newtonian physics, is rather trivial).

 

[Tomaz Kristan]

ObsessiveMathsFreak,

Very well.

1 - do you agree that the force from the RIGHT side is always finite, for every ball?

2 - do you agree that the force from the LEFT side is always SMALLER than TWICE the immediate left neighbor's gravity force? [As the immediate left neighbor contains half of the mass on the left side, for every sphere.]

What means, that we have a finite force everywhere?

Do you agree then, that those forces are all balanced by the surface reaction force of every ball?

(Now, is it normal, that the tex preview doesn't work? So I can't write down this as a bunch of formulae!)

 

[Tomaz Kristan]

To me an axiomatic physical theory is a mathematical theory + a rule to link the mathematical objects in it to observations

That's the idea, I am also certain. It's no doubt, in fact.

 

[Gelsamel Epsilon]

I think the problem lies in that you cannot have 'infinite balls'

 

[Tomaz Kristan]

I think the problem lies in that you cannot have 'infinite balls'

Of course. Only that Newtonism forgets to tell you that.

Everybody knows, that you can't have the infinite number of (even ever smaller) balls in the real life.

Here, we are talking about an abstract theory, which is apparently inconsistent. What is still officially unknown and unheard of.

 

[radou]

Of course. Only that Newtonism forgets to tell you that.

So what?

P.S. I don't mean to sound rude, I'm just asking a simple question.

 

[ObsessiveMathsFreak]

1 - do you agree that the force from the RIGHT side is always finite, for every ball?

2 - do you agree that the force from the LEFT side is always SMALLER than TWICE the immediate left neighbor's gravity force? [As the immediate left neighbor contains half of the mass on the left side, for every sphere.]

I'm finding it difficult to follow in words.

What means, that we have a finite force everywhere?

But there are an infinite number of forces.

Do you agree then, that those forces are all balanced by the surface reaction force of every ball?

Are they? You haven't proven that yet. The reaction forces may or may not be cancelling everything out.

(Now, is it normal, that the tex preview doesn't work? So I can't write down this as a bunch of formulae!)

Please, do try. Otherwise I'm going to have to write them out instead.

 

[Gelsamel Epsilon]

Hmm, well I've been told numerous times that infinity amounts of things screws everything up.

Not only that but you cannot get "infinity" amount of things so the problem is irrelevant.

 

[Tomaz Kristan]

So what?

P.S. I don't mean to sound rude, I'm just asking a simple question.

It's still possible, that I am wrong. I don't see how, not do anybody else, but it could be, after all.

The other possibility is, that we will be forced to abandon those infinities, and everything will be under control, again.

I would like, that the infinity concept was illogical. Most people hate this idea, though.

Unwanted side effect could also be, that people will become more agnostic, scientifically. They will say: For more than 300 years, you had an error just before your noses, and you haven't seen it! How one can believe science?

That would be a bad thing to happen. In fact, science only harbored the magic (of infinity) for too long. Once we clean it, the science will be better than ever before.

 

[Tomaz Kristan]

Not only that but you cannot get "infinity" amount of things so the problem is irrelevant.

I repeat. Not in reality, that is quite common view lately. But in well established axiomatic systems, like Newtonian mechanics, the infinity is included as a vital part.

 

[Tomaz Kristan]

Left side forces:

2*F(x,x+1)>F(x,x+1)+F(x,x+2)+F(x,x+3)+...+F(x,x+y)+...

Is that true, ObsessiveMathsFreak?

 

[Tomaz Kristan]

F(x,x+1)=G*m(x)*m(x+1)/d(x,x+1)^2

D(x,x+1)=(10^-x+10^-(x+1))/2

 

[Tomaz Kristan]

m(x)=2^-x

So, is that true, that the force on every ball is finite, ObsessiveMathsFreak?

 

[ObsessiveMathsFreak]

What about the reactive forces? You're not solving for them.