A paradox inside Newtonian world

[Tomaz Kristan]

Three weeks ago I have constructed this apparent paradox inside the Newtonian world.



http://critticall.com/alog/Antinomy_inside_mechanics.pdf"

Am I wrong or not? What's your say?

- Thomas

 

[arildno]

You're calculations are wrong. That's all there is to it.
I'll see if I am going to bother to show you your computational mistakes.

 

[Galileo]

Hello Tomaz,

Gravity (an internal force) however, relentlessly works. The rightmost mass
point starts to fall to the left. That is obvious, since here is no force to
drag it right. Right?

That's correct.

What about that one on 1/10? The gravitational influence of the 1 kg point
9/10 away is much smaller, than the influence of the one at 1/100.

The one on the right is farther away than the one on the left, but the right one is also heavier. The gravitational force also increases with mass. But in this case, sure, the force the left one exerts is way higher than the one on the right.

We can prove this for every single point. So, it is clear, that all points
drifts to the left side, so does the center of gravity.

You can in general show that the force the particle on the left exerts is 25 times higher than force from the right one. But to determine the motion of a particle you have to consider the NET force on it, which is the sum of the gravitational forces from ALL other particles, not just the ones next to it. Take these into account in your analysis and you'll see that the center of mass at 10/19 meters will stay in place.

 

[Tomaz Kristan]

The NET force IS negative, it is pointing to the left.

We have only a finite number of masses on the right. Every one of them, more than balanced with those on the left.

For example. The mass at 1/10 is pulled to the right by the mass at 1, with 25 times less force, than to to the left, by the mass at 1/100.

Since that one at 1/100 is 4 times smaller than that at 1 - and 10 times closer.

Holds for any other also. Those big masses on the right are quickly outweighed by those smaller on the left, which are much closer.

Right?

EDIT: typo.

 

[arildno]

Haven't you heard of Newton's 3.law?
Add all the component F=ma together, the sum of the force terms necessarily reduce to 0 due to N 3.law, hence the right-hand side also equals zero, meaning that the C.M has acceleration 0.

So, your conclusion is dead wrong; all that remains is to do a bit of tedious calculation to show exactly where the flaw in your argument is located.

 

[Tomaz Kristan]

Of course I've heard about the Newton Third Law.

That's why, we have a paradox here. A theorem, which opposes this law.

The calculation is quite simple, however. We have LESS force to the left, as it had been only a mass twice as big in the neighboring point, as it is. As stated, it is dispersed almost to 0.

The force to the right is even much smaller, due to a greater distance. 10 times the distance, means 100 times less force, per mass unit.

Don't you see?

 

[Doc Al]

How silly. Distribute your masses any way you want. If gravity between the masses is the only force acting, they all move towards the center of gravity. So what? You don't seriously think that they all move left, do you?

 

[Tomaz Kristan]

The center of mass is at 10/19.

Do you think, that the mass at 1/10 goes there? Ignoring the forces at its left side?

 

[chronon]

Three weeks ago I have constructed this apparent paradox inside the Newtonian world.
http://critticall.com/alog/Antinomy_inside_mechanics.pdf"
Am I wrong or not? What's your say?
- Thomas

It looks to me like you're right; there does seem to be a paradox.

Newtonian gravity does have problems with infinity, e.g. it's possible to devise a system of bodies where one goes to infinity in finite time, and of course there's always the question of how an infinite universe can remain static. However, I've not come across your paradox before.

One thing to take into account is that you're only considering the instantaneous force acting on each particle. It might be interesting to consider what happens to the system over a period of time - my feeling is that as you get closer to the origin, the time spent traveling left by each particle would get shorter and shorter before it met its neighbour and started traveling to the right. Hence it might turn out that the centre of mass doesn't actually move.

 

[Tomaz Kristan]

chronon,

I have thought about that. That the half kilo mass could fall back, doe to the escaping masses on the left side and therefore prevailing gravitational pull from the biggest 1 kilo mass. But that would mean, that the mass center travels left even faster.

And also, you may initially put the solid rods with a negligible mass between every two consequent balls. Then remove only the rightmost one.

This way the shape on the left side and therefore forces are stable long enough. Only the biggest ball is slowly drifting left, eventualy passes the 10/19 mark, where the old center of gravity stood.

Smaller balls can't go right for the gravitational pull from the left, nor left for the strong solid rods between them.

We have a Newtonian version of the so called Infinity Hotel, weighting 2 kilograms. And doing essentially more troubles.

EDIT: typo

 

[chronon]

chronon,

I have thought about that. That the half kilo mass could fall back, doe to the escaping masses on the left side and therefore prevailing gravitational pull from the biggest 1 kilo mass. But that would mean, that the mass center travels left even faster.

But the masses will hit each other eventually. What I am saying is that the time taken for this will get shorter and shorter as you go towards the origin, and when the masses on the left have joined together they will start moving to the right. There may be no interval of time in which the centre of mass is actually moving to the left.

And also, you may initially put the solid rods with a negligible mass between every two consequent balls. Then remove only the rightmost one.

But if you join the masses on the left together then the local forces between them will be canceled out by the force from the rods, and only the force of the rightmost mass will remain. Hence the paradox will disappear.

 

[robphy]

So, what happens to the left-most mass?
The net-force on it points to the right.

It seems to resolve this, consider a finite number (N<infinity) of masses arranged according to your scheme. Then, take the N-> infinity limit.

 

[Hurkyl]

Although there is ostensibly zero mass at the origin, it is being flung rightwards with infinite force. When you figure out how to handle that singularity, it will almost certainly provide the momentum contribution that you're missing.

 

[russ_watters]

The calculation is quite simple, however. We have LESS force to the left, as it had been only a mass twice as big in the neighboring point, as it is. As stated, it is dispersed almost to 0.

Why don't you do the calculations and post the results. You're not likely to find someone here to do them for you here.

 

[Galileo]

Alright, the net force on every mass is indeed pointing to the left. The problem is that you have infinitely many particles and the forces are tending towards infinity. In this way, the expression for the TOTAL force is the system is a sum which will diverge if counted one way (particle wise), but will be zero if the forces are counted pairwise.

I'd say the way to solve the paradox is to say the situation is unphysical and the correct way to handle it is like robphy said and calculate it for a finite number of particles, then take the limit N-> infinity.

 

[Hurkyl]

I'd say the way to solve the paradox is to say the situation is unphysical

That doesn't work; the apparent paradox doesn't hinge on the assumption that it's a physical situation. (Of course, it does hinge on the assumption that this is an allowed mass distribution in the mathematical formalism)

 

[Tomaz Kristan]

Although there is ostensibly zero mass at the origin

There is NO mass at 0. Since every mass is at 10^N. None at 0 at the beginning.

And I did the calculations. No right pointing NET force.

And the case with solid rods between every pair of neighboring masses, except between the last two, is astonishingly simple. Let understand and agree with this simpler case first!

 

[Hurkyl]

There is NO mass at 0.

I'm pretty sure there is zero mass at the origin, not "NO mass".

And I did the calculations. No right pointing NET force.

I was talking about the gravitational field at the origin.

And the case with solid rods between every pair of neighboring masses, except between the last two, is astonishingly simple. Let understand and agree with this simpler case first!

That's easy. The left complex exerts a leftward gravitational force of approximately 1.122 G on the right mass. The right mass exerts a rightward gravitational force of approximately 1.122 G on the left complex. (assuming the rods have zero mass)

 

[Tomaz Kristan]

The left complex exerts a leftward gravitational force of approximately 1.122 G on the right mass.

So, you say, the half kilo mass is falling toward the right one with the same acceleration as every other? No tidal force inside the left complex?

The rod between two balls just keeps the distance. Doesn't glue them together!

So how this uniform acceleration is possible?

 

[Tomaz Kristan]

In fact, every ball is falling, as the combined force of all other balls demands.

And this is the net force, always pointing to the left, balanced by the rod, like standing on the floor. The Moon will not pull you up, Jupiter might, when close enough.

Here we have an infinite cascade of ever smaller, yet closer balls. Every sphere is tide down by all bellow.

And so to speak, Jupiter is falling down.

 

[Hurkyl]

No tidal force inside the left complex?

There is certainly a considerable amount of tension in the rods, but you already hypothesized that they are perfectly rigid, so they can withstand the stress without expanding. Your masses are point particles, so nothing is happening to them either.

Or, did you mean that your masses are unattached to the rods? In that case, the behavior is, as far as I can tell, identical to your original problem. (unless you suppose the existence of a "point rod" between the origin and your complex... in which case you will get the uniform acceleration I mentioned)

What does Jupiter have to do with anything?

 

[Galileo]

That doesn't work; the apparent paradox doesn't hinge on the assumption that it's a physical situation. (Of course, it does hinge on the assumption that this is an allowed mass distribution in the mathematical formalism)

Well, it IS a physical paradox, since it apparently violates conservation of momentum. Mathematically there's nothing weird; the proof that the sum of internal forces is zero doesn't work here because that sum is a divergent series.

I did the calculation. The total force can be gotten from:

$$F_T=G\sum_{i=0}^{\infty}(25)^i\left(\sum_{k=1}^i \frac{2^k}{(10^k-1)^2}-\sum_{i=0}^{\infty} \frac{50^k}{(10^k-1)^2}\right)$$
the rightmost sum is about equal to 1.1226 (in appropriate units).
This sum is clearly diverging. But as long as you have a finite number of particles, say N, then the first sum goes from 1 to N and the last one from 1 to N-i and you can rearrange the terms in the sum to see they cancel pairwise.

 

[Tomaz Kristan]

You can also rearrange those balls on the following way:

First, you have a 1/256 kg ball. Very small and dense, holding in the air. You hang a bigger ball of 1/128 kg on the bottom of that one. Since both spheres are small enough, the gravity will press them together. Still holding the top one by an external force.

Now, you can put another, two times bigger, beneath the two. Until you come to the 1 kilo, holding almost two kilos in the air, supporting only the highest and smallest one.

Now you extend this complex at the top with an ever smaller and more dense ball.

You do that for every natural N and there is exactly 2 kg construction, with no top ball, which hangs in the air with no external support at all! Every ball clings under the one above by the pure gravity.

However, the planet will slowly drift up. Very, very slowly of course.

And yes, this is only possible inside an ideal Newtonian world. Not here. What is a shame, anyway.

 

[Tomaz Kristan]

You can point this 2 kg construction of infinite number of spheres, each 10 times smaller by radius and 2 times smaller by mass to the nearest star. With wider end closer to the star and wait for that star - to be dragged over here!

Every ball of this meter or so long device will be coined to rest by those ever smaller balls with the enormous gravitational pull. When the star will come close eventually, the biggest ball will fly to that star, but you still have plenty of them in the row.

Of course, not in the real life but in the ideal world of Newtonian laws an Newtonian gravity it must work this paradoxical way.

EDIT: radius instead volume.

 

[arildno]

Assuming Galileo is correct, this has nothing whatsoever to do with "Newtonian physics", "Einsteinian physics" or any physics whatsoever.

It is simply a restatement of the mathematical fact that a merely conditionally convergent series, when rearranged, may give a different value to the infinite sum.

You would be able to generate similar "paradoxes" in just about any physical model in which a conditionally convergent series would appear.

 

[Tomaz Kristan]

I know, and I said, it is not a paradox inside the real world.

It is however a clear case of an antinomy, paradox or contradiction inside the abstract Newtonian world.

I don't care, why it arises. I care it is here and not noticed before. If you eliminate the infinity, you eliminate this paradox also.

But for now, infinity is a legitimate tool, I may use. Too bad, if leads to a paradox.

 

[ObsessiveMathsFreak]

The paradox has emerged because of the introduction of an "infinite" number of particles. With any finite number of particles, the system will be fine as eventually particles will be forced to the right. However, with an infinite amount of particles, problems begin to emerge.

To begin with, the center of mass calculation no longer makes sense. The center of mass is at 10/19 as already mentioned. However, a little calculation shows the gravitational potential energy of the system at the point x=0 to be;

$$U=\sum_{n=0}^{\infinity} -\frac{2^{-n}}{10^{-n}}$$

This sum diverges, i.e. the center of mass calculation is no longer valid for gravitational potential energy at least. This is a hint that something will go wrong later on.

If you compute the force on the particles, you will find that the force is indeed always to the left, but is a divergent sum. Moreover, as the particles are decreasing in size, their accellerations will also be divergent, growing without bound.

We have the center of mass given by;
$$R=\frac{1}{M}\sum_{n=0}^{\infinity} m_i r_i$$ where $$r_i$$ is the position of each particle.

Computing the accelleration of R gives us;
$$\frac{d^2 R}{dt^2} = \frac{d^2}{dt^2} \frac{1}{M}\sum_{n=0}^{\infinity} m_i r_i$$
so
$$\frac{d^2 R}{dt^2} = \frac{1}{M}\sum_{n=0}^{\infinity} m_i \frac{d^2 r_i}{dt^2} =$$

As noted before, even with the decreasing masses, the sum on the right hand side is divergent. What this means is that the accelleration of the center of mass R, is singular, infinite, not a number.

So when we pose the question; "For an infinite number of particles positioned in this fashion, how will the center of mass change?", the answer is; "We cannot say". It is invalid to say that it will move to the left, as the distance it moves in any finite time is incomputable. The center will not drift or jerk some finite distance. It will move an "infinite" amount in any finite non zero period of time.

There is a singularity in the answer which prevents us from computing the new position of the center of mass. We can compute the new position of any individual mass, but despite this, we cannot do the same for the center of mass.

Once upon a time I wrote a http://www.obsessivemathsfreak.org/wordpress/?p=5 on how indroducing infinities into mathematical equations can lead you into paradoxes. This is simply another one, albiet more complicatied than simpler integral ones.

 

[Tomaz Kristan]

Almost agree with you, ObsessiveMathsFreak!

Let we have the infinite number of solid balls on the X axis.

For every N=0, 1, 2, ... there is a ball with the radius of 10^-N meter and with the mass of 2^-N kg.

From right to left, from the biggest to ever smaller and more dense, they touch each other.

Far away from this two kilo and 10/9 meter long structure, there is a star on the same axis, on the positive side, several billion meters away.

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.

Another form of the same paradox, already mentioned here, with no moving parts, except for the star!

 

[Tomaz Kristan]

This paradox is constructed on a finite amount of mass. While this one http://adsabs.harvard.edu/abs/1989Ap&SS.159..169H"
- the so called Seeligers paradox, uses an infinite amount of mass inside infinite space.

I think, mine is at least as that important - if not proven wrong, of course.

Guess not!

 

[arildno]

I know, and I said, it is not a paradox inside the real world.

It is however a clear case of an antinomy, paradox or contradiction inside the abstract Newtonian world.

I don't care, why it arises. I care it is here and not noticed before. If you eliminate the infinity, you eliminate this paradox also.

But for now, infinity is a legitimate tool, I may use. Too bad, if leads to a paradox.

NO, it is not a paradox in an "abstract Newtonian world".
It is a reflection of the simple fact that the limit of a conditionally convergent series will depend on the order you sum the terms together.
There is nothing paradoxical about this, merely misplaced amazement due to ignorance of simple maths.

 

[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.