How Can the Newtonian Spherical Ball Be in Stable Equilibrium?

[Weran]

I was discussing a problem related to Einstein's universe model. As we all know its an unstable equilibrium. So a small fluctuation in the density would result in an either expanding or contracting universe. However, I read that

"There are static and stable solutions in Newtonian gravity theory: stars, planets, etc... Even large nebulae. The Newtonian universe could be a large spherical ball in a state of stable equilibrium. The instability occurs mainly when you assume an infinite distribution of homogeneous matter, and even there, the potential is ill-defined. The theory breaks down at this point, and we can't use the math to show anything."

How can the Newtonian spherical ball be in stable equilibrium? What's the difference between this and Einstein's Universe model?

 

[berkeman]

Welcome to the PF.

However, I read that

"There are static and stable solutions in Newtonian gravity theory: stars, planets, etc... Even large nebulae. The Newtonian universe could be a large spherical ball in a state of stable equilibrium. The instability occurs mainly when you assume an infinite distribution of homogeneous matter, and even there, the potential is ill-defined. The theory breaks down at this point, and we can't use the math to show anything."

We require reputable references for discussion of topics like this. Can you link to where you read this? Are you aware of any peer-reviewed journal articles that discuss this?

 

[Weran]

Can you link to where you read this?

It was in the physics stack exchange https://physics.stackexchange.com/q...the-only-way-to-show-that-the-universe-is-exp

Are you aware of any peer-reviewed journal articles that discuss this?

I am not aware of it and that is the main problem. Its kind of a dead-end question...I know, but I thought maybe some poeple in this forum heard something like this and they can help me.

 

[PeterDonis]

How can the Newtonian spherical ball be in stable equilibrium?

A Newtonian finite spherical ball. More precisely, a configuration of matter of finite extent, surrounded by vacuum extending out to infinity.

What's the difference between this and Einstein's Universe model?

In the Einstein static universe, the spatial geometry is not flat. In Newtonian physics, the spatial geometry is flat.

 

[PAllen]

Of course, the Newtonian ball must have several features for stability - pressure balancing gravity, and inability to radiate. The latter would seem to require something like absolute zero, but it is hard to square that with a source of pressure. So even the Newtonian ball is not really plausible for large distance and time scales (even with finite assumption).

 

[PeterDonis]

inability to radiate. The latter would seem to require something like absolute zero

In Newtonian physics, I don't know that there is any reason to require objects to radiate at finite temperatures. Of course, in Newtonian physics, there isn't really a microscopic model of matter at all; matter is just a blob of stuff described by density and pressure as a function of position. This model is self-consistent, but of course we have lots of evidence now that it isn't correct.

 

[Weran]

A Newtonian finite spherical ball. More precisely, a configuration of matter of finite extent, surrounded by vacuum extending out to infinity.

So even the Newtonian ball is not really plausible for large distance and time scales (even with finite assumption).

So I ll make some bulletpoints which one is correct ?

1 - A Newtonian finite ball is at stable equilibrium point and the size/scale does not affect this stability
2 - A Newtonian finite ball is at an unstable equilibrium point at all times
3 - A Newtonian finite ball is at the stable equilibrium point, however, the size/scale of this ball effects this stability. Hence after some scale, it becomes unstable.So I am right, the universe cannot be described by Newtonian finite ball (?).

 

[PeterDonis]

1 - A Newtonian finite ball is at stable equilibrium point and the size/scale does not affect this stability

A Newtonian finite ball can be in a stable equilibrium, if it is in hydrostatic equilibrium--pressure balancing gravity at every point.

2 - A Newtonian finite ball is at an unstable equilibrium point at all times

Not necessarily, see above. If the ball is not in hydrostatic equilibrium, it will not be in stable equilibrium. But if it is, it will.

3 - A Newtonian finite ball is at the stable equilibrium point, however, the size/scale of this ball effects this stability. Hence after some scale, it becomes unstable.

As far as I know there is no limit in Newtonian physics to the size of an object that can be in hydrostatic equilibrium; the size must be finite, but it can be any finite size.

So I am right, the universe cannot be described by Newtonian finite ball (?).

Our universe can't be, because our universe is not a ball of matter of finite size surrounded by empty space. But I don't think the OP of this thread is asking about our universe; it's just asking about what is possible in Newtonian physics.

 

[Weran]

But I don't think the OP of this thread is asking about our universe; it's just asking about what is possible in Newtonian physics.

I am the OP. Well both is okay for me. But I understand your point. Thanks

 

[PeterDonis]

both is okay for me.

Neither the Newtonian spherical ball nor the Einstein static universe apply to our actual universe, so the topic of this thread has nothing to do with modeling our actual universe. If you want to ask what model describes our actual universe, you should start a separate thread for that.

 

[Weran]

Neither the Newtonian spherical ball nor the Einstein static universe apply to our actual universe, so the topic of this thread has nothing to do with modeling our actual universe.

Oh umm yes I understand. Okay The main problem was this.

I claimed that by just using mathematics, we can prove that matter-filled static universe is at unstable equilibrium (Basically Einstein's Static universe model). Then I tried to prove this by using fluid equations etc (which is a simple thing to do). Then another person claimed that this is not an exact prove since a matter filled static universe is also possible for a finite size Newtonian ball (which is stable at all times for under certain conditions).

So here the main question is can someone mathematically prove that a matter-filled static universe is always unstable ?

There are lots of things to account I guess. So its not a clear-cut thing to answer.

 

[PAllen]

It just depends on what you mean by universe and by Newtonian physics. It is absolutely true that Newtonian physics allows an arbitrarily large, stable, ball of fluid. An infinite homogeneous state leads to self contradiction. My comment was more in line with "what would physicists in the second half of the 19th century - after Maxwell and Boltzman, before any notion of relativity - conclude about a model of the universe". They would see that a stable finite universe had great difficulties for thermodynamic reasons. Even more, the universe isn't fluid, and a large finite collection of gravitating bodies (i.e. galaxies) is generally unstable under pure Newtonian gravity, but this has to do with chaotic features of Newtonian gravity that Poincare first identified, not with a fluid model. Galaxies should occasionally eject from the 'ball', and this only adds to radiative instability. Thus they would have been very troubled by the state of affairs.

 

[PeterDonis]

can someone mathematically prove that a matter-filled static universe is always unstable ?

First you have to make up your mind which mathematical theory you want to use. If you use General Relativity, yes, you can prove this. If you use Newtonian physics, you can't.

 

[Weran]

First you have to make up your mind which mathematical theory you want to use. If you use General Relativity, yes, you can prove this. If you use Newtonian physics, you can't.

I see, could you share the approaches in simple terms or reccomend an article ?

 

[PeterDonis]

could you share the approaches in simple terms

What "approaches" do you mean? I've already listed the two theories you can pick from.

 

[Weran]

What "approaches" do you mean? I've already listed the two theories you can pick from.

Sorry thanks for the help