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Author Topic: Shell Theorem problem (split from What is Mass)  (Read 4020 times)

Offline imatfaal

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Shell Theorem problem (split from What is Mass)
« on: 11/07/2011 11:01:16 »
To coin an analogy from football, some of those bosons look more like plays than players.
Great line! - I am gonna steal that. 

Phract - you could try the videos of Leonard Susskinds - the theoretical minimum.  They are all available on line and range from introduction to einsteins relativity to qcd and particle models.  Good stuff - not too easy, but not too hard


 

Offline Phractality

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Shell Theorem problem (split from What is Mass)
« Reply #1 on: 13/07/2011 06:51:29 »
To coin an analogy from football, some of those bosons look more like plays than players.
Great line! - I am gonna steal that. 

Phract - you could try the videos of Leonard Susskinds - the theoretical minimum.  They are all available on line and range from introduction to einsteins relativity to qcd and particle models.  Good stuff - not too easy, but not too hard

I'm still working (sporadically) on my answer to CPT Arkangel's question about black holes, but I'm going to detour for a moment to point out a glaring falacy in Leonard Susskind's Cosmology Lecture 2. I certainly don't claim to be more knowlegible than Susskind, but nobody is infallible. Occasionally, a child sees thru the falacies of adult thinking, and a dummy occasionally pokes a hole in the logic of a genius.

Newton proved his shell theorem on the assumption that the spheres are finite and that infinite space beyond the outermost sphere is empty. He always referred to the gravity due to the mass inside the finite spheres, while tacitly assuming the absence of any mass outside of those spheres.

Beginning at about 42 minutes, Susskind explains Newton's shell theorem. Then, at about 54 minutes into the lecture, he tacity assumes that Newton's shell theorem is valid for infinitely large homogeneic spheres, surrounded by more infinitely large homogeneic spheres, with no outermost sphere and no empty space beyond. The remainder of the lecture is falacious. Susskind is not alone in making this glaring error; I think Einstein made it, too, and perhaps others before him. It is a key assumption underlying the big bang bunk.

If you remove a spherical section of an infinite homogeneous universe, the gravity due to the remainder is equal and opposite to the gravity due to the part you removed. This is true regardless of how large a sphere you remove.

Another approach to the problem is to calculate the gravity at the focus of a pyramidal subset of the universe. Each half of the pyramid is equivalent, so the gravity from one half cancels that of the other half. You can truncate the pyramid at +r and -r and prove that the two halves always cancel as r approaches infinity. If you remove a piece of that pyramid, the change in gravity at the focus will be equal to the gravity due to a mirror image (across the vertex) of the piece you removed. Dividing the whole 4π sterradians into pyramids yields zero gravity.

In an infinite homogeneous universe, every point is equivalent to every other point, so every point is at the same gravitational potential. The gravity field is the gradient of gravitational potential, so the gravity field is zero everywhere, and gravity does not have any tendence to slow the expanion of space.

The question remains whether clumping of matter tends to decrease gravitational potential. I suspect the answer is yes, and that may be responsible for the foamy large-scale structure of the universe, but it does not tend to pull the most distant galaxies closer together, it only works on neighboring galaxies. Unless the expansion of space decreases the lumpiness, it does not increase gravitational potential.

It has been claimed that this argument presented by Suskind and others proves that the universe is finite. In fact, it proves only that the shell theorem is valid only for a finite universe.

P.S.: As far as I know, that football analogy is my own original, created ad hoc for that post. I wouldn't mind getting credit when you use it; but I forgive you if treat is folklore.
« Last Edit: 13/07/2011 22:36:39 by Phractality »
 

Offline imatfaal

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Shell Theorem problem (split from What is Mass)
« Reply #2 on: 13/07/2011 13:17:50 »
Fract - I would have to think hard and long about the above.  Will have a go...
 

Offline yor_on

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Shell Theorem problem (split from What is Mass)
« Reply #3 on: 14/07/2011 23:46:55 »
"Another approach to the problem is to calculate the gravity at the focus of a pyramidal subset of the universe. Each half of the pyramid is equivalent, so the gravity from one half cancels that of the other half. You can truncate the pyramid at +r and -r and prove that the two halves always cancel as r approaches infinity. If you remove a piece of that pyramid, the change in gravity at the focus will be equal to the gravity due to a mirror image (across the vertex) of the piece you removed. Dividing the whole 4π sterradians into pyramids yields zero gravity. " Could you 'dumb that down' for me Phractality, I'm sort of loosing myself reading you there :) Too many descriptions to take in for one mind I'm afraid, well, mine at least.
 

Offline Phractality

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Shell Theorem problem (split from What is Mass)
« Reply #4 on: 16/07/2011 03:40:22 »
Must be ADD!  ??? I get thru a proof and then I have to do something else. When I get bact to it I have to start over with a different approach. Math ain't my thing, but I'll keep trying.  :-\
 

Offline yor_on

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Shell Theorem problem (split from What is Mass)
« Reply #5 on: 16/07/2011 19:01:07 »
I don't see it as the math, it's just very 'compressed' Phractality. And I'm sure you have it clear in your mind, but, :) I'm still out there, looking in, sort of. Don't worry, you do 'your thing' from a geometrical standpoint as I see it, and it's interesting to read any which way.
 

Offline PhysBang

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Shell Theorem problem (split from What is Mass)
« Reply #6 on: 16/07/2011 20:41:10 »
Newton proved his shell theorem on the assumption that the spheres are finite and that infinite space beyond the outermost sphere is empty. He always referred to the gravity due to the mass inside the finite spheres, while tacitly assuming the absence of any mass outside of those spheres.

Beginning at about 42 minutes, Susskind explains Newton's shell theorem. Then, at about 54 minutes into the lecture, he tacity assumes that Newton's shell theorem is valid for infinitely large homogeneic spheres, surrounded by more infinitely large homogeneic spheres, with no outermost sphere and no empty space beyond. The remainder of the lecture is falacious. Susskind is not alone in making this glaring error; I think Einstein made it, too, and perhaps others before him. It is a key assumption underlying the big bang bunk.
In the context of general relativity, one uses Birkhoff's theorem rather that Newton's.

The Friedmann or Friedmann-Lemaitre equations provide a solution to the Einstein Field Equation, so we know that the infinite solution or the finite but unbounded solution of homogeneous distribution of mass-energy works in general relativity. Birkhoff's theorem establishes that a spherically symmetric change in the distribution of matter doesn't matter to what goes on outside that sphere, so over-dense regions of the universe are free to collapse without impacting the overall geometry of the spacetime.

 

Offline Phractality

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Shell Theorem problem (split from What is Mass)
« Reply #7 on: 17/07/2011 05:33:05 »
In the context of general relativity, one uses Birkhoff's theorem rather that Newton's.

The Friedmann or Friedmann-Lemaitre equations provide a solution to the Einstein Field Equation, so we know that the infinite solution or the finite but unbounded solution of homogeneous distribution of mass-energy works in general relativity. Birkhoff's theorem establishes that a spherically symmetric change in the distribution of matter doesn't matter to what goes on outside that sphere, so over-dense regions of the universe are free to collapse without impacting the overall geometry of the spacetime.

Here's a "simple proof of Birkoff's theorem"; it's Geek to me! I don't even understand the notations.

You, on the other hand, make it sound so simple! And your interpretation of it sounds perfectly reasonable. I don't see how that relates to the cosmological constant or the FRW equations, though.

Susskind's Cosmology Lecture 2, he derives the Friedman, Robinson & Walker (FRW) equations, by extending Newton's shell theorem to an expanding, homogeneous, isotropic space-time metric. He represented a finite piece of the metric on the drawing board and said something like, "This metric is infinite, and the shell theorem makes it want to collapse in on itself." Later, he concludes that it can't be infinite, which is supposed to prove that the universe is finite. I'm not saying that the universe is proven to be infinite; I'm just saying that the proof that it is finite is a falaceous proof.

Maybe Birkoff's proof is valid, maybe not. I can't judge it because it's written in a mathematical language that I can't read.
 

Offline PhysBang

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Shell Theorem problem (split from What is Mass)
« Reply #8 on: 18/07/2011 00:10:02 »
I think that there might be a singularity in the Newtonian case. That is, it might be impossible to cancel out the mass of an infinite universe using Newtonian theory. (I haven't worked it out, but I have seen this claimed in a few sources.) This doesn't happen with the GR case, though it is more complicated.

NB: I disagree with Susskind that the Principia is hard to read. On the contrary, I find the recent translation to be an amazingly good read and the geometrical methods are very informative.
 

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Shell Theorem problem (split from What is Mass)
« Reply #8 on: 18/07/2011 00:10:02 »

 

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