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Old 2010-06-17, 11:51   #1
davieddy
 
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Default Gravity Force of a Spherical Shell

As we know,
A) the field inside is zero
B) outside it is the same as if the mass was located at the centre.

The "grown-up" way of explaining this is symmetry and
Gauss' Flux Theorem (easy to understand and derive without
calculus).

However, it is easy to see that the field inside is zero:
a chord through a point P inside the sphere has the same
angle of incidence at each end. Consider a narrow cone with apex P.
The mass in the cone goes as distance^2 compensating for
the inverse square force law. So the fields due to the masses
at opposite ends of the cone cancel at P.

Can B be explained as elegantly?

Hint: ccorn is temporalily ineligible for this puzzle

David

Last fiddled with by davieddy on 2010-06-17 at 12:08
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Old 2010-06-17, 12:02   #2
xilman
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Quote:
Originally Posted by davieddy View Post
As we know,
A) the field inside is zero
If the shell is stationary, certainly. If the shell is rotating, will there be frame dragging within it?

This is a genuine question on my part. I don't (yet) know the answer.

Paul
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Old 2010-06-17, 13:39   #3
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Originally Posted by xilman View Post
If the shell is stationary, certainly. If the shell is rotating, will there be frame dragging within it?

This is a genuine question on my part. I don't (yet) know the answer.

Paul
Actually, is the field inside a stationary shell zero in the case of GR with a non-zero cosmological constant?

I argue that it is not. The CC manifests itself as a discrepancy from a strict inverse square law, something which the classical proofs require.

Paul

Last fiddled with by xilman on 2010-06-18 at 08:00 Reason: Added "non-zero"
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Old 2010-06-17, 20:17   #4
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Quote:
Originally Posted by davieddy View Post
Hint: ccorn is temporalily ineligible for this puzzle

David
As is xilman
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Old 2010-06-18, 02:03   #5
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Quote:
Originally Posted by davieddy View Post
B) outside it is the same as if the mass was located at the centre.

< snip >

Can B be explained as elegantly?

Hint: ccorn is temporalily ineligible for this puzzle
If I were to answer, it'd be classically (non-relativistically), after looking up Isaac Newton's proof. While reading Principia I admired how he used geometry to prove so many things I'd not previously thought could be connected.

So I won't answer (though realizing I've just given others a hint).

Last fiddled with by cheesehead on 2010-06-18 at 02:07
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Old 2010-06-18, 03:41   #6
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Originally Posted by cheesehead View Post
If I were to answer, it'd be classically (non-relativistically), after looking up Isaac Newton's proof. While reading Principia I admired how he used geometry to prove so many things I'd not previously thought could be connected.

So I won't answer (though realizing I've just given others a hint).
I know Newton did it, but how precisely?
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Old 2010-06-18, 04:50   #7
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Quote:
Originally Posted by davieddy View Post
I know Newton did it, but how precisely?
It's in here somewhere:

http://www.archive.org/stream/newtonspmathema00newtrich

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Old 2010-06-18, 05:20   #8
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Originally Posted by cheesehead View Post
THX

Of course Newton is famous for gravity, calculus and was presumably
intimately familiar with Greek geometry.

The problem is a good exercize in integration: take your pick
how you set the problem up. Surface area between planes h apart
= 2 pi R h is tempting. Or rings at angle theta from either the centre
or the point outside where you are calculating the field.

I am after something neater!

David

Last fiddled with by davieddy on 2010-06-18 at 05:24
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Old 2010-06-23, 16:43   #9
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Quote:
Originally Posted by davieddy View Post
The problem is a good exercize in integration: take your pick how you set the problem up. Surface area between planes h apart
= 2 pi R h is tempting. Or rings at angle theta from either the centre
or the point outside where you are calculating the field.

I am after something neater!
Solution there (Post 22). If you can simplify it further, please do so.
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Old 2010-06-24, 16:49   #10
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Quote:
Originally Posted by davieddy View Post
I know Newton did it, but how precisely?
Extremely precisely. But not entirely rigorously according to modern standards. Newton had already invented the integral calculus at the time he published the Principia, but would not publish it for some time yet. So he was forced, in order to be understood, to couch his arguments in long-winded geometric reasoning which makes his proofs difficult reading for anyone but experts. (Check it out... it's actually really interesting, but dense, reading.) And I'm sure it wouldn't stand up to modern scrutiny as rigorous mathematics.
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Old 2010-06-24, 21:23   #11
davieddy
 
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Quote:
Originally Posted by ccorn View Post
Solution there (Post 22). If you can simplify it further, please do so.
Well, this solution is essentially geometric, simple and
not altogether unrigorous, and is IMO elegance itself.

I meant "Can you tell me precisely how Newton did it?"

David
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