Gravity Force of a Spherical Shell

[davieddy]

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

[xilman]

Originally Posted by davieddy

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

[xilman]

Originally Posted by xilman

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

[davieddy]

Originally Posted by davieddy

Hint: ccorn is temporalily ineligible for this puzzle

David

As is xilman

[cheesehead]

Originally Posted by davieddy

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).

[davieddy]

Originally Posted by cheesehead

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?

[cheesehead]

Originally Posted by davieddy

I know Newton did it, but how precisely?

It's in here somewhere:

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

[davieddy]

Originally Posted by cheesehead

It's in here somewhere:

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

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

[ccorn]

Originally Posted by davieddy

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.

[sichase]

Originally Posted by davieddy

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.

[davieddy]

Originally Posted by ccorn

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