Re: Forces in a finite bounded universe

From: George Jones (george_llew_jones_at_yahoo.com)
Date: 11/11/04 

Date: Thu, 11 Nov 2004 11:57:44 -0500

<mmeron@cars3.uchicago.edu> wrote in message
news:Sdykd.7$45.1807@news.uchicago.edu...
> In article <cmu38r$8te$1@pc-news.cogsci.ed.ac.uk>, richard@cogsci.ed.ac.uk
(Richard Tobin) writes:
> >In article <w5wkd.391929$D%.191225@attbi_s51>,
> >Sam Wormley <swormley1@mchsi.com> wrote:
> >
> >> Yes, you are correct, Mati, but is yours a better answer to the OP's
> >> question at the level of which the question was asked, "Does the g
> >> gravitational attraction fall off faster than 1/r^2 ?"
> >
> >Here's a slightly more concrete question: in the 3-d spherical space
> >of Riemann, does a point feel any gravitational attraction from a
> >mass at the antipodal point?
> >
> >(Or is there no antipodal point?)
> >
> At this point GR should be used to describe the situation, and my GR
> is very rusty. So, any takers?

Here's my 2 cents worth.

I'm not sure that the setup is possible in GR.

Here's why. First, pin down the space. The surface of the Earth is
approximately S^2, the 2-d surface of a sphere in 3-d. I think Richard
Tobin wants space to be S^3, the 3-d "surface" of a hypersphere.

Now place 2 masses, maybe one fairly large and the other a small test
mass, "opposite" each other in S^3. {Does this have coordinate-
independent meaning?). How does the test mass respond to the presence
of the other mass?

Now, in GR, Einstein's equation relates geometry to the distribution
of mass-energy, so we can't just plop down any old masses into any old
spaces. Einstein's equation constrains things.

If our large mass is a point, or is spherically symmetric, then the
solution (ignoring the effects of the small test mass on geometry) is
Schwarzschild, which does not have S^3 for spatial slices. The test mass
certainly "feels" the other mass, but we can't place the 2 masses
"opposite" each other.

Closed Robertson-Walker universes do have S^3 as spatial slices, but are
usually taken to have the energy-momentum tensor of a perfect fluid.
Maybe a point masses could be inserted perturbatively into this, but I'm not
sure. To me, it doesn't sound easy.

While we're on the subject of force: did you (or anyone else) see the
interesting (and maybe slightly provocative) October Physics Today
article on (the fading from view of) force, by Frank Wilczek?

The article is available for free (no membership required) at

http://www.physicstoday.org/vol-57/iss-10/p11.html

Regards,
George

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