Frederick Sparber wrote:
Will a sphere within a sphere (a ball-bearing in a transparent hollow
sphere)
due to the gravitational attraction between them, center itself during
free fall?
First, in Newtonian gravitation:
Inside a uniform spherical shell there's no gravitational field (no
field due to the shell, that is), so the ball bearing will ignore the
shell and fall normally. It won't move to the center.
The force of gravity exerted by the inner ball on the entire surface of
the shell will cancel, and the shell won't accelerate as a result of the
ball bearing's field -- so it, too, will therefore fall normally, and
won't "center itself" around the ball bearing. You can check this by
integrating the field of a point particle over an offset spherical shell
(and noting that the ball bearing consists of a big blob of point
particles), or you can just use conservation of momentum to argue that
since the ball bearing doesn't feel a force from the shell, the shell
must not feel a (net) force from the ball bearing either.
Now, in relativity:
It's a whole lot harder, but I think the answer's the same, based on
this very sloppy argument: Within the chamber in the big sphere, the
stress/energy tensor is zero (assuming the lights are off, and ignoring
the contribution of the ball bearing). If the stress tensor is zero,
then the Ricci tensor must be zero too, by Einstein's field equation.
And if the Ricci tensor is zero, then a small ball of initially comoving
particles won't change in volume as time goes by (though it _may_
deform), since that's what the Ricci tensor measures. What that says is
that there aren't any points of attraction or repulsion in empty space,
but there may be tidal effects. Tidal effects => a saddle point in the
field. Inside the chamber, the field due to the shell must be
spherically symmetric, so we _can't_ have a saddle point in the center
of the sphere. Therefore, particles can't be attracted to the center of
the sphere (nor repelled from it).
Since momentum is still conserved in GR, I _think_ we can again argue
that the big sphere can't be pulled "to the center" by the small sphere,
either, in that case.
*http://en.wikipedia.org/wiki/Le_Sage%27s_theory_of_gravitation*
_*http://en.wikipedia.org/wiki/Introduction_to_general_relativity*_
Or?
Fred <http://en.wikipedia.org/wiki/Introduction_to_general_relativity>
