Re: Gravity inside a spherical shell in non-Euclidean space
- From: Igor <thoovler@xxxxxxxxxx>
- Date: Fri, 25 Apr 2008 09:23:07 -0700 (PDT)
On Apr 24, 5:41 pm, George Smith <gsm...@xxxxxxxxxxxxxxxxxxxxx> wrote:
It is well-known that the gravity from an isotropic spherical shell of
matter in flat 3-space cancels throughout the interior volume, so an
object stationary at some point in this volume is in free-fall and does
not tend to "fall" either to the center of the shell's enclosed volume
or to the nearest part of the shell surface.
What I am curious about is the limit for when these conditions hold. In
particular, is it dependent on the force being inverse-square? If so, it
should fail for, say, a four-dimensional analogue, or if the space is
not Euclidean. Gravity would fall off as an inverse-cube in the former
case, and in the latter case either slower than inverse square (with
focusing back to a high intensity) or exponentially quickly (for a
hyperbolic space).
In particular, what is the behavior of gravity in a spherical shell in a
hyperbolic 3-space?
To ensure that this is well-defined, let's say that the hyperbolic
3-space is perfect vacuum except that at all of, and only, the points
reached by starting at a given center-point and traveling along a
geodesic for between a and b metres, where 0 < a < b; on this set of
points the mass density is, say, that of silicate rock.
It's probably safe to say that gravity still cancels at the exact center
of the sphere. The question is what happens at a point, say, halfway out
to the shell, say along one of those geodesics at a distance of a/2.
Even for the Newtonian inverse-square gravity it isn't clear to me what
will happen. The system is axially symmetric about a line throuh the
center point and the a/2 point, so it seems likely the gravity is either
null, directed straight in to the center, or directed straight out to
the nearest point on the surface.
As for which it is, I've tried figuring it out. The tricky thing is that
it's not clear how to translate between coordinate systems in such a
space. The only coordinate system I can easily give it is a spherical
polar one. Centering this about the sphere's center and assuming the
point of interest to be halfway to the south pole leads to the question
of just how wide a "circle of constant latitude" is at a given latitude,
for example. Centering it about the point of interest leads to trying
to figure out the sphere's translation to the coordinate system; or at
least, how far the sphere surface is from the point in any given direction..
There may even be multiple tangential geodesics at a single point; I'm
not completely certain. Certainly the lack of unique parallel lines
renders anything as simple as an xyz Cartesian coordinate system moot.
Last but not least, there's no guarantee that the Newtonian answer will
be the correct one in a curved space. So if anyone knows what GR has to
say about such a shell in a hyperbolic 3-volume (of constant negative
curvature) I'd like to know. (Assume the shell has no tension or
pressure inside, only mass-density; a "dust" solution. A spherical shell
of dust in an otherwise-empty expanding universe with no cosmological
term, say; that would have a negative curvature, since it's
approximately empty and therefore well below the critical density omega.)
Failing that, I'd at least like to know what happens to
normally-inverse-square forces in spherical shells in non-Euclidean
spaces, or higher-dimensional ones.
Does Gauss's Law still hold in non-Euclidean space?
.
- References:
- Gravity inside a spherical shell in non-Euclidean space
- From: George Smith
- Gravity inside a spherical shell in non-Euclidean space
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