Re: Gravity inside a spherical shell in non-Euclidean space

Darwin123 wrote:
In my mechanics course, taken oh so many years ago, we did the
problem for force laws of arbitrary exponent. Yes, the cancellation of
force inside the volume of the sphere only occurs for the inverse
square case. The other power laws result in cancellation only at the
exact center of the sphere. The force of gravity will be significant
away from the center.
I think an inverse cube law would result in attraction toward the
shell (a la hollow earth theory). An inverse 1.5 would result in a
small shrinking attraction toward the center.

This is roughly what I thought would happen, but are you assuming a Euclidean space with non-inverse-square law?

For example, in the hyperbolic case, the falloff is exponential instead of inverse-power. An e^-r falloff in Euclidean space certainly results in the sphere walls attracting things in the interior, as the far part of the sphere exerts a weaker attraction than before, relative to the near part. But in the hyperbolic geometry case, the far part is also larger than it is in the Euclidean case.

Interpreting the curved space as the presence of an existing gravity field does, as another poster suggested, predict tides affecting the objects inside the shell, however. In the case of a hyperbolic geometry of uniform negative curvature, it corresponds to an expanding cosmos well below the critical density, which suggests that objects inside the sphere will (seem to) be attracted to its surface, and that the shell will be under considerable tension.

Ultimately, the big question is what the curvature of *time* will look like in the vicinity of the shell's inner surface.
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