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Imagine there's a sphere of radius $R$ which has constant density $\rho$ and you can stand anywhere inside of the sphere. Wherever you stand within the sphere, you'll only feel gravity coming from the mass at smaller radii than the point at which you're standing. That is, you could be standing at some point with radius $r$ and if somebody come along and added more mass to the sphere so that the radius was extended to $R'>R$ you would feel no effect because you're insensitive to it.

Puzzle: What happens if somebody comes by and adds mass to the sphere until it extends out to infinity? Now the mass distribution is uniform everywhere in space so you'd expect no gravitational pull in any direction, by symmetry. What happens? Do you suddenly stop feeling a force? If you have an issue with extending the mass to infinity then imagine that we've added a point to $\mathbb R^4$ so that we're on $S^4$.

What's the intuitive way to understand what's happening in this limit? What's the technical explanation? Is it a boundary value issue?

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Suggestion to the question (v1): Limit the scope of the question to Newtonian gravity in $\mathbb{R}^3$ or $(S^1)^3$ space for clarity. If you like this question you may also enjoy reading this, this, and this Phys.SE posts –  Qmechanic Sep 7 '13 at 13:37
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2 Answers 2

I think it is a matter of symmetry: It does not matter how big you make the mass sphere, it is still a sphere, there is an asymmetry with respect to the center (assuming the probing point is not at the center of the sphere) as the sphere has rotational symmetry. In the case of the full space all points become equivalent and there is no force whatsoever because there is translational symmetry. You cannot go from one group of symmetry to another in a continuous fashion. The symmetry group of the infinite space includes rotations around any axis and translations. You may want to check about the Inönü-Wigner contractions.

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Well, the facetious answer is that there is a maximum mass that any stable object can have, if made out of ordinary matter--if you add more and more matter to an object, general relativity tells us that it the object will eventually become unstable and collapse gravitationally, a result first derived by Chandrasekhar in the case of matter stablized by electron degeneracy pressure.

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