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zinq
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Some time ago I tried to define classical inverse-square gravity on a 3-dimensional (cubical) torus T3: the quotient space obtained by identifying opposite faces of a unit cube. (Or more rigorously, the quotient space
of R3 by its subgroup Z3 of integer points.)
I assumed there was a unit point mass (call it mass A) at (0, 0, 0) and tried to define the force on another unit point mass (call it mass B) at an arbitrary point (x, y, z). My method — admittedly naïve — was to sum up the inverse-square forces from a copy of mass A at every integer point
acting on mass B. I hoped that — if summed cleverly — the forces from all these copies of mass A would mostly cancel, and the triply-infinite sum would converge.
That is, I was trying to sum the force terms
over the array of all integer points
But I tried every method I could think of: spherical shells, cubical shells, etc. . . . and clearly none of these summation schemes converged.
QUESTION: Is there a smart way to arrange for these forces to converge? Or perhaps this force method is just wrong and I should be using a potential function method instead?
Or perhaps it is known that there is no way to define classical gravity on a 3-torus? (And what about a 2-dimensional torus — and what about an n-torus for a dimension n greater than 3 ?)
T3 = R3/Z3
of R3 by its subgroup Z3 of integer points.)
I assumed there was a unit point mass (call it mass A) at (0, 0, 0) and tried to define the force on another unit point mass (call it mass B) at an arbitrary point (x, y, z). My method — admittedly naïve — was to sum up the inverse-square forces from a copy of mass A at every integer point
(K, L, M) ∈ Z3
acting on mass B. I hoped that — if summed cleverly — the forces from all these copies of mass A would mostly cancel, and the triply-infinite sum would converge.
That is, I was trying to sum the force terms
-(x-K, y-L, z-M) / ((x-K)2 + (y-L)2 + (z-M)2)3/2
over the array of all integer points
(K, L, M) ∈ Z3
But I tried every method I could think of: spherical shells, cubical shells, etc. . . . and clearly none of these summation schemes converged.
QUESTION: Is there a smart way to arrange for these forces to converge? Or perhaps this force method is just wrong and I should be using a potential function method instead?
Or perhaps it is known that there is no way to define classical gravity on a 3-torus? (And what about a 2-dimensional torus — and what about an n-torus for a dimension n greater than 3 ?)
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