So, in the classic arcade game Asteroids, you move in a game field where the top and bottom edges are identified and the left and right edges likewise, topologically a torus. I'm interested in how such a game world might accommodate gravity.
One model I might use is to tile the plane with copies of the game field, and then resolve the forces in the plane (e.g. as if it were a slice of physical space). Then the question of determining gravity between two objects reduces to the question of determining the force felt on an object in the plane by an infinite grid of other objects (say at integer co-ordinates; the force felt on the object by all the copies of itself seems like it should be zero, by symmetry).
However, there's a problem here: at distance approximately $r$, we have $O(r)$ objects each contributing a force that is $O(1/r^2)$, for a total of $O(1/r)$. Summing these contributions over all distances is a harmonic series, which diverges.
Intuitively, however, it seems like the situation isn't so bad as that, because the total contribution from the points at distance approximately $r$ is much much less than the sum of individual contributions, because of the high degree of cancellation. We might hope that for large $r$, the points almost lie on a circle, so cancel each other almost perfectly.
Then again, I know that the absolute values diverging means that I can pick an order to sum the forces in to make the result whatever I like, including infinite, and it seems hard to argue that computing the forces nearest the candidate point first is particularly better than any other sort of order. Even if I do decide that the right thing is to take the limit as $r \to \infty$ of the forces within distance $r$, that's a pretty scary sum that I don't see a good way to get a handle on.
Can anyone offer me:
- asymptotic estimates on the sum of the forces at distance approximately $r$, after cancellation, that might prove that at least one form of this question has an answer?
- any techniques that can make the sum itself more tractable?
- any thoughts on what the inverse-cube case would look like, where it seems like convergence would be absolute?
- any other thinking anyone else has done on the topic of gravity or similar forces on quotient spaces?
The physical answer would be that a truly infinite grid of point masses is unphysical anyway. If the grid is not infinite but merely very large, it will have a center somewhere, and the field at any particular point will depend on its position relative to the center.
And if you insist on making it infinite anyway, Newtonian gravity breaks down (though Newton himself argued that such a grid would be gravitationally stable due to its symmetry), and we have to move to general relativity anyway. GR would then make the entire grid start to contract uniformly under its own gravity and end up with a Big Crunch not long after we set it up. (Since the GR equations are local, this is what it predicts would happen to a finite torus-shaped universe too).
However, for a game we should probably be satisfied with something that looks like gravity locally. The physicist would then say, forget the fields -- the potential is what actually exist, so let's just try to find a potential that satisfies the PDE for a free-space inverse-square potential away from the lattice points and has a potential well of the desired steepness around the lattice point itself.
Requiring that the solution fits together across the glued-together edges would then hopefully determine a unique solution, up to an additive constant, and we can then take the gradient of that solution to find the actual field to use.
(I suspect in practice this could be done by summing potential contributions from the infinitely many point masses at each point in the space. This will give rise to a divergent series, but with an appropriate regularization one could hope to get a sum that haves well enough to satisfy the PDE anyway).
This wouldn't work for a three-dimensional lattice, since Gauss's law would then require that the total mass in each lattice cell is zero, but I think there would be some hope for an infinite two-dimensional lattice in an otherwise empty 3D space.