Imagine a Newtonian universe consisting of an infinite number of equal masses equidistantly arranged at rest along a ray pointing to the right. Each mass other than first will experience a smaller gravitational force to the left and a greater (but still finite, as it turns out) gravitational force to the right. As a result, the whole ray of masses will shift to the right, but getting compressed as the masses further out will experience less of a disparity between the left-ward and right-ward forces. There is something intuitively bizarre about a whole collection of particles starting to move in one direction under the influence of their mutual gravitational forces. It sure looks like a violation of conservation of momentum. Not that such oddities should surprise us in infinitary Newtonian scenarios.
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I don't understand. Surely if they're all the same mass and equidistant from each other, and there's an infinite number of them in a straight line, then by symmetry, the resultant force on any one mass ought to be zero? Where does the imbalance come from?
It's a ray, not a line. The force to the right is always bigger than the force to the left.
What causes the force to be greater on one side than the other? It doesn't seem like it can be masses (because of symmetry considerations). Or is the force just greater on one side by construction?
There are more masses to the right than to the left of every mass. For each mass, there are infinitely many masses to the right and only finitely many to the left.
Got it. Thanks. :)
It sure looks like a violation of conservation of momentum. Not that such oddities should surprise us in infinitary Newtonian scenarios.
Conservation of momentum is only granted for closed physical systems and furthermore the general momentum of any closed system with finitely many particles can be associated with the center of mass.
I think, these are the two essential problems in your given physical example here.
How does an infinite ray constitute as a closed physical system and regardless of closed or not where is the actual center of mass of that infinite ray?
I think, it would also be interesting to think about an example, where the physical system with infinitely many particles is definitely closed. Where might there be the center of mass and does the conservation of momentum still apply to that physical system?
There is no meaningful center of mass in infinite systems.
But infinite systems seem to be just as apt to be closed as finite ones: there is no external influence.
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