Two Newtonian point particles, A and B, with mass 1kg are at rest separated by a distance of 1m. They are influenced only by the other’s gravitational attraction. Describe their future motion. In particular do they ever return to their original positions, and after how long?
Note that your force grows unboundedly in N, so close to zero you have things that are arbitrarily heavy compared to their distance. So what this paradox really is about, is alternating series’ that grow with N, and whether we can say that they add up to zero.
If we call the force between the first two bodies f12, then the series of internal forces on this system of bodies (using negative to denote vector component towards zero) looks like -f12+f12-f23+f23-f13+f13-f34..., where, again, each new term is bigger than the last.
If you split this sum up by interactions, it’s (-f12+f12)+(-f23+f23)+(-f13+f13)..., so “obviously” it adds up to zero. But if you split this sum up by bodies, each term is negative (and growing!) so the sum must be negative infinity.
The typical physicist solution is to say that open sets aren’t physical, and to get the best answer we should take the limit of compact sets.
The same can be said of unit masses at every whole negative number.
The arrow that points to the right is at the same place that the additional guest in Hilbert’s Hotel goes. Such unintuitiveness is life when infinities/singularities such as the diverging forces acting on your points are involved.
Try this
A different question in the same vein:
The collision of two “infinitely small” points is quite another problem. Have some similarities, too.
For two points on a colliding path, the action and reaction force are present and of equal size and oppposite directions.
My example can have finite size balls or zero size mass points, but there is no reaction force to be seen. At least, I don’t.
Note that your force grows unboundedly in N, so close to zero you have things that are arbitrarily heavy compared to their distance. So what this paradox really is about, is alternating series’ that grow with N, and whether we can say that they add up to zero.
If we call the force between the first two bodies f12, then the series of internal forces on this system of bodies (using negative to denote vector component towards zero) looks like -f12+f12-f23+f23-f13+f13-f34..., where, again, each new term is bigger than the last.
If you split this sum up by interactions, it’s (-f12+f12)+(-f23+f23)+(-f13+f13)..., so “obviously” it adds up to zero. But if you split this sum up by bodies, each term is negative (and growing!) so the sum must be negative infinity.
The typical physicist solution is to say that open sets aren’t physical, and to get the best answer we should take the limit of compact sets.
The same can be said of unit masses at every whole negative number.
The arrow that points to the right is at the same place that the additional guest in Hilbert’s Hotel goes. Such unintuitiveness is life when infinities/singularities such as the diverging forces acting on your points are involved.
I think the point is still that infinities are bad and can screw you up in imaginative ways.
Agree.