That said, the hypothetical you give is cool and I agree the two principles decouple there! (I intuitively want to save that case by saying the COM is only stationary in a covering space where the train has in fact moved a bunch by the time it stops, but idk how to make this make sense for a different arrangement of portals.) I guess another thing that seems a bit compelling for the two decoupling is that conservation of angular momentum is analogous to conservation of momentum but there’s no angular analogue to the center of mass (that’s rotating uniformly, anyway). I guess another thing that’s a bit compelling is that there’s no nice notion of a center of energy once we view spacetime as being curved ( https://physics.stackexchange.com/a/269273 ). I think I’ve become convinced that conservation of momentum is a significantly bigger principle :). But still, the two seem equivalent to me before one gets to general relativity. (I guess this actually depends a bit on what the proof of 12.72 is like — in particular, if that proof basically uses the conservation of momentum, then I’d be more happy to say that the two aren’t equivalent already for relativity/fields.)
I think the point about angular momentum is a very good way of gesturing at how its possibly different. Angular momentum is conserved, but an isolated system can still rotate itself, by spinning up and then stopping a flywheel (moving the “center of rotation”).
Thank for finding that book and screenshot. Equation 12.72 is directly claiming that momentum is proportional to energy flow (and in the same direction). I am very curious how that intersects with claims common in metamaterials (https://journals.aps.org/pra/abstract/10.1103/PhysRevA.75.053810 ) that the two can flow in opposite directions.
That said, the hypothetical you give is cool and I agree the two principles decouple there! (I intuitively want to save that case by saying the COM is only stationary in a covering space where the train has in fact moved a bunch by the time it stops, but idk how to make this make sense for a different arrangement of portals.) I guess another thing that seems a bit compelling for the two decoupling is that conservation of angular momentum is analogous to conservation of momentum but there’s no angular analogue to the center of mass (that’s rotating uniformly, anyway). I guess another thing that’s a bit compelling is that there’s no nice notion of a center of energy once we view spacetime as being curved ( https://physics.stackexchange.com/a/269273 ). I think I’ve become convinced that conservation of momentum is a significantly bigger principle :). But still, the two seem equivalent to me before one gets to general relativity. (I guess this actually depends a bit on what the proof of 12.72 is like — in particular, if that proof basically uses the conservation of momentum, then I’d be more happy to say that the two aren’t equivalent already for relativity/fields.)
I think the point about angular momentum is a very good way of gesturing at how its possibly different. Angular momentum is conserved, but an isolated system can still rotate itself, by spinning up and then stopping a flywheel (moving the “center of rotation”).
Thank for finding that book and screenshot. Equation 12.72 is directly claiming that momentum is proportional to energy flow (and in the same direction). I am very curious how that intersects with claims common in metamaterials (https://journals.aps.org/pra/abstract/10.1103/PhysRevA.75.053810 ) that the two can flow in opposite directions.