Be careful. Physics seems to be translation invariant, but space is not. You can drop the ball in and out of the cave and its displacement over time will be the same, but you can definitely tell whether it is in the cave or out of the cave. You can set your zero point anywhere, but that doesn’t mean that objects in space move when you change your zero point. Space is isotropic. There’s no discernible difference between upward, sideways, or diagonal, but if you measure the sideways distance between two houses to be 40 meters, a person who called your “sideways” their “up” will measure the distance between the houses to be 40 meters up and down. You can do everything here as you can do there, but here is not there. In the absence of any reference point, no point in space is different from any other point, but in the absence of any reference point there’s no need for physics, because if there was anything to describe with physics, you could use it as a reference point.
I suppose you could try to define space as the thing you can move around in without changing your physics, but the usual strategy is to define physics and derive conservation of momentum from the fact that your physics is translation invariant.
Be careful. Physics seems to be translation invariant, but space is not. You can drop the ball in and out of the cave and its displacement over time will be the same, but you can definitely tell whether it is in the cave or out of the cave. You can set your zero point anywhere, but that doesn’t mean that objects in space move when you change your zero point. Space is isotropic. There’s no discernible difference between upward, sideways, or diagonal, but if you measure the sideways distance between two houses to be 40 meters, a person who called your “sideways” their “up” will measure the distance between the houses to be 40 meters up and down. You can do everything here as you can do there, but here is not there. In the absence of any reference point, no point in space is different from any other point, but in the absence of any reference point there’s no need for physics, because if there was anything to describe with physics, you could use it as a reference point.
I suppose you could try to define space as the thing you can move around in without changing your physics, but the usual strategy is to define physics and derive conservation of momentum from the fact that your physics is translation invariant.
Formally, I mean that translation commutes with time-evolution. (Maybe “translation-equivariant” would be a better term? Idk, am not a physicist.)
I guess my story could have been better written to emphasize the commutativity aspect.