Saying that a moving ladder “fits” means that the start of the ladder is in the garage at the same time that the end of the ladder is. If the ladder is moving and contracted because of relativity, these two events are not simultaneous in all reference frames. Thus, you cannot definitely say that the moving ladder fits—whether it fits depends on your reference frame. (In another reference frame you would see the ladder longer than the garage, but you would also see the start of the ladder pass out of the garage before the end of the ladder passes into it.)
Why have that definition of “fit”? I could eqaully well say that fitting means that there is a reference frame that has a time where the ladder is completely inside.
If you had the carage loop back so that the end would be glued to the start you could still spin the ladder inside it. From the point of the ladder it would appear to need to pass the garage multiple times to oene fit ladder lenght but from the outside it would appear as if the ladder fits within one loop completely. With either perspective the one garage space enough to contain the ladder without collisions. In this way it most definetly fits. Usually garages are thought to be space-limited but not time limited. Thus the eating of the time-dimension is a perfectly valid way of staying within the spatial limits.
edit: actually there is a godo reazson to priviledge the rest frame oft he garage as the one that count as ragardst to fitting as then all of the fitting happens within its space and time.
Why have that definition of “fit”? I could eqaully well say that fitting means that there is a reference frame that has a time where the ladder is completely inside.
In that case, the ladder fits.
From the point of the ladder
Each rung of the ladder has a distinct reference frame. “From the point of the ladder” is meaningless.
If the ladder point of view is ildefined so is the garage point of view as the front and back of the garage have distinct reference frames. Any inertial reference frame is equally good. The ladder is not accelerating thus inertial. In the sense that we can talk of any frame as more than a single event or world line the ladder frame is perfectly good.
In the normal example, where the ladder is straight and moving forward, it has only one reference frame. Strictly speaking, each rung has a different reference frame, but they differ only by translation.
From what I understand, you modified it to a circular ladder spinning in a circular garage. In this case, each rung is moving in a different direction, and therefore at a different velocity. Thus, each rung has its own reference frame.
ah, I meant to glue the end and start together without curved shape/motion. But I guess that is physically unrealisable and potentially more distracting than explanatory.
Actually that’s not a big deal. Technically you need general relativity to do that, but it’s just a quotient space on special relativity. In any case, it works out exactly the same as an infinite series of ladders and garages.
There is one thing you have to be careful about. From the rest frame, the universe could be described as repeating itself every, say, ten feet. But from the point of view of the ladder, it’s repeating itself every five feet and 8.8 nanoseconds. That is, if you move five feet, you’ll be in the same place, but your clock will be off by 8.8 nanoseconds.
Actually from the point of view of the ladder the universe still repeats at every ten feet. It is just that from it’s point of view it takes the space of two carages at any one instant.Both the garage and ladder are in a state of rest and show equally good times. Yes they read different but doesn’t mean they are in error.
I am not sure whether it would see other instances of itself. I only spesified a spatial gluing and not that the garage be split into timeslices. I guess that the change of the point of view has changed some of that gluing to be from future to past. For if the ladder would be too long the frontend would not crash to the same ladder time backend but to a future one. (ignoring the problem of how you would try to slide the ladder into too small a hole in the first place)
Actually from the point of view of the ladder the universe still repeats at every ten feet.
No, it does not. I think I messed up before and it’s actually 20 feet and 8.8 nanoseconds. From the the point of view of the garage, the coordinates (0 ft, 0 ns) and (10 ft, 0 ns) correspond to the same event. From the point of view of the ladder, the coordinates became (0 ft, 0 ns) and (20 ft, 8.8 ns). They still have to be the same event.
The universe is definitely repeating itself to be off by a certain time, and the distance it is off by is not ten feet.
The ladder sees the carage length contract. That is less than 10 feet. The ladder doesn’t see itself contract that puts the limit on the repeating of the universe.
Are you sure the ladder point equivalences are not (0 ft, 0ns) and (20 ft, −8.8ns)?
Saying that a moving ladder “fits” means that the start of the ladder is in the garage at the same time that the end of the ladder is. If the ladder is moving and contracted because of relativity, these two events are not simultaneous in all reference frames. Thus, you cannot definitely say that the moving ladder fits—whether it fits depends on your reference frame. (In another reference frame you would see the ladder longer than the garage, but you would also see the start of the ladder pass out of the garage before the end of the ladder passes into it.)
Why have that definition of “fit”? I could eqaully well say that fitting means that there is a reference frame that has a time where the ladder is completely inside.
If you had the carage loop back so that the end would be glued to the start you could still spin the ladder inside it. From the point of the ladder it would appear to need to pass the garage multiple times to oene fit ladder lenght but from the outside it would appear as if the ladder fits within one loop completely. With either perspective the one garage space enough to contain the ladder without collisions. In this way it most definetly fits. Usually garages are thought to be space-limited but not time limited. Thus the eating of the time-dimension is a perfectly valid way of staying within the spatial limits.
edit: actually there is a godo reazson to priviledge the rest frame oft he garage as the one that count as ragardst to fitting as then all of the fitting happens within its space and time.
In that case, the ladder fits.
Each rung of the ladder has a distinct reference frame. “From the point of the ladder” is meaningless.
If the ladder point of view is ildefined so is the garage point of view as the front and back of the garage have distinct reference frames. Any inertial reference frame is equally good. The ladder is not accelerating thus inertial. In the sense that we can talk of any frame as more than a single event or world line the ladder frame is perfectly good.
In the normal example, where the ladder is straight and moving forward, it has only one reference frame. Strictly speaking, each rung has a different reference frame, but they differ only by translation.
From what I understand, you modified it to a circular ladder spinning in a circular garage. In this case, each rung is moving in a different direction, and therefore at a different velocity. Thus, each rung has its own reference frame.
ah, I meant to glue the end and start together without curved shape/motion. But I guess that is physically unrealisable and potentially more distracting than explanatory.
Actually that’s not a big deal. Technically you need general relativity to do that, but it’s just a quotient space on special relativity. In any case, it works out exactly the same as an infinite series of ladders and garages.
There is one thing you have to be careful about. From the rest frame, the universe could be described as repeating itself every, say, ten feet. But from the point of view of the ladder, it’s repeating itself every five feet and 8.8 nanoseconds. That is, if you move five feet, you’ll be in the same place, but your clock will be off by 8.8 nanoseconds.
Actually from the point of view of the ladder the universe still repeats at every ten feet. It is just that from it’s point of view it takes the space of two carages at any one instant.Both the garage and ladder are in a state of rest and show equally good times. Yes they read different but doesn’t mean they are in error.
I am not sure whether it would see other instances of itself. I only spesified a spatial gluing and not that the garage be split into timeslices. I guess that the change of the point of view has changed some of that gluing to be from future to past. For if the ladder would be too long the frontend would not crash to the same ladder time backend but to a future one. (ignoring the problem of how you would try to slide the ladder into too small a hole in the first place)
No, it does not. I think I messed up before and it’s actually 20 feet and 8.8 nanoseconds. From the the point of view of the garage, the coordinates (0 ft, 0 ns) and (10 ft, 0 ns) correspond to the same event. From the point of view of the ladder, the coordinates became (0 ft, 0 ns) and (20 ft, 8.8 ns). They still have to be the same event.
The universe is definitely repeating itself to be off by a certain time, and the distance it is off by is not ten feet.
The ladder sees the carage length contract. That is less than 10 feet. The ladder doesn’t see itself contract that puts the limit on the repeating of the universe.
Are you sure the ladder point equivalences are not (0 ft, 0ns) and (20 ft, −8.8ns)?
It depends on which direction it’s moving. I didn’t bother to check the sign.
Thinking about it now, if it’s going in the positive direction, then it should be (20 ft, −8.8ns). You are correct.