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Saw
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I am trying to ascertain whether the 4th paradox proposed by Zeno (the Stadium) can be used to refute the idea of quantized space and whether such conclusion is affected by SR.
Let us assume:
- Two equal blocks moving towards each other: block A from left to right and block B from right to left.
- The size of each block is the minimum spatial quantum or atom of space: it is indivisible. It could be the Planck Length, if you wish.
- In event 1 the left edge of block B is lined up with the right edge of block A.
- In event 2 the left edge of block B is lined up with the left edge of block A.
Under Galilean Relativity:
In the reference frames of either A or B, the transit from event 1 to event 2 means that the other block has moved one atom of space. That poses no problem for the quantization of space.
We now bring up Zenos’ 4th paradox. We consider a block C, at rest in a reference frame C where A and B are moving with equal velocities and in opposite directions. In that frame C, each of blocks A and B has moved, in the time lapse from event 1 to event 2, only one half-block, half-space atom. Hence looking at things from this coordinate system does pose a problem for the quantization of space.
To sum up, space quantization seems to be at odds with the relativity of space (in the sense of distances traversed), even under Galilean Relativity.
If we now consider Special Relativity…, that does not seem to alter the conclusion, does it?
To start with, in frame A block B has actually moved along the length of block A (a space atom), but in frame B block A is length-contracted, it is less than a space quantum long and it has displaced only that shorter length at event 2… So the problem for space quantization arises even earlier. And in frame C, both A and B are length-contracted, both are less than a space atom long and between events 1 and 2, each of them travels half that distance…
Conclusion: whenever you bring up reference frames, the idea of quantized space suffers; even more under SR, where more values are relative. Would you agree to that?
Let us assume:
- Two equal blocks moving towards each other: block A from left to right and block B from right to left.
- The size of each block is the minimum spatial quantum or atom of space: it is indivisible. It could be the Planck Length, if you wish.
- In event 1 the left edge of block B is lined up with the right edge of block A.
- In event 2 the left edge of block B is lined up with the left edge of block A.
Under Galilean Relativity:
In the reference frames of either A or B, the transit from event 1 to event 2 means that the other block has moved one atom of space. That poses no problem for the quantization of space.
We now bring up Zenos’ 4th paradox. We consider a block C, at rest in a reference frame C where A and B are moving with equal velocities and in opposite directions. In that frame C, each of blocks A and B has moved, in the time lapse from event 1 to event 2, only one half-block, half-space atom. Hence looking at things from this coordinate system does pose a problem for the quantization of space.
To sum up, space quantization seems to be at odds with the relativity of space (in the sense of distances traversed), even under Galilean Relativity.
If we now consider Special Relativity…, that does not seem to alter the conclusion, does it?
To start with, in frame A block B has actually moved along the length of block A (a space atom), but in frame B block A is length-contracted, it is less than a space quantum long and it has displaced only that shorter length at event 2… So the problem for space quantization arises even earlier. And in frame C, both A and B are length-contracted, both are less than a space atom long and between events 1 and 2, each of them travels half that distance…
Conclusion: whenever you bring up reference frames, the idea of quantized space suffers; even more under SR, where more values are relative. Would you agree to that?