Zeno's 4th Paradox & Quantized Space: SR & GR Effects

[Saw]

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?

 

[JesseM]

I think there are problems reconciling the Planck scale with relativity, leading to proposals for modifications such as doubly special relativity...apparently one such proposal is de Sitter invariant special relativity, about which the wikipedia article claims "whereas in all doubly special relativity models the Lorentz symmetry is violated, in de Sitter relativity it remains as a physical symmetry". And the abstract of the paper at http://arxiv.org/abs/0709.3947 says:

The properties of Lorentz transformations in de Sitter relativity are studied. It is shown that, in addition to leaving invariant the velocity of light, they also leave invariant the length-scale related to the curvature of the de Sitter spacetime. The basic conclusion is that it is possible to have an invariant length parameter without breaking the Lorentz symmetry. This result may have important implications for the study of quantum kinematics, and in particular for quantum gravity.

 

[Saw]

I think there are problems reconciling the Planck scale with relativity, leading to proposals for modifications such as doubly special relativity...apparently one such proposal is de Sitter invariant special relativity, about which the wikipedia article claims "whereas in all doubly special relativity models the Lorentz symmetry is violated, in de Sitter relativity it remains as a physical symmetry".

Thanks a lot. I have quickly read the references and I gather (but I am not sure) that in this de Sitter relativity theory there is... not only invariant speed of light but also an invariant length? Hence would it be compatible with quantized space? In particular, do you know how it would respond to the Stadium paradox as stated above?

 

[JesseM]

Thanks a lot. I have quickly read the references and I gather (but I am not sure) that in this de Sitter relativity theory there is... not only invariant speed of light but also an invariant length? Hence would it be compatible with quantized space? In particular, do you know how it would respond to the Stadium paradox as stated above?

I'm not really familiar with this area, but maybe someone with more knowledge of GR could look over the arxiv paper I linked to and follow it sufficiently to answer your questions...anyone?

 

[Ken G]

It seems to me that any such discussion of a minimum length scale would need to bring in quantum mechanics as well, because quantum mechanics teaches us that we cannot blithely talk about the "events" when various parts of a tiny particle line up, without providing the observational means to establish that indeed such a lining up has occured. It also teaches us that spectacular amounts of energy would be needed to establish that, and the gravitational consequences of that measuring event would need to be included if one wishes to discuss the impact on quantized space. Is there any reason to think de Sitter relativity would work is such an environment? I'm not sure this kind of question is one that modern physics is (anywhere near) ready to really be able to say much that is useful about, though it's always interesting to speculate.

 

[phyti]

- 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.

If this describes a minimum change of state, wouldn't A and B swap positons?

 

[atyy]

http://arxiv.org/abs/gr-qc/0605006