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I've found the following paradox, and I wonder how to resolve it.

Two discs are floating in space, call them A and B. They are at a fixed distance D, coaxial, and rotate at the same speed. Each of them has a hole near the border.

The position of the hole in disc B lags behind the position of the hole in disc A, by a small amount of time. This time is exactly equal to the time it takes light to traverse D.

This means that a laser pulse that gets through hole A is going to get through hole B, and hit a detector on the other side, but the size of the holes is such that there is very little margin for error.

Now: an observer passes along this contraption, moving in the axial direction at a sizeable fraction of the speed of light.

Due to Lorentz contraction, the distance between A and B is going to be smaller in the observer's frame of reference. Plus, the rotation of the discs is going to be slower, due to time dilation.

Either of these effects would be enough to prevent the laser pulse from passing through hole B: it's still traveling at the same speed in the observer's frame of reference, but it has less ground to cover, and on top of that the other disc won't have rotated enough to put the hole in its path. So the detector doesn't get hit!

It's illogical for the detector to be hit or not hit depending on the observer. What am I missing? How to resolve this?

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    $\begingroup$ I don't understand fully. Does the observer move in the axial direction (perpendicular to the discs) or parallel to the discs? $\endgroup$ – md2perpe Jul 7 at 18:55
  • $\begingroup$ The observer moves in the axial direction. I'll clarify the original question. $\endgroup$ – F. Polo Jul 7 at 20:52
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    $\begingroup$ Then the relativity of simultaneity comes into play changing the angle between the two holes. $\endgroup$ – md2perpe Jul 7 at 21:11
  • $\begingroup$ You begin with the observer moving in the axial direction. Then you say the rotation of the discs is going to be slower, due to time dilation. It will not. The spinning of the disks is normal, at right angles, to the axial motion where the time dilation will occur. I have seen this as a cylinder with a groove in the side which spins fast enough for a pulse of light to make it from one end to the other. Nice idea for a paradox but the rotation will not change being at right angles to the motion of the observer. $\endgroup$ – Elliot Jul 9 at 15:53
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Expanding on Dale's answer, by shifting your frame of reference, the relative alignment of the two disks changes, since what is "simultaneous" changes!

If we take disk A as the origin, then the relative-simultaneous (undilated) time of disk B shifts under a frame-velocity shift of $v$ by $\beta \frac{x}{c}$, where $x$ is the (non-contracted) displacement to disk B and the usual Lorentz-transformation definitions of $\beta = v/c, \gamma=1/\sqrt{1-\beta^2}$. Disk B therefore is "now rotated ahead" of what it was before the coordinate transformation by the amount it rotated in a time of $ \beta \frac{x}{c}$.

The time it takes for the beam to traverse from A to B is now reduced by the spatial dilation (by a factor of $1/\gamma$) and by the movement of disk B during the travel time (by a factor of $1/(1+\beta)$); the rotation of Disk B is also slowed by time dilation (by a factor of $1/\gamma$). The pre-transformation rotation time of Disk B when the beam was traversing the distance was $\frac{x}{c}$, while the new time is $\frac{1}{\gamma^2}\frac{1}{1+\beta}\frac{x}{c}=\frac{1-\beta^2}{1+\beta}\frac{x}{c}=(1-\beta)\frac{x}{c}$, which is a reduction of $\beta \frac{x}{c}$ - this exactly cancels out the Relativity of simultaneity shift above!

This cancellation is guaranteed by the conservation under any Lorentz transformations of the spacetime interval between the beam passing through the hole in disk A and the hole in disk B - that is, the beam passing through hole A then hole B always aligns with what happens during the traversal from hole A to hole B, no matter what your inertial frame of reference is.

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It's illogical for the detector to be hit or not hit depending on the observer. What am I missing? How to resolve this?

The key to resolving almost all relativity “paradoxes” is the relativity of simultaneity. Conceptually it is the most difficult part of special relativity and so it is the part that gets neglected most often. That is the case here. You accounted for time dilation and length contraction, but forgot to account for relativity of simultaneity.

One other thing is that in any frame where the disks are moving the distance that the light travels is different from the distance between the disks. By the time the light moves the distance D’ the far disk has moved. Nevertheless, the key issue is the relativity of simultaneity

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    $\begingroup$ Would be nice to see this quantitatively. $\endgroup$ – lalala Jul 8 at 7:54
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I'm self-answering because the "click" moment for me was when, after reading all other answers, I realized that this scenario is actually a cunningly-disguised variant of the well-known one where two lightning bolts simultaneously strike the opposite ends of a train.

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    $\begingroup$ @Albert said “it is experimentally proven that one way speed of light is anisotropic in rotating frame of reference”. This is not true. As you correctly mention earlier, simultaneity is conventional and therefore so is the one way speed of light. It cannot be experimentally proven, only assumed. Assuming simultaneity and assuming a one way speed of light are logically the same thing. You cannot have it both ways. You cannot claim that the usual isotropic convention cannot be confirmed by experiment but your anisotropic convention can. Both are conventions and not experimentally provable $\endgroup$ – Dale Jul 8 at 15:55
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    $\begingroup$ @Albert please feel free to post a question about this topic. Your misconceptions are far too great to address in comments. $\endgroup$ – Dale Jul 8 at 17:02
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    $\begingroup$ The one way speed of light is purely a convention. Just as simultaneity is. They are, in fact, the same convention. If you are not willing to address your misconception in a venue where it can effectively be addressed, that is fine by me. I just wanted @F.Polo to be aware that your comment here was incorrect $\endgroup$ – Dale Jul 8 at 18:09
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    $\begingroup$ (cont) Professor John D. Norton, an authority on the science of Albert Einstein and the philosophy of science, has some nice info on The Conventionality of Simultaneity. $\endgroup$ – PM 2Ring Jul 9 at 4:46
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    $\begingroup$ @Albert Lorentz Ether Theory is mathematically equivalent to Special Relativity. A luminiferous ether is certainly not required. I assume you do not intend to promote some other, non-mainstream, ether theory, since that would be off-topic... $\endgroup$ – PM 2Ring Jul 9 at 4:53

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