Resolve the Bug-Rivet Paradox - Strange SR Paradox

[zefram_c]

Does anyone know of a resolution to the bug and rivet paradox? It is outlined here:

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/bugrivet.html#c1

My first guess is that the rivet does not reach the bug, and the rivet's predictions are wrong because it undergoes infinite acceleration. However, I would like to see a more formal solution.

 

[pmb_phy]

Does anyone know of a resolution to the bug and rivet paradox? It is outlined here:

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/bugrivet.html#c1

My first guess is that the rivet does not reach the bug, and the rivet's predictions are wrong because it undergoes infinite acceleration. However, I would like to see a more formal solution.

The invalid assumption, which makes it a paradox, is that the rivet is rigid. It is not. To determine if the bug gets squashed requires calculation. Give it a try using the Lorentz transformation equations and note that no signal can travel faster than c. Note that such to resolve such a paradox imply maximum rigidity (i.e. contortions in metal travel at c) and other silly things. In reality the rivet would snip off the head and pass right through the metal destroying it, melting/vaporizing the bug in the process.

I highly suggest following the derivation in that page. I've scanned over it and it seems that close attention needs to be payed. Use the Lorentz transformation as well as the way they calculate it there. I am unable to sit long enough to read these pages closely and am unable to read it now (herniated disk, can't sit for more than a minute or so)

Notice in particular this part - from Rivet frame of reference

The bug disagrees with this analysis and finds the time for the rivet head to hit the wall is earlier than the time for the rivet end to reach the bottom of the hole. The paradox is not resolved.

There is an invalid assumption implied here. It implies that once the rivet end reaches the wall that the rivet head stops simultaneously. That can't happen. If it did then we'd be able to communitcate faster than light.



<ignore previous response. I had another paradox in mind, i.e. rivet contracted length was same as rivet hole length>

 

[DW]

The invalid assumption is that the rivet is rigid. It is not. In one frame one end stops before the other and in another frame each end stops at the same time. One has to account for the finite elasticity. Its invalid to assume that the rigidity is infinite.

Correction - in neither frame do they both stop at the same time. The picture 1 is not the scenario. It is a comparison of their proper lengths and even if it was the scenario they are not even the same length in that picture. To answer the question actually asked, yes the bug gets squished and absolutley does so according to both frames and yes I can give a rigorous solution. According to the frame of the wall the lip of the rivet hits first. The impulse speed stopping the matter of the rivet is limited to the speed of light. Therefor a signal can not travel the length of the rivet from right to left in a time less than that which can be solved from $$\Delta t_{signal} = \frac{\frac{1}{\gamma }L_{0-rivet} - v\Delta t_{signal}}{c}$$. The minus sign is chosen consistent with their choice of negative sign on v in picture 2. The time it would take for a particle traveling at the rivets speed to reach the bug if it starts a distance $$\frac{1}{\gamma}L_{0-rivet}$$ inside the hole at the moment the lip hits is $$\Delta t_{particle} = \frac{L_{0-hole} - \frac{1}{\gamma }L_{0-rivet}}{-v}$$. Again the sign chosen consistent with that from picture 2. All you have to do from here is solve for the time from the first equation and plug in the numbers and you'll see that the first time calculated is longer than the second time calculated. That means that in order for the lip impacting to somehow stop the rivets particles at the left surface from impacting the signal would have to travel faster than the speed of light. Yes faster than c information transfer would lead to a paradox for this special relativistic scenario, but the impulse speed is less than c for any real matterials so there is no paradox. The exact same kind of analysis can be done for picture 3 to show that it would require faster than c information transfer in order to stop the lip from impacting. So the answer is that in both frames the rivet deforms however necessary so that both ends impact. The more interesting question is what is the final state of the rivet after impact. After oscillations damp down, is the proper length longer than it initially was as you would think from pictures 1 or 2 or is it less than it initially was as you would think from picture 3. In order to answer that question one needs to know the elasticity and dampening properties of the matterial that the rivet is made of.

 

[zefram_c]

Thanks guys. My feeble excuse is that in all of the SR related courses I took and books I read, none put any special emphasis on the fact that SR excludes perfectly rigid bodies as the rived is implicitly assumed to behave. In fact I don't think any of them even mentioned as much.

 

[pmb_phy]

Thanks guys. My feeble excuse is that in all of the SR related courses I took and books I read, none put any special emphasis on the fact that SR excludes perfectly rigid bodies as the rived is implicitly assumed to behave. In fact I don't think any of them even mentioned as much.

Your welcome.

Einstein wrote a paper on this very early on. That paper was called

On the Inertia of Energy Required by the Principle of Relativity, Albert Einstein, Annalen der Physik 23 (1907). The subsection, No. 3, is called Remarks concerning the dynamics of the rigid body.

 

[meteor]

Curiously, even a physicist of the value of Max Born stumbled with it, he devised a model for a rigid body in special relativity