length contraction, light speed and a bomb

imsmooth

This was probably asked some time ago. It involves a paradox of relativity and I hope someone can answer it for me:

Imagine a solid cylinder (call this the plunger) with a bar welded to one end making a T. The plunger fits just perfectly into a hollowed cylinder of the same length, and the end of bar welded on the end prevents the plunger from going in any further. There is a button at the end of the hollow cylinder that, when pressed, triggers a bomb. It is just far enough from the fully engaged plunger so that it does not get pressed.

Now if we send the plunger forth at the speed of light into the hollowed cylinder, someone standing next to the hollowed object will see the length of the plunger contract. When it enters the cylinder it will be too short to press the button. However, someone riding on the plunger will see the hollow cylinder contract. Its length is contracted enough that the plunger can reach the button, and detonate the bomb.

So what happens? Both can't happen.

The_Duck

First, can we say that the plunger goes at almost lightspeed, instead of lightspeed? I don't know how to accelerate plungers to lightspeed.

Let's think about the situation in the frame of the stationary hollow cylinder, where it looks like the plunger should not possibly be able to hit the button. I believe that to resolve "paradoxes" like this one you must reconsider your implicit assumption that the plunger is perfectly rigid. You would like to imagine that the front end of the plunger stops the instant the back end catches on the edge of the cylinder. But the "message" that the back has stopped cannot be communicated forward along the plunger (via the electric forces between the atoms composing the plunger) faster than the speed of light. So the front will keep moving for a bit even after the back has stopped. If the "message" "telling" the front of the plunger to decelerate doesn't have time to catch up to the front end before the front end hits the button, the bomb will certainly explode.

yuiop

I accept the perfect rigidity is not compatible with SR and deformation of the plunger can explain why the button is activated in this case... but I see a deeper paradox when the scenario is viewed from different frames....


I accept this part too and this represents the observations made by an observer that is at rest with the cylinder (Frame C). In frame C the plunger stretches and this could be detected by stress/strain devices mounted on the plunger. In this frame the proper length of the plunger increases.

Now consider the point of view of an observer at rest with the plunger (Frame P). In frame P, the cylinder is moving and length contracted and the button is trivially activated. After the collision between the plunger end and the cylinder base, the restraining T bar has not yet made contact with the open rim of the cylinder. In frame P the collision causes the plunger to compress and the proper length of the plunger decreases.

The paradox is now this. All observers should agree on the proper length of the plunger, but in this scenario, frame C sees an increase in the proper length and frame P sees a decrease in the proper length, so we have something that superficially does not agree with relativity. The plunger is either physically compressed or physically stretched but both at the same time is impossible. I know all these sort of paradoxes can be resolved, but the solution seems to be subtle in this case and I wonder if anyone has any suggestions?

Passionflower

The bomb will most likely be detonated. The plunger enters the cylinder near light speed and is only stopped by the bar. Two things have to be taken into account:

1. The side of the plunger where the bar is located cannot instantly stop and will deform the cylinder.
2. More importantly even if the plunger would stop instantly where the bar is located the other end will not stop instantly, at most it will stop later by: length * speed of sound.

So most likely the bomb will be detonated.

If everything is perfectly rigid and non deformable (which is impossible) then the bomb would not be detonated.

PeterDonis

You haven't specified the problem precisely enough. You need to decide between two possible scenarios, 1 and 2 which I'll describe first from the cylinder frame and then from the plunger frame:

Cylinder Frame

(1) The plunger is length contracted to a large enough extent that, even taking into account the time delay between the bar at the back stopping the back end, and the information that the back end has stopped propagating to the front end, the front end stops before it hits the button. This is the scenario implicit in your sentence, "When it enters the cylinder it will be too short to press the button."

(2) The plunger is length contracted, but not enough to prevent it from stretching, once the back end stops, sufficiently to hit the button. This is because of the time delay between the bar stopping the back end and the information that the back end has stopped propagating to the front end.

Plunger Frame:

(1) The cylinder is length contracted, but when the bar at the back end of the plunger hits it, it sends an expansion wave down the cylinder that pushes the button away from the plunger. This expansion wave propagates fast enough to keep the plunger from hitting the button before it has stopped.

(2) The cylinder is length contracted sufficiently that the plunger's front end can reach the button before the expansion wave set off by the bar hitting the back end can reach the button. This is the scenario implicit in your sentence, "Its length is contracted enough that the plunger can reach the button, and detonate the bomb."

So the apparent "paradox" is due to you implicitly thinking of scenario 1 when looking at things from the cylinder frame, but implicitly switching to scenario 2 (which of course is a different scenario, not compatible with scenario 1) when looking at things from the plunger frame.

Both of these scenarios are perfectly consistent within themselves; the question is which one will actually happen in practice (since of course they're mutually exclusive). In order to specify the problem properly, you have to specify which scenario will actually happen (or, equivalently, specify the actual material properties of the plunger and the cylinder--stiffness, sound speed, etc.--precisely enough to allow one to calculate which scenario will actually happen).

sylas

Assume the plunger moves at 0.6c, in which case γ is 1.25. Assume the cylinder is 1 meter long.

From the point of view of the cylinder, the moving plunger is 80 cm long.

When the T bar hits the cylinder, the far end of the cylinder is still 20cm from the button. However, the rules of physics mean that at BEST, the forces slowing the rest of the plunger move down the plunger at the speed of light. (In practice, it is a lot less than this.)

The end of the plunger is 20 cm from the button, and moving at 0.6c. It will get to the button in about 1.1 nanoseconds.

The consequences of the T hitting the cylinder is 100 cm from the button, and moving at c (or less) will take at the very least 3.3 nanoseconds to get as far as the button.

If the only thing stopping the plunger is the T bar, then the button will get pressed.

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From the point of view of the plunger, the cylinder is 80 cm long, and hence the button will be pressed before the T bar hits the cylinder.

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You will get the same result no matter what v you use. As long as the plunger has a proper length that is the same exactly as the cylinder, and pressing of the button is exactly at that precise distance, then the T bar cannot prevent the button being pressed.

There is no paradox here. The relativity of simultaneity is not a paradox -- merely counter-intuitive. In both frames, the same conclusion is reached that the button is going to be pressed. If you did not get the same conclusion then you would have a paradox; but in fact both frames give the same results for what events occur. The only differences are where and when -- which is no more a paradox than having things appear to your left rather than your right when viewed from a different direction.

Cheers -- sylas

PS. On the answers given previously.

The term "proper length" refers to the length in the frame where the object is at rest. There is no contraction of proper length. Contraction of length from the view of the moving frame goes hand in glove with changes in the notion of what is simultaneous at the front and the back. (yuiop)

Rather than "perfect rigidity", you can think in terms of infinite rigidity... which means thinking that the speed of sound in the material is the speed of light and a compression wave in the material is capable of having it spring back towards undeformed length at the speed of light. You still get deformations, but you don't get paradoxes. The paradoxes only arise when you propose influences moving through material instantaneously, or else at more than light speed.

PeterDonis has the right idea -- except that in the plunger frame, the button was already pressed even before the back end hits the cylinder. The question of deforming the cylinder never arises. You can assume the cylinder does not deform at all.

PeterDonis

This raises a good point--if the problem specifications are such that the events "bar on back end of plunger hitting cylinder" and "front end of plunger hitting button" would be spacelike separated if they occurred, then the question of deformation of the cylinder doesn't even arise. But of course the deformation of the *plunger* is still a crucial factor.

That said, I'm not sure the problem specifications *require* that for any v, "in the plunger frame, the button was already pressed even before the back end hits the cylinder," because as I read the OP, the plunger's length at rest is not supposed to be *exactly* the same as the cylinder length; it's supposed to be shorter, enough so that if both plunger and cylinder were sitting at rest, the bar at the back of the plunger would prevent the front of the plunger from hitting the button:

This means that there may a *little* leeway for the plunger to stop before it hits the button, depending on the precise values of v and the rest length of the plunger compared to that of the cylinder.

DaleSpam

This is also known as the bug and rivet paradox, and is one of my favorites.

sylas

Not really. In the plunger frame, the plunger is significantly longer than the cylinder, because of length contraction of the cylinder. This means that the plunger reaches the end of the cylinder and presses the button well before the T-bar hits.

As soon as you say they have the same proper length and that the relative velocity is relativistic, all the leeway is gone completely.

In the cylinder frame, the plunger touches the button at the same time as the T-bar hits the other end of the cylinder -- and because it takes non-trivial time for the button end to have any response to the effects of the T-bar stopping, the button is well and truly pressed before the T-bar can have any effect.

This corresponds precisely to what I tried to describe earlier about length contraction.

What is length contraction in the one frame corresponds precisely to a change in simultaneity for the location of the two ends in the other frame. Change in length is the precisely same thing as change in simultaneity; it is simply the same thing being considered from a different perspective.

Cheers -- sylas

PeterDonis

Ok, I see now. Even if the proper length of the plunger is slightly shorter, as soon as the relative velocity is such that the cylinder, in the plunger frame, is length contracted by an amount larger than the difference in proper lengths, there's no leeway any more. So even if, say, the plunger was 10 cm shorter than the cylinder (to give a significant gap when both are at rest), as soon as the length contraction factor is less than 9/10, which corresponds to a relative velocity greater than 1/sqrt(10) or about 0.3c, the plunger will hit the button before the cylinder hits the bar, in the plunger frame.

sylas

Yes, that's it!

Using the title "bug and rivet", I had a look back in the archives. There's a good discussion of this particular problem from 2004 in the thread: What is the resolution of the The Bug-Rivet Paradox paradox in special relativity?.

The last post of that thread, by Janus, also works through the problem precisely as you have done here, with a plunger (or rivet) that has a proper length shorter than the cylinder (or hole), and yet which will crush the bug or explode the bomb no matter how strong it might be, as long as the velocity is such that gamma is more than the ratio of proper lengths.

Cheers -- sylas

yuiop

I think PeterDonnis has touched on the correct resolution of this sub paradox in that some of the events are spacelike and so are not causally related and the sequence of events is observer dependent. However there are a couple of things I would like to clear up.

For orientation assume the T part of the plunger and the open end of the cylinder to be on the right and likewise the closed end of the cylinder with the button and the face of the plunger that activates the button to be on the right. Also assume that the bomb explodes in both frames and that the transmission of physical compression/expansion forces through the materials is slower than the initial relative velocities of the cylinder and plunger. The observers for a given reference frame are inertial and initially co-moving but do not accelerate after the collisions. Finally assume that the proper lengths of the cylinder and the plunger are such that when they are both at rest relative to each other and the T bar is in contact with the open end of the cylinder that the plunger is too short to activate the button.

Consider the sequence of events as seen in the two different frames.

In the plunger frame (P):

Initially the plunger is at rest and the cylinder is moving with constant velocity to the right. The cylinder is significantly shorter than the plunger.

P1) Button pressed / bomb explodes.
P2) Left end of the plunger compresses due to the collision and the plunger starts to accelerate to the right and starts length contracting. The cylinder slows down and starts to lose length contraction and is also physically stretched at the closed end. Overall the length of plunger decreases relative to the length of the cylinder.
P3) T bar makes contact with the open end of the cylinder.
P4) T bar end of the plunger starts to stretch and the open end of the cylinder starts to compress due to the collision. The T bar end of the plunger starts to accelerate (and length contract) and the open end of the cylinder starts to slow down and lose length contraction.

Finally the cylinder comes to rest and the plunger moves away to the right with constant velocity equal to the initial velocity of the cylinder (assuming the plunger and cylinder have equal rest mass). The plunger is now significantly shorter than the cylinder.

In the cylinder frame (C):

Initially the cylinder is at rest and the plunger is moving with constant velocity to the left. The plunger is significantly shorter than the plunger.

C1) T bar makes contact with the open end of the cylinder.
C2) T bar end of the plunger starts to stretch and the open end of the cylinder starts to compress due to the collision. The T bar end of the plunger starts to slow down and lose length contraction and open end of the cylinder starts to accelerate and length contract. Overall the length of plunger increases relative to the length of the cylinder.
C3) Button pressed / bomb explodes.
C4) The closed end of the cylinder starts to accelerate to the left and the closed end of the cylinder is physically stretched (assuming the middle part of the cylinder has not yet started moving). The left end of the plunger having made contact with the closed end of the cylinder starts to compress.

Finally the plunger comes to rest and the cylinder moves away to the left with constant velocity equal to the initial velocity of the plunger (assuming the plunger and cylinder have equal rest mass). The cylinder is now significantly shorter than the plunger.

Now if we consider all the events to be spacelike separated then the sequence of events in one frame should simply be a time reversal of the events in the other frame but this is not the case. For example the T bar making contact with the open end of the cylinder is the first event in frame C but it not the last event in frame P. Presumably the explanation is that there is a mixture of timelike events (that must retain the same temporal order in any reference frame because they are potentially causally related) and spacelike events that can be any order. C1 and C2 are causally related in frame C, so they retain the same temporal order (P3 then P4) in frame P. On the other hand, the T bar making contact with the cylinder (C1) and the button being depressed (C3) in frame C, are not causally related and happen in the reverse order in frame P (event P3 preceded by event P1) Does that seem about right? If not, would anyone like to clean up and refine the sequence of events described above?

I am aware that there is no contraction of proper length due to transformations. I was simply talking about Newtonian change in proper length due to physical forces such as a car being shorter after a cllision in a traffic accident to to permanent deformation of materials when I referred to changes in proper length. I realise there are some ambiguities about the the definition of proper length of an object during a collision when different parts of the same object have different velocities and different states of acceleration at a given instant in an a given inertial reference frame so I have avoided the use of the changes of proper length in this second treatment. I simply use stretch and compress to mean changes in length due physical forces (that may or may not be permanent) in the the Newtonian sense and the term length contraction to refer to changes in measured length due the observers relative velocity in the SR sense.

The question of deformation does arise in the plunger frame, but in this frame it happens after the button is pressed while in the cylinder frame it happens before. Any permanent physical deformation (eg T bar bent and scored by the collision) must happen in all reference frames.

sylas

Sure. I agree (since I think this is part of what you are saying?) that the resolution of this paradox can be identified unambiguously. It isn't actually a paradox, and like all the various relativity paradoxes, the solution is obtained simply by applying relativity correctly. If a paradox appears to arise, this is -- always -- because someone is not applying physics correctly and is making an error.

"P2" is not an event. It refers to a series of distinct events all along the length of the plunger. Also, the plunger does not "contract". It extends. One end with the Tbar hits the cylinder and slows rapidly. (As rapidly as you like; there is no physical limit on accelerations.) But the other end is still moving, and will continue to move until the effects of the collision progress along the plunger. The effects eventually decelerate all parts of the plunger, but not all at the same time, and while one end slows and the other continues as before, the proper length INCREASES -- as does the measured length, as measured in any frame you like.

In the plunger frame (inertial frame in which the plunger is initially at rest), the cylinder rams into the T-bar, and EXTENDS that end of the plunger to the right.

As the effects of the collision continue to occur, you will have rebounds and deformations and all kinds of things which require more information to resolve. This is not actually part of the original problem.

In this puzzle, we would usually assume that the plunger is much lighter than the cylinder; but it hardly matters. In the inertial frame, the plunger (initially at rest) ends up moving right, and the cylinder (initially moving right) ends up still moving right, but perhaps more slowly. The cylinder will tend to be compressed in the collision and the plunger extended, and this is a change in proper lengths. You can impose an elasticity constraint on the materials so that plunder and cylinder rebound to their same proper lengths, if you like, it makes no difference to the original question of whether or not the button gets pressed.

The button DOES get pressed. This is the only correct resolution to the original question.

Whether the plunger and the cylinder end up with changes in proper length is a matter of how the material responds to deformations -- which is not part of the original problem. You cannot assume an answer about how lengths end up after the collisions without additional information of this kind.

Perhaps. You don't actually have to assume any compression of the cylinder. It can be as incompressible as you like -- which means that the accelerations/decelerations of the plunger as as large as you like.

The essential point to answer the original question is simply this. The button pressing end of the plunger will hit the button well before the effects of the collision start to have any impact at all.

So the answer to the original puzzle is "the bomb explodes". Finis. Everything else is beyond the scope of the question and depends on assumptions not given.

Cheers -- sylas