Meter stick slides over a meter wide hole at a high speed

[Hiero]

Consider a meter stick sliding along its length towards a hole which is a meter wide in the direction of motion. Suppose the meter stick moves with speed so that the gamma factor is 10.

Then in the stick’s frame, the hole is 10cm and should be easy to cross. But in the hole’s frame, the stick is only 10cm and so should easily fall in.

Does the stick fall or not? Why?

I just can’t seem to wrap my head around this scenario. (Seems similar to the ”ladder paradox” but that is just an example of the relativity of simultaneity; I don’t see how that will save us in the stick-hole version.)

 

[Orodruin]

Instead of looking at a stick that is sliding along, which would need acceleration to fall down - leading to necessarily looking at the acceleration process and understanding that the stick is necessarily deformed in at least one of the frames, consider a situation where there is a one meter stick oriented horizontally and falling vertically towards a plate that is moving with a gamma factor of 10. The plate has a 1 m hole in its rest frame. This hole in the original frame will be 10 cm due to length contraction and the stick will not be able to fall through.

If you Lorentz transform this to the plate rest frame, then the x-distance between the stick end-points will indeed be 10 cm, but the stick will no longer be horizontally oriented, it will be tilted. Due to this and to the movement of the stick in the horizontal direction, the stick will not be able to pass through the hole in this frame either - the same conclusion as for the original frame.

 

[Hiero]

@Orodruin, I understand that scenario. To be quantitative, let’s say the stick moves vertically at speed v_y in the first frame (where it’s horizontal). In the plate’s frame the meter stick will have Δy = v_yv_x/c^2 meters but also a vertical speed of v_y/ϒ_x and so the time between when the top and bottom ends reach the plate is ϒ_xv_x/c^2 in which time it moves horizontally by v_x times that. So altogether we need the the gap to be 1/ϒ_x+ϒ_xv_x^2/c^2 = ϒ meters long which is more than one, hence they still collide.I don’t quite understand in what ways this is equivalent to the original setup. Are you saying the stick will cross the hole? I don’t see the connection.

 

[Hiero]

Actually, I think I see part of your point. The motion of falling down the hole tends to rotate the rod in the right direction for the forward-end to stay above the forward edge.

So then in the stick’s frame, it will bend into the hole a little, despite being a “perfectly rigid” stick? Is there then subtler bending in the hole’s frame as well?

If I wanted to analyze this within the realm of special relativity, is it possible? At least to some good approximation?

 

[Orodruin]

So then in the stick’s frame, it will bend into the hole a little, despite being a “perfectly rigid” stick?

A perfectly rigid stick simply does not exist. The reason to look at the other scenario is to avoid the entire problem of specifying exactly how the rod is allowed to accelerate.

If I wanted to analyze this within the realm of special relativity, is it possible? At least to some good approximation?

Yes, but you will have to accept that a perfectly rigid object does not exist and that the answer will depend on what you specify for how the rod accelerates

 

[pervect]

I'd strongly suggest thinking about a simpler paradox first. When the OP understands the simpler paradox, they can maybe consider coming back to this one. They're very similar, the simpler paradox only involves inertial motion, not accelerated frames of reference and/or gravity. The simpler paradox is "The Barn and Pole Pardox", and there's a lot written about it, for instance http://math.ucr.edu/home/baez/physics/Relativity/SR/barn_pole.html

 

[Hiero]

A perfectly rigid stick simply does not exist.

Can’t we say something like it’s the most ideal rigidity? In other words it’s the least amount of bending for the scenario? I’m just wondering because this bending has nothing to do with any stiffness constants, right?@pervect Of course I understand that problem; I even mentioned it in the OP (under a different name, “ladder paradox,” but I linked the details).
That’s a trivial example of the frame dependence of simultaneity.

Not to be rude, but how do you think I could correctly analyze Orodruin’s situation in post 2 but not understand the barn thing...Anyway I am not seeing how the stick will make it over the hole... it won’t right?

 

[Ibix]

If you imagine doing this at non-relativistic speeds then the rod will tip as it moves over the gap and start to tumble. Handling objects that start to spin in relativity is complex because they cannot behave rigidly and you have to specify a material model. That goes double for a rod sliding off a table, where part of it is supported and part of it isn't, so it must flex. That's why we keep ducking the problem.

I can offer another slightly modified version. This happens explicitly in zero g. The rod is very thin compared to its length, as is the tabletop, and it is traveling very close to the table. As it passes over the gap in the table it is struck simultaneously (in the table frame) along its length by an anvil which is moving perpendicular to the table in the table frame, knocking the rod through the hole in the arbitrarily thin tabletop in arbitrarily short time. What does that look like from the rod frame?

 

[Hiero]

What does that look like from the rod frame?

Well the anvil will be angled downwards and the impulse will no longer be simultaneous, the front of the rod gets hit down first. I believe the front end (or any point) gets hit when it’s the same fraction of the way along the gap as in the first frame.
Am I missing any key details?

 

[Ibix]

Am I missing any key details?

In the table frame the whole rod is hit simultaneously. In the rod frame, as you say, the rod is hit at different times along its length. You might like to think about whether or not it'll suffer any bending stresses.

 

[Hiero]

@Ibix indeed the front end will be want to bend down. So going back to the original problem, is the result that in either frame the stick does not cross the gap? In the stick’s proper frame it falls in by this bending effect. Is this right?

To be honest I thought the answer was supposed to be that the stick will cross. I’ve been so confused trying to figure how it will come back up in the hole’s frame. If it’s actually the case that the rod falls in the hole, then I am much more content with neglecting the quantitative details.

I am curious though, is there any concept of ideal rigidity? Of course “perfect” rigidity is out the window, but is there some concept of “most rigid”? (Any elementary references on rigidity in SR would be cool.)

 

[jartsa]

Consider a meter stick sliding along its length towards a hole which is a meter wide in the direction of motion. Suppose the meter stick moves with speed so that the gamma factor is 10.

Then in the stick’s frame, the hole is 10cm and should be easy to cross. But in the hole’s frame, the stick is only 10cm and so should easily fall in.

Does the stick fall or not? Why?

I just can’t seem to wrap my head around this scenario. (Seems similar to the ”ladder paradox” but that is just an example of the relativity of simultaneity; I don’t see how that will save us in the stick-hole version.)

The stick bends in both frames. The frames disagree about the bending force.

So how about if we say that on the stick there is a mechanism that makes the first 10 cm of the stick 'weightless'. Like for example a small physicist standing on the stick on the 15 cm mark and pulling on a rope attached on the front of the stick.

In the frame where the hole is ten times wider the downwards force is ten times smaller. That might explain something. I am referring to the special relativistic transformation of transverse forces: F`= F/gamma.

 

[pervect]

<<snip original confusing stuff>>

So, let us proceed, knowing that you understand the simpler case of the barn and pole. , I will assume that you understand that it's the relativity of simultaneity that explains the apparent paradox. (If you don't, we may have to revisit this earlier problem, as it will then be obvious that we do not share the same understanding of it).

What's not obvious, though, is how relativity of simultaneity applies to the new problem. The short answer is that the relativity of simultaneity drastically changes our understanding of rigid bodies, rigid body mechanics, and rigid body motions in special relativity.

In Newtonian physics, rigid bodies have six degrees of freedom - three translational degrees of freedom, and three rotational degrees of freedom. In special relativity, there is no such thing as a rigid body, which is Orodruin's point in #5. We do have something called a Born-rigid body in special relativity, but it's not the same as a rigid body.

The reason a Born rigid body is not the same as a rigid body is that Born rigid bodies cannot change their state of rotation without violating the rigidity conditions.

So the short answer to your problem is that you are (apparently) incorrectly trying to apply techniques that used to work in Newtonian physics, techniques built upon rigid bodies, to a relativistic situation where rigid bodies no longer exist in exactly the same sense.

There are two routes one could take from this point. The easier route is to study what happens to something like a steel bar, which is not perfectly rigid. Because it's not perfectly rigid, steel bars can rotate just fine in special relativity - it's just that when they change their state of rotation, they must change their shape. They must deform. When we allow the bar to deform, there's no problem.

The point of Orodruin's #5 is that we can understand how the steel bar deforms when it goes over the hole in the floor with a bit of work. With just a litlte more work, we can come to the understanding that the speed of sound in the bar serves as a quantitative measure of it's rigidity, and have a nice neat answer that because the speed of sound in steel is about 5km/second, our notions of the steel bar remaining "rigid" as it goes over a hole in the floor start to become very poor approximations well before we reach relativistic velocities. If we study how a steel bar moving at 10 times the speed of sound in steel, 50km/secm, responds to the situation of going over a hole in the floor, using purely Newtonian methods, we will see that it's better to treat the bar as if it had no rigidity. With a bit of imagination, we can look for things with low speeds of sound. The low-speed-of-sound realm will give us some insight into the relativistic behavior as well - some of the details will change, but studying how a slinky drops



will give us a lot of insight into the low-speed-of-sound realm. Imagining a collection of slinky's going over the hole, each separate slinky representing one section on the bar, will give us better insight into it's behavior at 50km/sec than thinking of it as rigid.

The other route one could take is to study Born rigidity in more depth. This can be rather interesting, but it may not answer your original question, because in the end we'll come to the conclusion that Born rigid objects don't change their state of rotation.

 

[Hiero]

Easily, I'm afraid

By this you mean the rod makes it easily over the hole?

It's clear, on a qualitative level, how bending could cause the rod to fall in the hole, in the rod's frame.

I cannot understand, on a qualitative level, how any bending could save the rod from falling in the hole, in the hole's frame.

So if the answer is that the rod passes the hole, please say that explicitly, because I cannot fathom it.

Anyway thanks for your response; I will be sure to look into "born rigidity," even if it doesn't help with this problem.

 

[Orodruin]

So if the answer is that the rod passes the hole, please say that explicitly, because I cannot fathom it.

Again, the problem of your setup is acceleration. Without specifying when the rod starts accelerating and how different parts of the rod accelerate, there is no definite answer to your question.

 

[PAllen]

Well, let's ask how far a hockey puck falls under 1 g crossing a 1 meter hole, moving at .999 c. The answer is about 200,000 times less than the radius of one hydrogen atom. Normal equilibrium interactions between a solid body and air are much larger than this. Thus the puck would not have any 'awareness' of the hole at all. This underscores @Orodruin 's point that the features of this problem are all dependent on what acceleration profile you specify.

 

[pervect]

By this you mean the rod makes it easily over the hole?

Nope, that's not what I meant at all. I'll just snip that and focus on the rest.

It's clear, on a qualitative level, how bending could cause the rod to fall in the hole, in the rod's frame.

I cannot understand, on a qualitative level, how any bending could save the rod from falling in the hole, in the hole's frame.

In the hole frame, the rod bends, and the tip of the rod hits the edge of the hole. I usually stop the analysis there, because I start to imagine the consequences of the tip of the rod hitting the edge of the hole at relativistic velocities.

In the rod frame, we seem to agree that the tip of the rod hits the edge of the hole.

So, in either frame, the tip of the rod hits the edge of the hole.

 

[Hiero]

Well, let's ask how far a hockey puck falls under 1 g crossing a 1 meter hole, moving at .999 c. The answer is about 200,000 times less than the radius of one hydrogen atom.

That’s a good point. Perhaps I should have made it a kilometer stick! That should be a million times what you said so like 5 hydrogen atoms? Light sure is fast huh.

Again, the problem of your setup is acceleration. Without specifying when the rod starts accelerating and how different parts of the rod accelerate, there is no definite answer to your question.

Well to avoid the complexities of rotation Ibix mentioned, let us think of a false floor over the hole, which will be pulled away the moment the back end goes over the edge, in the hole’s frame.

The entire stick will begin accelerating at once in that frame, similar to Ibix’s distributed impulse example.
With that, do we need to specify anything else?

As a side question - I’m honestly not sure how to handle gravity. The thing I’m unsure of is that a downwards acceleration in the stick’s frame should be 1/ϒ^2 times smaller in the hole’s frame. So one frame is g the other is g/ϒ^2 ? I don’t feel like this is right, but I’ve never thought of gravity beyond Newton.

 

[sweet springs]

Though I am afraid it is just a duplicate of above discussions:

In Hole system
Top end of Rod touches Hole bottom and Bottom end of Rod enter Hole top are simultaneous.

In Rod system
When Top end of Rod touches Hole bottom, Bottom end of Rod does not enter Hole top yet.
When Bottom end of Rod enter Hole top, Top end of Rod smashes hit by Hole bottome or penetrate Hole.

So Rod and Hole cannot keep sound in the process. Bottom end of Rod cannot stop entering Hole. Elastic signal from Rod top "Don't push me anymore!" cannot reach Rod end to prevent it. The two or at least one must be broken.

 

[Orodruin]

Well to avoid the complexities of rotation Ibix mentioned, let us think of a false floor over the hole, which will be pulled away the moment the back end goes over the edge, in the hole’s frame.

What makes you think that this avoids rotation in the rod frame? Also, you are probably thinking of using gravity to accelerate the rod through. Don’t. It will just add another layer of complication and the closest you will get in SR is a non-inertial reference frame with a constant proper acceleration in the vertical direction. It would be much better for your own sanity not to care about the acceleration mechanism and instead just specify a scenario fully in one frame. You can then Lorentz transform to the other to find out what the situation looks like there.

I don’t feel like this is right, but I’ve never thought of gravity beyond Newton.

The point is that you shouldn’t think of gravity in SR. How gravity is described in relativity is given by GR.

 

[pervect]

Well to avoid the complexities of rotation Ibix mentioned, let us think of a false floor over the hole, which will be pulled away the moment the back end goes over the edge, in the hole’s frame.

I'm not sure of why you are doing this, and I also want to point out the hidden frame dependence. The event "when the back end goes over the edge in the hole's frame" is well defined, but you say that the floor is pulled away at "the same instant of time". The idea of the 'same instant of time' over an extended spatial region is frame-dependent because of the relativity of simultaneity. "The same instant of time" in one frame of reference, plotted on a space-time diagram, is different than "the same instant of time" in another frame.

This is key point in understanding the pole-in-barn paradox, something that you mentioned earlier that you were familiar with when I asked. It's also something that a lot of people get wrong. Thus it's hard for me to tell if you actually are familiar with the relativity of simultaneity, or whether you only think you are familiar with it.

As a side question - I’m honestly not sure how to handle gravity. The thing I’m unsure of is that a downwards acceleration in the stick’s frame should be 1/ϒ^2 times smaller in the hole’s frame. So one frame is g the other is g/ϒ^2 ? I don’t feel like this is right, but I’ve never thought of gravity beyond Newton.

You'll be missing out on at least two other effect. If we have a rocketship accelerating to simulate gravity, and we have a block or train sliding across it's floor, a gyroscope attached to the sliding block will precess. This is called Thomas precession.

See for instance https://arxiv.org/abs/0708.2490v1

As an application where both the reference frame and the gyroscope accelerates we will consider a gyroscope on a train that moves along an upward accelerating platform as shown in Fig. 3.

Figure 3: A train at two consecutive times moving with velocity v relative to a platform that accelerates upward. A gyroscope with a torque free suspension on the train will precess clockwise for v >0.

A second effect, which I shall loosely call 'curvature of the track', is described in the above paper in figure 11.

A graduate level treatment that systematically reveals all the effects of a frame of reference associated with a train moving across the floor of an accelerating spaceship involves tensors and Christoffel symbols. If you are familiar with these graduate level techniques, I can point to a few long threads where the topic is discussed in some mathematical detail, and a few metrics are derived. (Different people have different approaches). If you don't have that background, working through https://arxiv.org/abs/0708.2490v1 is probably more viable attempt to gain some understanding of the two effects I mentioned above.

 

[Ibix]

@Ibix indeed the front end will be want to bend down. So going back to the original problem, is the result that in either frame the stick does not cross the gap? In the stick’s proper frame it falls in by this bending effect. Is this right?

To be honest I thought the answer was supposed to be that the stick will cross. I’ve been so confused trying to figure how it will come back up in the hole’s frame. If it’s actually the case that the rod falls in the hole, then I am much more content with neglecting the quantitative details.

I am curious though, is there any concept of ideal rigidity? Of course “perfect” rigidity is out the window, but is there some concept of “most rigid”? (Any elementary references on rigidity in SR would be cool.)

Here is a 2+1d Minkowski diagram of the example I gave - the rod (blue) is just in front of the table (green, with a gap) at the beginning of the experiment, moving in the +x direction at 0.6c. It is struck by a force in the y-direction that pushes it below the table as it passes the gap. I haven't illustrated the anvil, so all you see is the effect of the impact - the "kink" in the blue worldsheet.

You were wondering what that looked like in the frame where the rod is stationary. Here's the answer:

Now you can see that the kink, which was parallel to the x axis, is not parallel to the x' axis - it's not a simultaneous strike in this frame. In fact, the rod kind of ripples through the hole. The ripple is faster than light, though (not a problem - there's no direct causal dependency of one part of the impact on any other part), so there's never any bending stress on the rod - each adjacent element of rod is already in motion by the earliest time it could become aware that the next-door element started moving.

This is basically why you can't have a general model of rigidity in relativity - something that is a rigid motion in one frame isn't necessarily rigid in any other. You can define things like Born rigidity, but they can't apply to things with changing proper acceleration.

Does that help?

 

[PAllen]

Just a fine pint clarification of @Ibix lasr post: Born rigid acceleration is possible for any acceleration profile at all including arbitrary changes in 4-acceleration (including direction). However, once the profile is specified for one point of a born rigid object, the motion of the whole object is determined. Any change in some other part of the body different from this, which would be needed to account for rotation, is not possible.

 

[Herman Trivilino]

I am curious though, is there any concept of ideal rigidity?

Think of the rod as being made of delicate glass. It breaks whenever it's bent, and you are free to choose the amount of bending it can sustain before breaking. The smaller your choice, the closer we get to your ideal. However small your choice, the rod makes it through the hole unbroken.

 

[PAllen]

Think of the rod as being made of delicate glass. It breaks whenever it's bent, and you are free to choose the amount of bending it can sustain before breaking. The smaller your choice, the closer we get to your ideal. However small your choice, the rod makes it through the hole unbroken.

Can you clarify? Seems to me that most variants of the scenario have the rod breaking in all frames. For example, any scenario where the rod tilts as it goes through the hole must involve breakage unless the you allow a lot of deformation. If you take the anvil suggestion instead, the impact along different parts of the rod is not simulaneous in the rod frame, so again it must break, unless you allow substantial deformation. If you plug in numbers you need something more like ideal chewing gum rather than ideal glass to avoid breakage.

 

[PAllen]

Now you can see that the kink, which was parallel to the x axis, is not parallel to the x' axis - it's not a simultaneous strike in this frame. In fact, the rod kind of ripples through the hole. The ripple is faster than light, though (not a problem - there's no direct causal dependency of one part of the impact on any other part), so there's never any bending stress on the rod - each adjacent element of rod is already in motion by the earliest time it could become aware that the next-door element started moving.

I must disagree with part of this. By any standard definition (e.g. shear tensor) as well as physical common sense, there is enormous shear stress in your scenario, and this feature is invariant. The FTL motion of the kink simply means that the shear, instead of propagating (limited to speed of sound in material), is causally independently induced at each point along the rod, not that there is no shear. Note that in the MCIF of a given rod element, you will have adjacent elements have acceleration of trillions g, and relative speed of large fraction of c.

Any attempt to do this with real materials would result in a rod being converted to a compressed plasma moving at high speed downward (in the original rod rest frame) once the process is complete. Note, this downward moving plasma line will be rotated relative to the original rod (per the original rod rest frame).

Note, that the illustration provided does not account for the fact that after the anvil accelerates a rod element to near c relative to the original rod, there is no proposed counter anvil to decelerate an element, so every rod element ends up continuing to move at high speed downward relative to the original rod. That is, retraction of the anvil does nothing to stop the downward motion induced on each rod element.

 

[Orodruin]

To be honest, I think it is rather meaningless to discuss this issue with ”real” materials for the reasons already mentioned (also, stress energy tensor calculations are typically beyond introductory courses). The interesting stuff is how a series of events in one frame appears in another. Questions such as ”does it break” should preferably be postponed for later.

 

[Herman Trivilino]

Can you clarify? Seems to me that most variants of the scenario have the rod breaking in all frames.

I was thinking that in the variant posed by the OP the rod makes it through the hole unbroken.

My point is that if the rod doesn't bend in its own rest frame, then it doesn't break. That's invariant regardless of how delicate the rod.

 

[PAllen]

I was thinking that in the variant posed by the OP the rod makes it through the hole unbroken.

My point is that if the rod doesn't bend in its own rest frame, then it doesn't break. That's invariant regardless of how delicate the rod.

For the rod not to bend in its own rest frame, the hole must be bigger than the rod per the rod rest frame. That means the hole must be much bigger than the rod in the hole rest frame. The OP actually describes a case where the hole is too small in the rod's initial rest frame, in which case it cannot get through the hole without bending in the rod rest frame, and this establishes that it can't get through the hole without shear in any frame (despite appearances in the hole rest frame).

 

[Ibix]

I must disagree with part of this. By any standard definition (e.g. shear tensor) as well as physical common sense, there is enormous shear stress in your scenario, and this feature is invariant. The FTL motion of the kink simply means that the shear, instead of propagating (limited to speed of sound in material), is causally independently induced at each point along the rod, not that there is no shear.

Hm. The problem as I stated it was one given to us as undergrads (you are, of course, correct that there should be another anvil to stop the y-direction motion). I think it was just presented as a variant on the rod and barn problem, but we certainly ended up arguing about whether or not the rod broke. We concluded that it didn't on the grounds I stated before, that each subsequent atom of the rod is in motion before it can communicate with its neighbours.

Unfortunately I've been sketching out the stress-energy tensor for the rod in my head and it does seem likely that there will be space-space off diagonal elements in it, which implies shear stresses. So you may be right. I need pen and paper.

As an amusing aside, I've been googling stress and strain tensors and now I see adverts offering me counselling services.