Hockey Puck Paradox: Will Antman Fly or Cruise?

[Grinkle]

Credit to this thread -

https://www.physicsforums.com/threads/flash-open-manhole-cover-paradox.936632/

I did not think this one up myself.

Per suggestion from @pervect I am creating a thread for this.

Antman is sitting on top of a hockey puck trying to hide from Hulk. Hulk has seen him and smacks the puck with a hockey stick custom manufactured by Stark Enterprises to withstand the stresses incurred when being swung at relativistic speeds.

The puck shoots across the ice at 0.9999c and along the way it encounters a man hole. Antman is not concerned because this particular puck has a 3 inch diameter, and he sees the man hole is only 1.6" wide.

Wasp is hovering above the open man hole and is very worried, seeing the tiny puck about to fall slightly into the manhole as it moves over it, catch the far edge of the man hole with the bottom of the puck and spin end over end, launching Antman into the air.

So, does Antman gently cruise over the tiny opening, or does the puck dip slightly as it crosses the large hole and catch on the far edge?

Antman will end up fine either way, that outcome is never in doubt.

 

[Nugatory]

This is the classical pole-barn paradox in a slightly different but well-discussed variant: https://en.wikipedia.org/wiki/Ladder_paradox#Man_falling_into_grate_variation

You can also google for "relativity grate paradox"; one of the first hits will be from here: https://www.physicsforums.com/threads/about-man-falling-into-grate-paradox.684882/

 

[Grinkle]

Am I picturing the resolution properly?

From the frame of the puck, the puck bends into the man hole as soon as its front edge crosses the lip of the man hole, so the puck sees itself deforming along the z-axis and 'pouring' itself into the hole.

From the frame of the hole, the puck doesn't pour in, it just falls in.

If that is a close enough description, I have another question, but want to make sure I'm not veering off-track already.

 

[Dale]

This is a tough one. It requires some material theory or local criterion for falling. The problem is that the usual idea that the hockey puck falls only if it is entirely unsupported is non local. That cannot be a valid criterion.

@Grinkle I think your “pours” vs “falls” explanation is consistent with Rindler. But I think that the “falls” criterion is wrong.

 

[Grinkle]

the usual idea that the hockey puck falls only if it is entirely unsupported is non local

And you (I think) answered my next question which was to ask if this is really analyzable as a SR problem which is how I was thinking of it; the answer being no. The local criteria for falling makes the frames non-inertial when the puck begins to be unsupported and starts to accelerate along the z-axis (I think).

 

[jartsa]

Am I picturing the resolution properly?

From the frame of the puck, the puck bends into the man hole as soon as its front edge crosses the lip of the man hole, so the puck sees itself deforming along the z-axis and 'pouring' itself into the hole.

From the frame of the hole, the puck doesn't pour in, it just falls in.

If that is a close enough description, I have another question, but want to make sure I'm not veering off-track already.

We have a very short puck in one frame and a 'normal' puck in the other frame.

The puck receives impulses on its different points, the impulses are not simultaneous, the impulses cause the puck to bend and to accelerate.

The front front of the 'normal' puck falls a distance x, as the puck bends.

The front front of the short puck falls the same distance x, as the whole puck falls. If we want to be completely exact the puck bends a little bit too.

The disappearance of the supporting force is the 'impulse'.

The bending angles may be the same in the two frames, but not the bending distances.

 

[PeroK]

And you (I think) answered my next question which was to ask if this is really analyzable as a SR problem which is how I was thinking of it; the answer being no. The local criteria for falling makes the frames non-inertial when the puck begins to be unsupported and starts to accelerate along the z-axis (I think).

The problem is clear from the rest frame of the manhole. The puck cannot be considered a rigid body at relativistic speeds, so it could be modeled as a loosely coupled set of particles, each particle in turn falls into the hole, in a "pouring" fashion of sorts.

This analysis then transforms non paradoxically to the puck frame.

The problem is, however, slightly absurd, as the total vertical displacement of each particle will be practically negligible in the time taken to cross the manhole. Less than a single molecule, perhaps.

 

[Grinkle]

The problem is, however, slightly absurd

I'm going to be chuckling over this bit of British understatement for quite a while. :-)

 

[PAllen]

Here is a description as close as I can figure to the 'intent' of the 'paradox'. First, as noted in the thread linked by @Nugatory in #2, trusting the order of magnitude calculation I did there - plus or minus a couple - one would need to posit hundreds of trillions of g force to cause a millimeter of downward motion as the puck crosses the manhole. Observations:

1) If this g force were applied over the whole region, with the surface in which the manhole was cut made of unobtainium, the puck (irrespective of magical friction-less surface property) would be reduced to a molecular layer by the force. I assume the puck is normal material. To fix this issue, we posit there is 1 g except inside the manhole (who knows how this is possible), where the super gravity is. The surface still needs to be made of unobtainium to have a sufficiently rigid edge as the puck reaches the manhole.

2) Then, the description of what happens in the manhole rest frame is: puck reaches edge and each element of its material past the edge follows a parabolic trajectory across the hole, with the bottom millimeter hitting the far edge and being sliced off. Much later, the puck will vaporize under the the heat generated from deformation and slicing, but this will happen so slow compared to the traversal time that it will be many meters passed the manhole. The sequence of events here is puck as a whole passes the near edge, bending to have a parabolic lower surface as it passes the edge, with the part not yet at the edge still flat (because it can't even 'know' parts further along have crossed the edge). The whole puck, with parabolic bottom edge, moves parabolically to the far edge. Then the puck starts getting a millimeter sliced off. Much later, the puck has vaporized.

3) The description from the puck is that a skinny hole approaches. As the hole traverses the underside of the puck, a wave of destruction passes. First, a portion of the puck is sucked into the hole, moves a millimeter down, and this millimeter gets sliced off. The puck passed the hole is thinner by a millimeter than the part that that has not been reached by the hole. Much later, the puck vaporizes. The sequence of events for each element of the puck is the same as (2), but it happens in a propagation along the puck rather than each phase completing for the whole puck before the next phase completes for the whole puck.

In the picture shown from the Rindler paper in the other related thread (the one on Flash running), the one inaccuracy is that the in the hole frame, the idea of the deformation and parabolic shape of lower edge are not shown. Yet this feature is a necessary consequence - each element acting independently under a constant g force follows a parabolic trajectory. This is shown in the rod frame, but it is equally true in the hole frame.

 

[pervect]

My $.02. The problem doesn't explicitly say that we should attempt to analyze it in terms of rigid bodies, but it seems to be set up to invite an attempt at such an analysis. The problem is that the closest equivalent to a rigid body in special relativity, a Born rigid body, is not allowed to change it's state of rotation.

This means that an unmodified description of a Born rigid body wouldn't be able to handle a puck tipping over (rotating) when one edge of the puck is supported and the other is not, which state of affairs we would expect to apply a net torque to the puck. And much of the problem is about what happens in this very case, where one side of the puck is supported, and the other is not. So a Born-rigid body approach doesn't seem suitable from the start, if the puck starts to rotate we have to abandon the Born rigid description. It's possible that there is some sort of approximate rigidity condition we could apply, but it's not especially clear how. So I thinik it would be better to avoid such a restriction in the first case, and do an analysis that does not assume that the puck is rigid.

Some interesting subtle aspects arise here. We can handle, to some extent at least, the problem of a puck sliding on the floor of an accelerating spaceship in special relativity (to mimic gravity without invoking GR), but it has some interesting and unexpected consequences which are confusing enough on their own, without introducing the issue of what happens when there is a hole in the floor. Specifically, a gyroscope mounted on such a sliding puck will precess due to Thomas precession. Furthermore, a Lorentz transform of the shape of the spaceship floor to a momentarily inertial frame co-moving with the puck will change the shape of the spaceship floorfrom flat to curved when one does a Lorentz transform.

Thus I I think a good treatment of the problem would need a puck with an associated material model , as the rigid body approximation just won't work, and would involve considering how the puck deformed when it reached the hole. The demands on the material model of having a puck capable of being accelerated to light speed by impact with a hockey stick are severe and make the problem rather unrealistic - it would be advisable to use a more reasonable model based on known materials rather than introducing some new material that doesn't have any known physical basis.

If we attempt such an analysis, I believe we would find that there isn't any appreciable difference between a steel puck (or one made of any other known material) and one made of jello. This basically arises from the fact that we can quantify the strength/weight ratio of a material which controls how much it deforms on application of an external force via the speed of sound in the material. And the speed of sound in steel is far, far, far less than the speed of light. The speed of sound n jello is even lower than in steel, but for practical purposes both are negligible in this problem. The same issue (the non-relativistic speed of sound in steel or any other known material) explains why it's unrealistic to consider a puck able to withstand a relativistic impact from a hockey stick. Realistically, both the puck and stick would be destroyed.

I suppose the last and simplest point is that this isn't a B level or I level problem that's good for a beginner at relativity to attempt to tackle, in spite of the colorful comic book themes which seem to suggest it should be simple. But there may be a good and reaosnably simple lesson here nonetheless, for a reader without extensive knowledge of special relativity, Born rigidity, and Thomas precession. The simple lesson here is that rigid objects and special relativity don't really get along very well. There are simpler and more common examples that teach this same lesson than this "paradox", though, for instance pointing out that pushing on one end of a long stick does not allow one to send signals faster than light.

 

[PeterDonis]

if this is really analyzable as a SR problem which is how I was thinking of it; the answer being no. The local criteria for falling makes the frames non-inertial

SR can handle non-inertial frames just fine. What it can't handle is gravity--curved spacetime. But you can avoid that by setting the problem in a rocket that is accelerating in flat spacetime, so the "falling" of the puck through the hole is due to the acceleration of the rocket, not to spacetime curvature. This version of the scenario is within the domain of SR.

the total vertical displacement of each particle will be practically negligible in the time taken to cross the manhole

That depends on the acceleration involved. At a 1 g acceleration, yes, what you say is true (IIRC this came up in the other thread that spawned this one as well). But there's no reason why the acceleration has to be 1 g. You could set the problem in a rocket accelerating at millions or billions of gs and it would be perfectly consistent. (Of course we have no way of building such a rocket in practice, but this is a thought experiment.)

 

[Dale]

I think a good treatment of the problem would need a puck with an associated material model

Same here. However, I actually don’t know anything about relativistic material models. So I would tend to model this in the rest frame of the puck.

In that frame the dropping of the puck would set up a transverse shear wave in the material. But the front of that wave would propagate far faster than the speed of sound in the puck, so I don’t know what that means as far as the motion goes.

Similarly with the other end. The compression wave from the impact would propagate much slower than the front.

 

[PAllen]

If you want the puck to move a measurable amount down the hole in the time it takes for traversal, the required g force is such that a material model is irrelevant. Bonds effectively do not exist. A pressureless dust model would be highly accurate for the very short traversal period. To analyze what happens later, over molecular time scales, only then you need a material model.

 

[pervect]

If you want the puck to move a measurable amount down the hole in the time it takes for traversal, the required g force is such that a material model is irrelevant. Bonds effectively do not exist. A pressureless dust model would be highly accurate for the very short traversal period. To analyze what happens later, over molecular time scales, only then you need a material model.

I tend to agree, though I called the pressureless dust model a "jello" model.

 

[Grinkle]

But you can avoid that by setting the problem in a rocket that is accelerating in flat spacetime, so the "falling" of the puck through the hole is due to the acceleration of the rocket, not to spacetime curvature.

I am no doubt missing something obvious - how is this not contrary to the equivalence principle?

 

[PeterDonis]

how is this not contrary to the equivalence principle?

I didn't say you would get a different answer if you set the problem in a gravitational field vs. an accelerating rocket. I just said that you can model the problem using SR and avoid any question of whether curved spacetime affects the answer, even though there is a non-inertial frame involved, contrary to your claim that SR cannot be used in non-inertial frames.

Whether or not the EP applies is a different question; it depends on whether the effects of spacetime curvature--tidal gravity--would be detectable in a patch of spacetime of the size necessary to model the problem. I think you could find a suitably large patch in a curved spacetime in which the "g force" was high enough, while still having negligible tidal gravity, by putting the experiment in a rocket "hovering" at a low altitude above the horizon of a supermassive black hole. Then the proper acceleration of the rocket could be very large, while still having negligible tidal gravity during the span of time it takes to run the experiment, because tidal gravity near the horizon of a black hole varies inversely with the hole's mass. In such a case, the results of the experiment would be indistinguishable from the results of an experiment run inside a rocket accelerating in flat spacetime with the same proper acceleration, as the EP requires.

 

[PeroK]

That depends on the acceleration involved. At a 1 g acceleration, yes, what you say is true (IIRC this came up in the other thread that spawned this one as well). But there's no reason why the acceleration has to be 1 g. You could set the problem in a rocket accelerating at millions or billions of gs and it would be perfectly consistent. (Of course we have no way of building such a rocket in practice, but this is a thought experiment.)

Just so I've got this straight. We have a superhero aboard a rocket ship, accelerating at billions of gs, hitting a hockey puck at close to the speed of light across an ice rink that has an open manhole in it!

 

[PeterDonis]

Just so I've got this straight. We have a superhero aboard a rocket ship, accelerating at billions of gs, hitting a hockey puck at close to the speed of light across an ice rink that has an open manhole in it!

Relativity: it's not for the faint of heart.

 

[Sorcerer]

I hate to be "that guy," but if it's traveling at a relativistic speed, wouldn't it just fly over the hole regardless of size? And if we posit that there is some sort of super gravity, wouldn't friction slow the puck down significantly before it even gets to the hole? And if that happens, why wouldn't the front of the puck hit the back of the hole, causing a deformation? And if the super gravity is only at the hole, aren't we already on the verge of leaving physics in the first place? Not to mention, the puck seems awfully close to an ideal rigid body, which is forbidden anyway, right?

 

[PeterDonis]

if it's traveling at a relativistic speed, wouldn't it just fly over the hole regardless of size?

It depends on the proper acceleration of the surface the puck is traveling on, and which has the hole. For a given relativistic speed and a given diameter (in its rest frame) of the hole, you can calculate what proper acceleration would be required for the puck to fall fast enough to go through the hole. (This was discussed in the previous thread linked to by the OP.) Proper acceleration is unbounded in relativity, so this calculation will always give an answer that is possible in principle, given the laws of physics. Whether it's possible in practice is of course a different question, but if we're talking about superheroes and hockey pucks moving at relativistic speeds we're clearly not restricting ourselves to practical possibilities.

 

[Sorcerer]

It depends on the proper acceleration of the surface the puck is traveling on, and which has the hole. For a given relativistic speed and a given diameter (in its rest frame) of the hole, you can calculate what proper acceleration would be required for the puck to fall fast enough to go through the hole. (This was discussed in the previous thread linked to by the OP.) Proper acceleration is unbounded in relativity, so this calculation will always give an answer that is possible in principle, given the laws of physics. Whether it's possible in practice is of course a different question, but if we're talking about superheroes and hockey pucks moving at relativistic speeds we're clearly not restricting ourselves to practical possibilities.

Well, what about this:

Supposing it falls, either way the puck is going to be deformed in both frames. Since there is no limit on proper acceleration (and gravity, I suppose), then in order for the puck to not be deformed and/or fall through the ice on its way to the hole, there has to be a normal force pushing up on the puck. But as soon as that normal force is gone, the "supergravity" on the front combined with the normal force on the back is going to deform the puck. And, since we're talking about relativity here, and gravity's effects are not instant, then the front of the puck is going to experience a massive downward net "force" long before the back of the puck; meanwhile, the back of the puck is still going to be experiencing the normal force, so there should be some serious deformation of the puck, and this should happen in both frames, shouldn't it? The obvious result is that the buck bends, and that's how it falls into the hole.

Does that make sense?

EDIT- also the puck can't be infinitely rigid, so deformation seems reasonable. Moreover, I'm pretty confident that if you actually did the calculations, the only way for the puck moving at relativistic speeds to not skip over the hole is for the gravity to be so strong that the puck gets deformed, either as it falls into the hole or as it crashes into the back of the hole and gets stuck. Is that reasonable?

 

[PeterDonis]

Supposing it falls, either way the puck is going to be deformed in both frames.

Yes. More precisely, from the standpoint of an inertial frame, the puck starts out deformed--under stress--because of the normal force of the surface pushing up on it; and when that force gets withdrawn, the puck relaxes to its unstressed state.

since we're talking about relativity here, and gravity's effects are not instant

"Gravity" is irrelevant here; we are assuming that we are working in a small enough patch of spacetime for the equivalence principle to apply, so we can model everything as happening inside an accelerating rocket in flat spacetime.

the front of the puck is going to experience a massive downward net "force" long before the back of the puck; meanwhile, the back of the puck is still going to be experiencing the normal force

You've misstated this. The front of the puck will stop experiencing the normal force, and start free-falling, while the back of the puck is still experiencing the normal force. And yes, that will deform the puck--the front of the puck will be relaxing to its unstressed state while the back is still stressed by the normal force.

there should be some serious deformation of the puck, and this should happen in both frames, shouldn't it?

Yes. But it can look very different in the two frames. In the rest frame of the hole, the puck, since it is strongly length contracted, can still look almost like it's just falling rigidly; you would have to look closely to see it, and it is not necessary to see it to explain, in that frame, why the puck falls through the hole; just length contraction is enough.

In the rest frame of the puck, however, the deformation of the puck is crucial in explaining why it still goes through the hole--or, as it would properly be described in this frame, how the far edge of the hole can accelerate upward fast enough to get above the leading edge of the puck.

Note, btw, that both of these frames are non-inertial, but the rest frame of the hole is a fairly simple non-inertial frame--it's just Rindler coordinates. The "rest frame" of the puck, however, is much more complicated--in fact the puck as a whole does not have a "rest frame" while the experiment is taking place, since parts of it will be moving relative to other parts. You have to pick one particular point inside the puck--say its center of mass--and use the rest frame of that point.

 

[pervect]

I hate to be "that guy," but if it's traveling at a relativistic speed, wouldn't it just fly over the hole regardless of size? And if we posit that there is some sort of super gravity, wouldn't friction slow the puck down significantly before it even gets to the hole? And if that happens, why wouldn't the front of the puck hit the back of the hole, causing a deformation? And if the super gravity is only at the hole, aren't we already on the verge of leaving physics in the first place? Not to mention, the puck seems awfully close to an ideal rigid body, which is forbidden anyway, right?

The basic issue is that the a model of the puck as a rigid body is incompatible with special relativity. So the question is, when you make your statements about the puck, are you basing them on the puck being a rigid body, or some other criterion (for instance, replacing rigid pucks with born rigid pucks). If the former, there are issues with relativity. We can explain more if needed, I suppose. If you're not modelling the puck as a rigid body, some discussion of how you are modelling it needs to take place before we can even attempt to find an answer. For instance the born-rigid puck might be an intersting case, but I'm not sure of the answer at this point. If the answer isn't of any interest because you're doing something else, it'd be a waste to spend too much time analyzing it.

 

[PAllen]

Adding just a bit to @pervect above, born rigid analysis of the situation would lead to things most would find very silly, but required by born rigidity. For example, as a born rigid puck slides over the manhole edge, it cannot begin tilting, as rotation cannot be initiated while maintaining born rigidity. You would have to posit that the puck could not start moving down till it was all passed the edge, then it could move down as a whole. On reaching the other side, if it hits, it cannot flip, for the same reason. You would have to posit that it simply stops before starting to fall straight down. How is all this possible? Of course it isn’t - born rigidity is all about magical application of forces to all material elements of a body so as to maintain global requirements.

 

[Laurie K]

I don't think a hockey puck that tilts, i.e. a wheel that is initially moving in a straight line while lying flat on its side, is equivalent to a wheel rotating around an axle which is the usual example given for Born rigidity. The rotation of a wheel around its axle is 90 degrees to the direction of movement while any tilting of the puck is in the direction of movement and limited to less than one 'rotation' unless it hits the outer edge and flips.

Meanwhile some kids with a hockey puck and a hole on a frozen lake work out that you don't have to go anywhere near relativistic speeds to stop the puck falling into the hole.

 

[PAllen]

I don't think a hockey puck that tilts, i.e. a wheel that is initially moving in a straight line while lying flat on its side, is equivalent to a wheel rotating around an axle which is the usual example given for Born rigidity. The rotation of a wheel around its axle is 90 degrees to the direction of movement while any tilting of the puck is in the direction of movement and limited to less than one 'rotation' unless it hits the outer edge and flips.

Meanwhile some kids with a hockey puck and a hole on a frozen lake work out that you don't have to go anywhere near relativistic speeds to stop the puck falling into the hole.

Please read the proofs of Herglotz-Noether theorem before commenting. Any rotation about any axis, by any amount, cannot be initiated in a Born rigid object.

Of course this is all implausible. However, to get the intended effect, it has been noted much earlier in the thread that trillions of g would be needed to get a puck moving relativistically relative to a manhole to fall through. Thus, that is what has been assumed for the thread. Issues with this have been pointed out - that any material would collapse to an atomic film under such forces, thus assumptions like born rigidity or the 'field' turning on only over the hole.

 

[Sorcerer]

Adding just a bit to @pervect above, born rigid analysis of the situation would lead to things most would find very silly, but required by born rigidity. For example, as a born rigid puck slides over the manhole edge, it cannot begin tilting, as rotation cannot be initiated while maintaining born rigidity. You would have to posit that the puck could not start moving down till it was all passed the edge, then it could move down as a whole. On reaching the other side, if it hits, it cannot flip, for the same reason. You would have to posit that it simply stops before starting to fall straight down. How is all this possible? Of course it isn’t - born rigidity is all about magical application of forces to all material elements of a body so as to maintain global requirements.

Sounds almost as nonsensical as a true rigid body ;). I've never actually heard about born rigidity, but in the case of this example it seems fairly unrealistic.