Flash open manhole cover paradox

[Alan McIntire]

The superhero "Flash" races over a 3' diameter open manhole at 0.999 the speed of light. I realize that this exceeds escape velocity on earth, so assume the running is being done on Terry Pratchett's "Discworld"
https://en.wikipedia.org/wiki/Discworld

From Flash's point of view, the manhole opening is only 1.6" wide, so while running, he plants his foot long right foot over the hole with no problem, and continues on to catch the villain.

From the reference frame of the open manhole, Flash's foot is only 0.54 inches long, so he trips on the open manhole and breaks a leg. Fortunately, being the Flash, he will heal super fast.

According to Wolfgang Rindler,

http://copaseticflow.blogspot.com/2013/06/wolfgang-rindler-and-rod-vs-hole.html

there's no such thing as a rigid body, so part of Flash's foot will bend into the hole, Flash's foot will break on the hole from both the viewpoint of the manhole cover and of Flash.

I think Flash will put the rear of his foot down before the manhole in his own frame, the hole will fit under his foot, and the front of his foot will come down, all in that order. Flash's foot will NOT bend downward.
I don't think Rindler was right in this specific case.

From the manhole cover point of view, Flash's heel will hit the ground before he comes to the manhole cover, he will skid over the opening, and the front of his foot will come down after skidding over the manhole cover. From both Flash's and the manhole cover point of view, Flash's foot NEVER rests on the ground when completing a step, but is constantly skidding, burning rubber from the bottom of his boot.

So what does happen that both the Flash an open manhole observer can agree upon? Was Rindler right, am I right, or is there a third possibility?

 

[Dale]

Isn’t Flash’s foot at rest with respect to the ground? So there should not be any length contraction where the rubber meets the road. His foot will easily fit in the manhole.

 

[pervect]

I agree with Dale. If we assume Flash has good traction, friction is preventing Flash's foot from moving relative to the ground at the moment of contact. Now, I suppose we could imagine that Flash's foot was sliding over the ground if he wasn't getting enough traction. This would be like running on ice, where your foot is sliding out from under you. With zero friction, you'd get no traction and be unable make progress, but with some non-zero amount of friction you could theoretically run, I suppose. I wouldn't want to try it though, I imagine I wouldn't be coordinated enough, my feet would slide out from under me, and I'd fall. It's easier to analyze what happens with tires, a situation where the tire is not getting enough traction and slides relative to the ground due to a lack of enough friction between the tire and the ground is commonly called "burning rubber". This also suggests some issues that Flash would have with a hotfoot if his foot was sliding over the ground at relativistic velocities in the presence of friction. And we've intimated some of the problems that arise if one assumes there is no friction.

Once we specify whether or not Flash's foot is in fact sliding relative to the ground, we can then proceed to analyze the problem further. One might consider trying to analyze a simpler problem that's easier to specify, like a wheel. We could optionally imagine the wheel having feet with a constant proper length mounted mounted along the circumference, I suppose, and then we'd see another vairant of the Ehrenfest paradox. You'll find a lot of literature on the Ehrenfest paradox. I am thinking that a full discussion of Ehrenfest's paradox would be a bit of a digression at this point, I am hoping at this point to explain the motivation of suggesting that Ehrenfest's paradox is releveant. The critical question here is how many of these "feet", presumed to be of constant proper length, would fit around the circumference of the rotating wheel.

 

[Grinkle]

Can the paradox be stated like -

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.

Of course the Marvel Universe does not necessarily follow the same laws of physics as the DC universe ... ;-)

 

[pervect]

Can the paradox be stated like ((this))

It is related, but I'd say it probably best be discussed in a different thread as it's not really the same as the original

 

[Grinkle]

ok - thanks, @pervect !

 

[Alan McIntire]

I think I'm right and Dale, pervect, and Rindler are all wrong. c = 299,792,548 meters per second.
0.999c =299,792,755.45 meters per second.
From the hole's perspectrve, since 1 yard = 0.9144 meters, Flash will be over the hole for

.9144/299,792755.45 = 3.05 /10^9 seconds. The distance fallen will be
1/2*16*t^2 feet. That works out to a whopping fall of 2.44 /10^17 meters. I'm sure the wear and tear is more on our shoes for every step than that insignificant amount. As I said, from Flash's perspective, he will put his foot over the small crack, will continue skidding a bit as he runs, and will go on to catch the thief.
From the manhole perspective, Flash will put the rear of his foot down before reaching the hole, will quickly skid across the opening, and finally have the front of his foot over the opening after the hole after wearing 2.44/10^171meters =0.0000000000000000244 meters off the sole of his boot.

I think the moral of this story is whether you have a human, a car, or anything tearing along at relativistic speeds on the ground and not in space, thanks to differences in x, x', t, and t' , there is going to be a LOT of friction and skidding involved, with a lot of wear and tear on the wheels of a car, the boots of Flash, and the hockey stick Antman is riding on.

Grinkle, your figure for Antman, 0.9999 c rather than my 0.999 c, reduces the hole to 0.0001999 *36 inches = 0.0072 inches. Antman will spend a lot less time than Flash did over the hole, and the fall/ wear and tear on the hockey stick will be a lot less than 0.0000000000000000244 meters

 

[PeterDonis]

That works out to a whopping fall of 2.44 /10^17 meters.

In which case Flash doesn't fall through the hole in either frame and this whole discussion becomes irrelevant. But that doesn't make Dale, pervect, and Rindler wrong; it just means you've changed the scenario to one in which everything they've said (and everything you've said as well) doesn't apply.

If you really want to include the quantitative effects of gravity, you need to dial up the gravitational field to where it actually becomes significant (i.e., causes enough of a fall that in at least one frame the obvious "naive" calculation makes Flash's foot fall into the hole). I'll leave it to you to do the math and figure out how strong a gravitational field is required; but you've certainly made it obvious that it's many orders of magnitude more than 1 g.

I think the moral of this story is whether you have a human, a car, or anything tearing along at relativistic speeds on the ground and not in space, thanks to differences in x, x', t, and t' , there is going to be a LOT of friction and skidding involved

No, the moral of your post is that if things can move at ultrarelativistic speeds, a 1 g gravitational field is negligible. There wouldn't even be any friction because there isn't enough gravity to make a difference in the tiny amount of time it takes for Flash and the hole to pass each other.

 

[Dale]

will continue skidding a bit as he runs,

Well, I think any skidding makes it confusing, but if there is any skidding then the foot should go backwards. And the speed of the foot while skidding would not generally be the same as the speed of the center of mass.

Skidding is counter-productive to running fast. So the Flash should skid as little as possible, IMO.

 

[peety]

If we assume he is skidding and ignore friction, isn't this the pole in barn paradox (but with not-fitting instead of fitting)? In his frame the hole is safely contracted; from the hole's point of view the heel and toe strike at different times and so again don't enter.

 

[Mister T]

This is also like the relativistic cookie cutter. In this paradox a sheet of cookie dough moves along a conveyor belt at high speed. The baker lowers a round cookie cutter and stamps out a circle to form a cookie.

https://www.physics.harvard.edu/uploads/files/undergrad/probweek/sol27.pdf

 

[Alan McIntire]

Thanks for that link, Mister T.
Peety, it's SIMILAR to to pole/barn paradox, but unlike the pole/barn paradox, Grinkle's hockeystick/Antman is constantly touching the ground. With discrepancies in x', x, t' and t, the only resolution for the hockey stick is plenty of wear and tear on the hockey stick due to friction with the ground- otherwise Antman and an observer stationary with the manhole cover see different things.

The Flash, rather than constantly touching the ground, will have his feet moving up and down something like a sine wave.

Since the difference in the up/down frame is constantly changing, Flash's feet are "accelerating". If we increase the gravitational field enough to make a noticeable difference, I suspect that
a "black hole" would appear, making it impossible to distinguish whether Flash's view that he could step over the hole, or the hole's point of view that the object falls in, is correct.
See
http://skfiz.wdfiles.com/local--files/archiwum-2005-06/Spotkanie SKFiz 24III2006 - Bartlomiej Szczygiel.pdf

There's a limit of c^2/a in terms of acceleration. Any rate higher than that would cause the object to break, even if it were made of the new element
"unbendable unbreak-ium"
I wonder if that puts an upper limit on flash's speed significantly LESS than c.From Flash's point of view, the times when the back, front, and middle of his foot hits the ground, assuming he's running flatfooted, is
0...0...0
From the manhole cover's point of view, it's something like
1...0..-1 where the units are something obscure like trillionths of a second.
So from the manhole cover frame the back of the foot is in the air AFTER Flash raised the rear of his foot in Flash's frame, and the front of the foot is in the air, not yet hitting the ground after the manhole cover.
As a result, Flash's foot will be curved in a sort of "U" in the manhole's frame.

 

[pervect]

The discussion seems to be floundering around the issue of how people walk :(.

Other than to repeat that I agree with Pete's model of the motion of the foot, and not Alan's (at least not so far as I understand it), I'm not sure what more can be usefully said.

There are some other interesting points that have been brought up, but the confusion about the details of the motion of the feet when people walk doesn't really seem allow for any useful discussion until it's cleared up. And there are no signs that the issue is being addressed ore even acknowledged by the original poster.

 

[Ibix]

floundering around the issue of how people walk :(.

The conversation is falling flat on its face, you mean.

the confusion about the details of the motion of the feet when people walk doesn't really seem allow for any useful discussion until it's cleared up.

It might be interesting to come up with a "realistic" animation of a stick figure running at relativistic speed. I suspect it requires a material model, sadly, since walking involves changing angular velocity of limbs, so can't be Born rigid.

 

[PeterDonis]

If we increase the gravitational field enough to make a noticeable difference, I suspect that
a "black hole" would appear

No, you can have an unbounded "gravitational field" (proper acceleration) and still be above the horizon of a black hole.

There's a limit of c^2/a in terms of acceleration.

This isn't a limit on proper acceleration; $a$ can be as large as you like. $c^2 / a$ is the distance "down" to the Rindler horizon for proper acceleration $a$. But no matter how large $a$ is, $c^2 / a$ is greater than zero.

I wonder if that puts an upper limit on flash's speed significantly LESS than c.

No, it doesn't. And in any case, Flash is moving perpendicular to the direction of acceleration.

 

[Grinkle]

The discussion seems to be floundering around the issue of how people walk :(.

Flash is almost certainly a toe-striker when sprinting. And he's always sprinting.

From:

http://livehealthy.chron.com/foot-strike-sprinting-6987.html

"The switch from heel strike to forefoot running in sprinters takes advantage of the body's natural mechanics."

Edit: The diagram in the OP shows a dramatic toe strike.

 

[lekh2003]

I'm just throwing it out there and this is not exactly helpful, but at the theoretical speeds Flash is running at, not being able to accurately determine the movement, position and length of his leg is the last of your problems. You are forgetting that the air molecules would not have time to move around his body and that he would start a violent nuclear reaction that would destroy the manhole before this problem could ever possibly occur. You already run into problems with this situation: https://moviepilot.com/posts/3771638 (don't know if this link is acceptable?) Reaching near-light speeds introduces all kinds of heat, fusion and durability issues.

I am, however, assuming that the discussion can be continued with the new restriction that the entire world is indeed perfectly durable and everlasting and air is non-existent.

 

[Alan McIntire]

Peter Donis, you're wrong on c^2/a.

See
.http://skfiz.wdfiles.com/local--fil...ie SKFiz 24III2006 - Bartlomiej Szczygiel.pdf

There IS an upper limit on how much a body can accelerate without coming apart;

For example, let's try 1 light year per year,
then (1light year/year)^2 / 1 light year per year per year acceleration = 1 year. If the ship is 1 light year long, the back end will never receive a signal that the front end is accelerating, and the ship will come apart - there's no such thing as indestructible.

a 6 ' human is about 1/164 million light seconds long. Accelerating at 164 milllion light seconds per second per second, the head will never receive the signal from the accelerating feet, and even Superman would come apart with that acceleration.

 

[PeterDonis]

Peter Donis, you're wrong on c^2/a.

See...

The article you linked to is a perfectly good discussion of Lorentz contraction and related topics, such as the well-known Bell Spaceship Paradox. But I don't see what it says that indicates I am "wrong on c^2/a". Can you be more specific?

There IS an upper limit on how much a body can accelerate without coming apart

There is an upper limit on how far below a given point with proper acceleration $a$ a single connected body can extend, yes; that's what the distance $c^2 / a$ tells you. But that does not impose any limit on how large $a$ itself can be. Just make sure the point with acceleration $a$ is at the bottom of the body.

a 6 ' human is about 1/164 million light seconds long. Accelerating at 164 milllion light seconds per second per second, the head will never receive the signal from the accelerating feet

This is true if the head has that acceleration. It is not true if the feet have that acceleration; in that case the head, to stay at the same distance from the feet, will have a smaller acceleration, by a sufficient amount that the distance from the feet to the head will be less than $c^2 / a_{head}$.

 

[Alan McIntire]

After thinking about it, I see I was wrong about the black hole- I suppose a Neutron Star would do it, as in Larry Niven's science fiction story "Neutron Star", where a character orbited too close to a Neutron Star and was pulled apart by tidal stresses.

Reread that link I gave you, remember that c^2/a limit. As a body accelerates towards a high gravity object, relativistic effects will appear, and as the object accelerates, it will SHRINK in the frame of the neutron star, while distances for the neutron star will shrink in the frame of the person falling. Refer to the "Bell's Paradox" article I referenced earlier, or similar articles on the internet. Think of the FRONT rocket as Flash's or Superman's foot, the BACK rocket as the Superhero's head, and the rope connecting the two rockets as Flash's or Superman's body. In "Bell's Paradox", the front and rear spaceships may accelerate at the same rate in the frame of the person originally at rest-the manhole cover, but in the frame of the Superhero, the feet start accelerating first, and the distance between head and feet gradually increases in the superhero frame. Of course that happens to a human jogger under normal conditions also, but our bodies are not rigid and can take an insignificant amount of relativistic stretching without coming apart.
A gravitational field strong enough to pull the center of Flash's foot a significant amount below the manhole cover, far enough to fall in over the short time he passes over the cover, would be strong enough to stretch him apart, as in "Bell's Paradox" and in Rindler's example.

Remember, Flash and manhole cover measure different things happening at different times, but after the experiment, they must agree on the net affect on Flash's body. Flash sees himself stepping on the manhole cover at an instant with no harmful effects, maybe an insignificant tugging on the center of his foot due to the gravitational field under the cover.

The manhole cover measures the length of Flash's foot shortened, and dropping below the rim under the tug of gravity. I suppose the cover sees the center of Flash's foot falling slightly below the hole's rim, but at the same time from the hole's frame, the back of Flash's foot is going up after pushing off the ground, and the same time as the front of Flash's foot is coming down to hit the ground after passing the manhole cover, making a slight "u" shape or bottom of a sine wave shape in Flash's foot.

 

[Dale]

Peter Donis, you're wrong on c^2/a.

Look at the units. c^2/a is not an acceleration, so it cannot be a limit of acceleration. Based on the units, what would it be a limit of?

 

[pervect]

The conversation is falling flat on its face, you mean.

*laughs*

It might be interesting to come up with a "realistic" animation of a stick figure running at relativistic speed. I suspect it requires a material model, sadly, since walking involves changing angular velocity of limbs, so can't be Born rigid.

My ambitions are more modest. They'd entail drawing stick figures of the motion of the non-relativistic case. But I'm not going to do the work, and perhaps I shouldn't be pessimistic but I'm not imagining the OP as doing the work either.

A reasonable goal without perhaps doing all the work of drawing the stick figures might be to think about the footprints a running man leaves (see above), and what this implies about the motion of the foot relative to the ground when the foot is contacting the ground..

Hint: if the foot were sliding on the ground while it touched the ground, the footprints would be smeared out by the relative motion.

I'm tempted to make remarks about the relativistic case, but I'll defer, because I don't think it's terribly productive to worry about the relativistic case until there is some agreement on the non-relativistic case.

[add]
In the absence of any agreement on the motions of a running man's feet, it could still be worthwhile to look at the relavistic version of simpler motions, such as the pole-barn paradox, or the relativistic sliding block

 

[Alan McIntire]

In reply to Peter Donis: c^2/a = (meters/sec)^2/(meters/sec^2) = meters which is the maximum length something can be, even Superman, before it automatically falls apart under gravitational acceleration, like the two spaceships (head and feet) and the connecting rope (body).

 

[Ibix]

In reply to Peter Donis: c^2/a = (meters/sec)^2/(meters/sec^2) = meters which is the maximum length something can be, even Superman, before it automatically falls apart under gravitational acceleration, like the two spaceships (head and feet) and the connecting rope (body).

As Peter says, that depends on how you apply the acceleration. Two rockets, one at your head and one at your feet, is not the same as one rocket at your feet. Your head will accelerate differently in the two scenarios.

 

[PeterDonis]

A gravitational field strong enough to pull the center of Flash's foot a significant amount below the manhole cover, far enough to fall in over the short time he passes over the cover, would be strong enough to stretch him apart, as in "Bell's Paradox" and in Rindler's example.

Have you done the math? If so, please show it. Just waving your hands is not enough.

 

[PeterDonis]

c^2/a = (meters/sec)^2/(meters/sec^2) = meters which is the maximum length something can be, even Superman, before it automatically falls apart under gravitational acceleration, like the two spaceships (head and feet) and the connecting rope (body).

This is not correct. Go back and read my post #19 again, carefully. $c^2 / a$ is a limit on a length, yes, but it's not the length you think it is.

As a general comment, as I hinted in my previous post just now, you are waving your hands and using heuristic reasoning instead of doing the math. You need to do the math. Your intuitions are not as reliable as you think they are.

 

[PeterDonis]

In "Bell's Paradox", the front and rear spaceships may accelerate at the same rate in the frame of the person originally at rest-the manhole cover, but in the frame of the Superhero, the feet start accelerating first, and the distance between head and feet gradually increases in the superhero frame.

This is not correct. In Bell's Spaceship Paradox, the front and rear ships start out at rest relative to each other, with their clocks synchronized, so when they start accelerating, they do so at the same time according to both of them. So that is not the reason why the distance between the two increases.

Again, you need to actually do the math instead of waving your hands and doing intuitive, heuristic reasoning.

 

[PeterDonis]

Btw, we have a PF Insights article on the Bell Spaceship Paradox which is relevant to this discussion:

https://www.physicsforums.com/insights/what-is-the-bell-spaceship-paradox-and-how-is-it-resolved/

 

[Alan McIntire]

Figure Flash is "racewalking". From Flash's point of view, letting H= heel of foot, C=center of foot, T= front of foot,
from Flash's point of view, he hits
H..C..T all within a foot of each other.

From the manhole point of view, rather than the ground being contracted in length, it's Flash's feet contracted in length. The ground "detects"
H.....C.....T more than a foot in length. From the ground point of view, Flash MUST be skidding.
From Flash's point of view, he hits the ground with kinetic energy of M0C^2( 1/sqrt (1-( v/c)^2) -1)

Both the ground and Flash must agree that the bottom of Flash's boot is shredding; from the ground's point of view it's because of skidding; from Flash's point of view a lot of the boot is being burned away due the kinetic energy each time his foot hits the ground.