GRED Answer: Fast train cars connected by string

[RJN]

GRED: Guess the Result of the Experiment of the Day: Fast train cars connected by string

Train cars sit on a circular track connected by taut strings. The train cars all begin to circle the track at once, faster and faster, eventually reaching relativistic speed. What happens to the strings?

Please post answers, comments, and discussion below. I will post what I believe to be the correct answer in a few days. OK, a few days have gone by, much discussion has subsided, and so I have posted my answer below. It is visible, though, just by clicking here: http://asterisk.apod.com/viewtopic.php? ... 25#p126497 .

The initial poll, where spoilers were not allowed, can be found here: http://asterisk.apod.com/viewtopic.php?f=32&t=20062 . If you are new to this GRED and want to ponder this question without seeing spoilers, please go there now instead of scrolling down.

GRED editor wanted: I would like GREDs to be a regular feature on the Asterisk where people send in their GRED suggestions to an editor who picks the good ones, posts them, and oversees GREDs generally. If you are interested in being that editor, please contact RJN.

- RJN

[Henning Makholm]

The strings break. Seen from the track's frame, the moving cars and moving strings all become relativistically contracted. Since there cannot become more cars or more pieces of string, there won't be enough cars and strings to fill out the entire track. Thus the strings must either break or stretch elastically, and the latter is not among the given options.

However, from the point of view of one of the passengers, won't it be the track that contracts, such that the neighboring car appears to move closer and therefore makes the string drop?

It is true that the track contracts, but that does not mean that the next car becomes closer. There's just going to be much more track (measured, for example, in number of sleepers/ties) between your car and the next one. Thanks to the relativity of simultaneity, the passenger is allowed to disagree with an observer on the ground about how many sleepers there are between his car and the next one.

Next objection: Assume that the cars are evenly spaced around the circle. If the passengers think there are more sleepers between each pair of cars, won't they think the entire circular track has more sleepers than when the cars were stationary? (Hint: remember that the Lorentz transformation connects only inertial frames).

[neufer]

The strings break.

Just as in the twin paradox where the accelerated frame twin to undergo time dilation in an absolute sense
the accelerated frame spinning train will also undergo Fitzgerald contraction in an absolute sense.

Twin paradox: the "little boy" on the carousel is actually the twin of the old man under the carousel.

[ChazNAus]

Not clear on when the "correct" answer will be posted - or was it already and I'm wasting my time (relatively speaking)?
Anyway if we assume the train tracks are capable of remaining stable even at relativistic speeds then it's the system of train cars and the connecting string(s) that are affected, not each piece separately. People on the train won't see a difference, outside observers will.
Of course if the tracks can't take these speeds then we'll have a train wreck by which all other train wrecks would be measured.

[Chris Peterson]

People on the train won't see a difference, outside observers will.

But the people in the trains will see the strings break, so how will they explain that? Clearly they will see some sort of difference.

[makc]

I don't think there's any reason for strings to break, except reaction to centripetal force. This one can be safely eliminated using sufficiently large circle. Place yourself in the car. With circle large enough, maybe a size of galaxy, it will be like a ride along straight line for you, with no significant acceleration. In this situation, a rope is at rest in your frame, and there's nothing to make it break.

But what about relativistic contraction, you think now? Well, the answer to that is very simple - in this accelerated frame geometry is not flat, and the ratio of circumference to diameter is no longer equal to π. Ta da. Look at me, I am so smart Nah, I cheated, actually. This is explained in one of 1st chapters of Einstein relativity book, and I just happen to remember it.

[Chris Peterson]

But what about relativistic contraction, you think now? Well, the answer to that is very simple - in this accelerated frame geometry is not flat, and the ratio of circumference to diameter is no longer equal to π.

Very plausible. Relativistic motion in a non-inertial frame is not easy to analyze, even though it is one of the simpler problems of GR. I know that the behavior of particles is different in linear and circular accelerators for just that reason.

[makc]

Don't take my word for it (oh yeah... chapter 23, one of the 1st, rright).

[Henning Makholm]

Not clear on when the "correct" answer will be posted - or was it already and I'm wasting my time (relatively speaking)?

For the last several instances (i.e. the ones I've observed), RJN has usually let the discussion progress for about a week before posting his answer.

[MAReynolds]

A simpler case is when the train cars are accelerated linearly. In this case, it has been shown that if the cars have identical accelerations, their separation increases, which, even in their frame, would mean that the string breaks. See ["The case of the identically accelerated twins"
by SP Boughn
American Journal of Physics 57 791-793 (1989).][/http://faculty.erau.edu/reynolds/ps303/TwinParadox1.pdf]

[MAReynolds]

Woops, I mean
"The case of the identically accelerated twins," by SP Boughn, American Journal of Physics 57 791-793 (1989).

[Henning Makholm]

But what about relativistic contraction, you think now? Well, the answer to that is very simple - in this accelerated frame geometry is not flat, and the ratio of circumference to diameter is no longer equal to π. Ta da.

That is actually an argument that the strings will break. Each passenger will agree with an observer on the ground about the distance between that passenger and the center of the circle (because the distance is at right angles to the two observers' mutual motion). So from the passenger's point of view, the radius of the circle did not change as he moved into a rotating frame. Therefore, according to your observation, the circumference of the track must have become longer in the rotating frame, meaning that there is not enough train to fit around it. Ergo, strings break.

Note that the rotating frame in which the entire train stands still is quite a different beast than the inertial frame in which some particular car is momentarily comoving. The rotating frame is not even locally inertial, and is not Lorentz-related to any inertial frame. In particular, the rotating and the comoving inertial frames may agree about the spatial coordinates of points in the comoving car, but not about simultaneity of events that happen at different ends of the car.

Also, seen from the inertial frame, all the other cars are actually in motion, at various crazy angles to the track which is moving as well, and analysis there becomes very complicated.

[Lars]

Is it part of the spec of the experiment that each car is connected to the one in front of it and behind it? I.e. that the cars are connected in a polygon?

If instead each car was connected only to a car directly across the circle from it, would that change the outcome?

[Henning Makholm]

If instead each car was connected only to a car directly across the circle from it, would that change the outcome?

Yes; the radius stays the same for all relevant observers.

(Hence my question in the original thread about whether there are more than two cars).

[Paul Gilmartin]

This is probably not the answer you're looking for, but for any
reasonable-diameter track not on the surface of a neutron star,
the centripetal acceleration will exceed the acceleration of
gravity long before relativistic speeds. So the strings will never
droop; they'll bow radially outward.

In fact, when the tangential velocity approaches
sqrt( Young's modulus / density ) (nominally the speed of sound),
the tensile strength of the strings will be exceeded and they
will break.

Oh! Did you mean massless strings?

I think this was a problem in Misner, Thorne, and Wheeler. I'll
need to look it up. Had to do with Lorentz invariants.

-- gil

[Vissie]

From external observation point of view the track remains the same but the cars and string decrease in apparent length as they approach relativistic speed. Centrifugal force will cause the strings to break.

[Henning Makholm]

From external observation point of view the track remains the same but the cars and string decrease in apparent length as they approach relativistic speed.

Yes!

Centrifugal force will cause the strings to break.

Centrifugal force is irrelevant -- we're assuming the strings are massless, and the centrifugal force on the cars themselves is taken up by the (unobtanium) track.

[Henning Makholm]

It turns out that this problem is famous enough to be named; it is Ehrenfest's paradox, and the Wikipedia article even contains a diagram showing foreshortened railway cars on a circular track!

(Predictably, the Wikipedia editors do not quite agree among themselves about the correct resolution, though the article itself as of this writing seems reasonably non-crackpotty).

[alter-ego]

It turns out that this problem is famous enough to be named; it is Ehrenfest's paradox, and the Wikipedia article even contains a diagram showing foreshortened railway cars on a circular track!

(Predictably, the Wikipedia editors do not quite agree among themselves about the correct resolution, though the article itself as of this writing seems reasonably non-crackpotty).

I've been wondering when that would be noticed. Be aware that the SR graphics/descriptions do not represent the more recent resolutions to the paradox. I think the Wiki article is helping the reader to visualize the paradox, not present the solution
There seems to be a common concensus among several papers, but the resolution/theory development is different. I can only read these to a limited depth, so I have to trust the authors/publications, but, in my opinion, the more "realistic" approach involves a metric derived with GR. I think I know the what the answer is, but like the Wiki artical said about the debate still goes on, I don't really know. Maybe I'll learn more and change my mind.

I like how you interact in these posts, and I suspect you are accelerating along the path to the best answer. For me, discovering that the problem is in fact the famous paradox really helped. Several very smart people have presented resolutions to this problem over the last 100 years. Now here we are innocently meandering about in the same arena! RJN really pulled out the stops for this problem. Good luck, Henning. I'm enjoying the show from the sidelines

[makc]

That is actually an argument that the strings will break. Each passenger will agree with an observer on the ground about the distance between that passenger and the center of the circle (because the distance is at right angles to the two observers' mutual motion). So from the passenger's point of view, the radius of the circle did not change as he moved into a rotating frame. Therefore, according to your observation, the circumference of the track must have become longer in the rotating frame, meaning that there is not enough train to fit around it. Ergo, strings break.

Ha, now you have a talent to make this sound too right to argue about. But then, how would things look in rotating frame? I mean if I am sitting in the car, what causes other cars to move away from me? Some sort of virtual space expansion? Ok, but how come it is limited to strings and not car itselves? Even the drawing in wikipedia is questionable - cars are shortened but strings are not - why? Shouldn't correct drawing look something like this:

0.GIF (2.34 KiB) Viewed 5198 times

[Henning Makholm]

But then, how would things look in rotating frame? I mean if I am sitting in the car, what causes other cars to move away from me? Some sort of virtual space expansion?

In the rotating frame, "space expansion" is the only possible answer. After all, if as you accelerate the circumference becomes larger, then the extra space must be added somewhere, which means everywhere at once.

One conclusion we can draw is that as described in the rotating frame the track itself is stretched as the cars increase in speed, contrary to what one would expect. It is allowed to do this because the rotating frame is not inertial and so is not Lorentz-related to the ground frame. One might think that the non-inertialness of the rotating frame only applies in the radial direction (to create a fictitious centrifugal force), but it also weird in the tangential direction. In particular, the time coordinate of the rotating frame is crazy; clocks at different ends of your railway car that are both synchronized to the rotating frame will not appear to the passengers to be synchronous.

Therefore, what you experience as a passenger is not well described by the naively rotating frame. We can fudge an adjusted rotating frame where the time coordinate is continuously offset such that it agrees with each passenger's ideas about simultaneity in his own vicinity (at least along the track) -- but then the time coordinates of the adjusted frame will not fit together seamlessly when you get 360° around the circle, and as a result you lose the opportunity to make arguments based on the entire train filling exactly one whole circle.

We're still wondering what makes the other cars appear to move away from you. The frame that best agrees with your experience if you stand in a car and measure the distances to the neighbor cars (say, with laser rangefinder) is the inertial frame that happens to move with you at the instant you make the measurements. This frame will lack the centrifugal acceleration you feel, but the error introduced will tend to zero if the circle is large enough and the cars numerous enough.

Obviously, what must happen is that you see the car in front of you accelerate faster than you and the one behind you accelerate slower. How can that be when all cars follow the same acceleration profile? It helps to imagine that the acceleration happens in discrete steps, such that the whole train stays at, say, .3c for some time before accelerating again. All the cars resume accelerating at the same time when viewed from the ground. But because you're already moving, due to the relativity of simultaneity, your laser rangefinder will show that the car in front of you began accelerating before your car does (the instrument is advanced enough to use its time-of-flight information to tell you how long ago the distance it displays were actually true). That gives it a head start. There must be such a head start at each speed during the acceleration (even if you did not stop for a while at that particular speed), and their cumulative effect is to add headway between your car and the next.

Ok, but how come it is limited to strings and not car itselves?

The strings are weaker than the cars, so they are the first to break.

The cars are strong enough not to break. The price of not breaking is that the front end and the back end of the car cannot have exactly the same acceleration at the same time. The back end of the car will need to accelerate more than the front end in order to keep the car in one piece. (This is not a specifically rotational effect; it is true also for linear acceleration). We had better specify that it is, for example, the frontmost powered axle of each car that follows a pre-chosen acceleration profile, and then the rest of the car must just follow it as best it can.

Even the drawing in wikipedia is questionable - cars are shortened but strings are not - why?

That one's easy: Wikipedia's version explicitly says that the strings are not strings but bungee cords, so they stretch instead of breaking.

Shouldn't correct drawing look something like this:

No, your drawing shows a circumference that is shorter than 2πr. It must be longer in order to capture the spatial curvature of the rotating frame.

[Henning Makholm]

There seems to be a common concensus among several papers, but the resolution/theory development is different. I can only read these to a limited depth, so I have to trust the authors/publications, but, in my opinion, the more "realistic" approach involves a metric derived with GR.

Yes. (Sort of -- you don't need the full GR; instead of solving the field equations we'd get the metric directly by choosing the coordinate transformation to/from the underlying flat space and then just transform the known Minkowski metric accordingly).

Arguably, the real content of the paradox is to understand the rotating metric properly. At least, all of the wrong solutions I can imagine turn out to be based on intuitive assumptions about the rotating frame that actually don't hold.

Several very smart people have presented resolutions to this problem over the last 100 years. Now here we are innocently meandering about in the same arena!

It is rather instructive, isn't it? As with all good paradoxes, the argument for the correct answer is simple and easy to follow. The real problem is to figure out what is wrong with the equally simple and apparently convincing arguments for a different answer.

[makc]

ok, let's approach this from different end. let's say there are N equally spaced marks on the track, and only one car moving; sitting at that car, you will see nearby marks closer to each other. in order for whole track to be longer, some distant marks have to get even more distant apart. I would be interested in the drawing of track with marks as it is seen from moving car.

[Henning Makholm]

ok, let's approach this from different end. let's say there are N equally spaced marks on the track, and only one car moving; sitting at that car, you will see nearby marks closer to each other. in order for whole track to be longer, some distant marks have to get even more distant apart.

Luckily enough railway tracks do come with equally spaced marks; they are called ties or sleepers. :–)

But you're conflating two different frames here. In the comoving inertial frame, you will see* the sleepers near you become closer. But that is not the frame in which the entire track becomes longer. On the contrary, you will see the circular track contracted into an ellipse with some of it being shorter than its proper length and other sections of it having the right length but being narrower. (If there were other cars on the track, you would see most of them opposite you on the circle, extra contracted because they would be moving backwards relative to you at speed 2v/(1+v²/c²)).

*Not with your eyes, of course, since I'm ignoring the lightspeed delay in seeing it.

The frame in which the entire track becomes longer is the non-inertial rotating frame. It is not one you will actually experience as a passenger because, as I said, its time coordinates are crazy. As you sit in the moving car you can measure the positions of different sleepers at various times, compute what the coordinates of these measurements should be in the rotating frame, and plot them on graph paper. You will then find that at a given rotating-frame instant (which, I emphasize, is a mathematical abstraction that does not match what you yourself experience) two neighboring sleepers are farther apart in the rotating frame than their ordinary rest distance. But this is purely an effect of the crazy time coordinate.

The best frame for describing the passenger's experience may be the "fudged" frame I alluded to previously. Unfortunately, its coordinate system cannot be extended all the way around the circle (nor very far on either side of the track), so it is not meaningful to ask about the shape or length of the entire track in that frame. The distance between sleepers will shrink in it, and neighboring cars will move away from you.

[makc]

The frame in which the entire track becomes longer is the non-inertial rotating frame. It is not one you will actually experience as a passenger because, as I said, its time coordinates are crazy. As you sit in the moving car you can measure the positions of different sleepers at various times, compute what the coordinates of these measurements should be in the rotating frame, and plot them on graph paper. You will then find that at a given rotating-frame instant (which, I emphasize, is a mathematical abstraction that does not match what you yourself experience) two neighboring sleepers are farther apart in the rotating frame than their ordinary rest distance. But this is purely an effect of the crazy time coordinate.

My brain is raped. You're saying that two sleepers are longer apart on paper, but if I put my measuring rod agains them, they are shorter apart. Well, if this is the case, your paper does not describe my experience and is, as you said, "mathematical abstraction" - so why consider it? Btw, what's that bit about "time coordinates are crazy"? Crazy or not, that's what we have, and we have to deal with what we have. Not abstractions.

edit: what I am saying here is that I will not rest until I see this situation from passenger point of view I already drew two or three drawings but I'm not happy about them yet. in all the cases, not only distance between sleepers nearby gets shorter, but whole damn track. I.e., consider this one:

0.GIF (2.64 KiB) Viewed 5177 times

red one is moving on track, green one is moving in tangent inertial frame; no matter what point on the outer track part you take, you see light coming back to red one earlier, so if he is not aware that he moves along the circle (think large circle around the galaxy) and thinks centrifugal force is just some gravity field, he'd conclude that all of the track is even shorter than it's to his green fellow.

this whole thing reminds me of another paradox where long moving train is being contained in short tonnel. that one is resolved by noting different order of events for observers, so I'd expect similar simple resolution to this one... yet I don't quite see it.