# Circular barn/pole?

1. Dec 7, 2015

### JVNY

Consider a train of proper length 100 at rest in an inertial ground frame. It is accelerated Born rigidly to the right on a straight track to 0.6c in the ground frame, after which is still has proper length 100 but has now ground length 80. It is then shunted onto a circular track of ground circumference 80. The front grazes the rear as the front exits the circle and returns to moving inertially to the right in the lab frame.

Is this just a circular version of the barn/pole paradox? The entire train is in the 80 circumference circle simultaneously in the ground frame. It seems consistent with Einstein's view of the circumference in the Ehrenfest paradox. On the other hand, if I analyze it from the train frame, an issue arises.

Say I treat the front as if it were momentarily at rest in a series of co-moving inertial frames (as if the circle were an polygon, with inertial frames parallel to the sides of the polygon). When the front changes direction to be at rest in the co-moving frame parallel to the first side of the polygon, the front undergoes infinite acceleration at a slight angle relative to the rest of the train. This should cause the train to deform (the front should stretch out away from the train) as in Bell's spaceship paradox. That is, the train does not just deform in the simple sense of bending. It also stretches out, as in Bell's paradox.

Would this occur? Or, by making the sides of the polygon smaller and more numerous could I reduce the stretching continually until in an actual circle there would be no stretching, so that the train would not be deformed? Then the scenario should be a circular version of the barn/pole paradox.

2. Dec 7, 2015

### A.T.

Yes. And unlike in the linear case, you cannot resolve it via relativity of simultaneity, because all frames agree that the whole train is in the loop at some time point (when front and end meet). The resolution is that space is non-Euclidean in rotating frames, so there is more circumference than 2*pi*r to fit the proper length of the train:
http://www.projects.science.uu.nl/igg/dieks/rotation.pdf [Broken]

Last edited by a moderator: May 7, 2017
3. Dec 7, 2015

### Staff: Mentor

Any time you find yourself thinking of "the train frame" when the train is moving non-inertially, beware. You are extremely likely to be making invalid assumptions.

The Bell spaceship paradox involves continuous, finite acceleration, not discrete, infinite bursts of acceleration. I don't see how there is any analogy here.

The question you are trying to answer is whether the train would, at some instant in the ground frame, fit exactly in the circular shunt with circumference 80 in the ground frame. In your previous thread, we assumed that the answer to that question was yes, and I don't see any reason why it wouldn't be. The proper acceleration of the train is perpendicular to its velocity at every point, so there will not be any "stretching" of the train; all the acceleration will do is change the train's direction.

No, it isn't, because the barn/pole paradox relies on inertial frames, and the train, while any portion of it is in the shunt, is not at rest in any inertial frame. (A particular point on the train can always be viewed as being momentarily at rest in some inertial frame, but there is no inertial frame in which the entire train is at rest while any portion of it is in the shunt.) So you can't just say "well, the train should be of length 100 in its rest frame, and the shunt should be length contracted in that frame, so how can the train fit into the shunt?" The train has no "rest frame".

4. Dec 7, 2015

### A.T.

It has no inertial rest frame. But there is a rotating frame, where the train is at rest and has a length of 100, while forming a circle of radius 80 / 2pi.

5. Dec 7, 2015

### pervect

Staff Emeritus
You've already run into trouble with your assumptions. While it's possible to accelerate the train, it's only possible to do so in a Born rigid manner on a straight track, not on a circular one. The wiki article on Born rigidity mentions this, https://en.wikipedia.org/w/index.php?title=Born_rigidity&oldid=615087209, oddly enough the current Wiki article on the Ehrnfest paradox does not, though as the Wiki article on Born Rigidity notes, Ehrnfest was the one who first noted the impossibility of making a disc rotate rigidly - which is why the "paradox" is known as the Ehrnfest paradox.

Trying to argue that the disk is a "train" doesn't solve the paradox, the motion is still rotational, and rotational motion cannot be Born rigid.

See also the sci.physics.faq on the issue, http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html. (I don't particularly care for the editors notes that were tacked onto the end of the original, though. The main article (sans notes) is a fair summary of the usual resolution, though it may be worth noting that there is a surplus of explanations of various degrees of popularity in the literature. Gron has several articles on the topic, the one cited in the sci.physics.faq, O. Gron, AJP Vol. 43 No. 10 pg 869 (1975) is probably the best-written and most-cited one, if you can track it down.

6. Dec 7, 2015

### A.T.

If I understand the OP correctly the acceleration along the track is on the straight part. On the circle there is only centripetal acceleration, perpendicular to the track. Why would that radial acceleration change the proper length of the train cars?

7. Dec 7, 2015

### Staff: Mentor

I assume you mean Born coordinates?

https://en.wikipedia.org/wiki/Born_coordinates

If you mean Born coordinates, then this is not correct. The train is "at rest" in Born coordinates, but its circumference is 80, not 100--i.e., the same as its circumference in cylindrical coordinates in the ground frame. This is easily verified from the line elements given in the Wikipedia page linked to above.

Rotational motion with a time-varying angular velocity cannot be Born rigid. But this scenario technically doesn't have time-varying angular velocity, in the sense of angular acceleration. It just has two points--entry into the shunt and exit from the shunt--where there is a discontinuity in angular velocity between zero and a fixed nonzero value. But this discontinuity can be made arbitrarily small, and I don't think it has a significant effect on the overall scenario. The constant angular velocity portion of the motion--i.e., the portion within the shunt, between the two discontinuities--can be Born rigid.

8. Dec 7, 2015

### A.T.

I mean a rotating frame, with the circular track center as origin, and the same angular velocity as the train on the circular track. The entire train is at rest in that frame, so its length in that frame is its proper length.

9. Dec 7, 2015

### Staff: Mentor

Please give the explicit math for the frame you're talking about. Did you look at the Born coordinates page that I linked to? It describes a frame that appears to meet your description; but the length of the train in that frame (meaning, the circumference of the circle it occupies) is what I said. Non-inertial frames don't work the same as inertial frames.

10. Dec 7, 2015

### A.T.

http://www.projects.science.uu.nl/igg/dieks/rotation.pdf [Broken] (Chapter 2)

Last edited by a moderator: May 7, 2017
11. Dec 7, 2015

### Staff: Mentor

These are Born coordinates; the length of the train in these coordinates will be 80, not 100. More precisely:

The train at a single instant of time (the instant at which the front of the train is just at the shunt exit and the rear of the train is just at the shunt entrance) in Born coordinates (the line element in the Wikipedia page, which is also given on page 4 of the paper you linked to) is described by a line element with constant $r$ and $dt = dr = dz = 0$; the line element is then integrated over the range $0 \le \varphi \lt 2 \pi$ to obtain the length of the closed spacelike curve $s$ that describes the train.

The only nonzero term in the line element is then $ds^2 = r^2 d\varphi^2$, and $r$ here is the radius of the circular shunt in the ground frame (because the $r$ coordinate is unchanged by the transformation from the ground frame to Born coordinates--we have placed the spatial origin of the ground frame at the center of the shunt and used cylindrical coordinates in the ground frame). So the length of the train is just $2 \pi r$, which is 80.

What may be confusing you is the spatial metric given in equation (5) of the paper you linked to, which has a "length contraction" factor in the $r^2 d\varphi^2$ term. However, this metric, as the paper carefully notes, describes "the spatial distance between two infinitesimally near points". It can be interpreted as "the spatial geometry seen by a rotating observer", but you have to be very careful about what that means. It does not mean that you can integrate this metric around a complete circle $0 \le \varphi \lt 2 \pi$ to get "the length of the train in the train frame". (The discussion of distances in the Wikipedia page I linked to earlier also goes into this.)

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12. Dec 7, 2015

### pervect

Staff Emeritus
Ah, I should have read the post more carefully, sorry. The train can accelerate rigidly on the straight track. The train (as a whole) obviously has to flex and bend when it makes the transition to the curved track, so that part of the motion isn't rigid (at least not in the Born sense, and probably not in any other sense). But as you point out the proper length of the train doesn't change in spite of the necessity for it's motion to be non-rigid.

It's rather interesting how a train manages to go around a curve and keep it's wheels on the track when you think about it. But it's quite off-topic :-), I did get distracted by the question though.

13. Dec 7, 2015

### JVNY

The consensus seems to be that the train fits into the 80 ground circumference circle without being deformed -- other than the non-relativistic bending caused by following the curved track.

So the scenario is like the barn/pole paradox in the inertial ground frame. But it is not like the barn/pole paradox in two other ways: (a) the resolution does not depend on the relativity of simultaneity, because there is no shared simultaneity for the parts of the train that are within the circle, and (b) there is no equivalent for the train of the fact that in the pole's reference frame the barn is length contracted.

Here is another hopefully clearer explanation of my concern about whether there might be some relativistic deformation as the train enters the circle (some deformation other than the non-relativistic bending). When the front of the train enters the circle, the rest of the train behind it is still moving to the right inertially. So by the "train frame" I meant the inertial frame in which the remainder of the train is at rest. Alternatively, one can say the inertial reference frame moving to the right at 0.6c that is co-moving with the rest of the train.

So in that inertial frame in which all parts of the train other than the front are at rest, the front of the train undergoes acceleration and moves away at the angle of the first side of the polygon.

I just mean that if the front accelerates away from the rest of the still-inertial train, then perhaps it stretches out or deforms the train. If this occurs when both the front and rear accelerate at the same rate in the ground frame (as occurs in Bell's paradox), then presumably it should also happen when only the front accelerates (in fact, it should stretch out even more, because the part of the train immediately behind the front is not accelerating at all). Put another way, if instead of having a train we have a row of point masses, what happens when the first point mass accelerates to move along the first side of the polygon? In the inertial frame in which all of the other points are at rest, does the second point in the row measure an increase in distance between it and the first point mass?

Or does that acceleration of the first point maintain the same distance between it and the second point but change their relative positions, thus causing the non-relativistic bending? That seems unlikely to be the case because the first point cannot change position instantaneously (or it would be moving faster than light).

14. Dec 7, 2015

### Staff: Mentor

Thinking of it as a polygon, as I've already said, is not necessary. What's more, it may be misleading, because if the front undergoes a discontinuous acceleration for an instant to change direction by a finite angle, its acceleration is not perpendicular to its velocity. But if it undergoes a discontinuous change in acceleration, from zero to the nonzero value that is just right to move it around the circle, its acceleration is always perpendicular to its velocity. That is what allows the front to accelerate while the rest of the train is still moving inertially, without causing any stretching of the train (at least to a first approximation--as I said, there will be some effect due to the discontinuity in acceleration, but I think it can be kept small enough to not change the overall behavior significantly).

No. In the Bell spaceship paradox, the acceleration is parallel to the velocity for both ships. In this case, the acceleration is perpendicular to the velocity. That makes a big difference.

15. Dec 8, 2015

### A.T.

It doesn't have to be single instant of time. The train can stay in the circle forever, so all the timing issues become irrelevant. You can even connect the front to the back of the train, so you have a rotating ring, which is forever at rest in the rotating frame. The length of each part of the train in the rotating frame is its proper length, so how can the total length of the train not be its proper length?

So in the rotating frame I have N train parts (all at rest forever), each is 100/N long (its proper length). But total length of the train is not N * 100/N = 100 ?

16. Dec 8, 2015

### Staff: Mentor

It does if you're trying to define the "length" of the train. That is the spacelike interval occupied by the train at a single instant of time.

Because you can't add all those infinitesimal proper lengths; they are not spacelike intervals all lying in a single surface of constant time. Each of those infinitesimal proper lengths is in a different spacelike hypersurface. The only global "length" you can define for the train as a whole is the one I computed in Born coordinates, which is 80.

Yes. Your implicit assumption that you can always add the N lengths to get a total length is not valid in the rotating frame. It is only valid in a frame in which all of the N lengths lie in the same spacelike hypersurface. An inertial frame covering the entire train meets this criterion; the rotating frame does not.

17. Dec 8, 2015

### pervect

Staff Emeritus
The length will still be 100, regardless of coordinates. But the method of calculating the length may not be obvious. I've seen two correct techniques. (There may be more). One of these techniques, used by AT's paper, and also by Ruggerio in his "Relative Space: Space Measurements on a Rotating Platform" http://arxiv.org/abs/gr-qc/0309020 involves the exchange of light signals. The approaches are very similar but the details vary. Ruggerio's method involves sending a light signal from one nearby observer "at rest" in the Born coordinate system to another nearby observer, and computing the proper time difference for a round trip. This is also called the "radar method" for obvious reasons. It should be noted that in generalized coordinates (such as in the rotating frame) the radar method only works for infinitesimal distances.

The end result is that the distance then is $(c/2)\, d\tau$, where $d\tau$ is the proper time for the round trip exchange of light signals between two nearby observers "at rest" (i.e. with constant Born coordinates).. This is probably the simplest technique, it does require one to know what proper time is and how to calculate it from coordinate time (something that I'm sure Peter knows, but I'm not so sure if the OP knows). I believe I've mentioned the Ruggerio paper to the OP in another thread. I'm willing to attempt to explain more to the OP about this approach if there are some specific questions, though I suspect that the references will do a better job than I will.

The less-simple method is to use projection operators. This is the approach that Wald uses, for instance, but Wald's exposition is so terse it's not very clear. It's also used by Eric Poisson in "A Relativists Toolkit", his exposition is clearer, but very abstract. There is also a PF thread on this, https://www.physicsforums.com/threads/origin-of-spatial-metric.251228/#post-1843733

The end result is that with a metric signature of (-+++), you get the induced spatial metric $h_{ab}$ by the equation $h_{ab} = g_{ab} + u_a u_b$, where $u_a$ and $u_b$ are the 4-velcoties of some observer with constant Born coordinates. $u^a$ will have components $(1/\sqrt{g_{00}},0,0,0)$, so $u_a$ will have components $g_{ab} \, u^b$. You let $a,b$ take on the range $0,3$ in peforming the sum, but throw away the $h_{0i}$ and $h_{i0}$ terms. The PF thread, I believe, uses a (+---) signature, just to make things a bit more confusing.

In any event, all the authors mentioned to date winds up with the same answer for the induced spatial line element, given by (5) in AT's reference and (14) in Ruggerio's reference, namely

$$ds^2 = dr^2 + \frac{r^2 d\varphi^2}{1-\omega^2 r^2/c^2}$$

18. Dec 8, 2015

### Staff: Mentor

Please define exactly what "length" you are computing. Ideally, I would like to see what spacelike curve it corresponds to in Born coordinates. The spacelike curve I gave ($t, r, z$ constant, $0 \le \varphi \lt 2 \pi$) has a length of 80, as I've already shown.

Please note the key word "nearby". I am not disputing that nearby measurements of length, using the methods you describe, give a "proper length" between nearby observers on the train which, if added up around the entire circumference of the train, would sum to 100. I am only disputing the adding up process, not the individual pieces. The adding up process does not make physical sense, because you are adding up a large number of spacelike line elements that are not collinear; they each lie in a different spacelike hypersurface. So calling the sum a "length" is, IMO, an abuse of terminology; it is not a physical "length" in any meaningful sense.

But this line element does not describe global "lengths" on the train. It only describes an "apparent" geometry formed by piecing together locally measured lengths in a way that does not correspond to any actual global length. (In more technical language, this line element describes the geometry of the quotient space of the congruence of worldlines describing the "rotating ring" formed by the train--i.e., by assuming that it is a ring rotating at the same angular velocity indefinitely. But that quotient space does not correspond to any spacelike hypersurface in the actual spacetime, so, as I said above, calling the circumference of the circle in the quotient space a "length" is, IMO, an abuse of terminology.)

19. Dec 8, 2015

### DrGreg

I disagree with your interpretation. The length of 100 in the quotient space is exactly the length you would get if a person on the train used a tape measure (assuming the train continues to circle indefinitely). I wouldn't say that's an "apparent" length but the true "rest length". No, it doesn't correspond to the length of a spacetime curve in a surface of Born-coordinate simultaneity, but why should it? Those surfaces aren't orthogonal to the congruence of worldlines of points in the train. Simultaneity of infinitesimal intervals only becomes an issue if you're trying to measure a distance that is changing over time. The train is at rest, and the length of each segment is constant over time, so whether you can choose a set of synchronised segments or not is irrelevant.

On the other hand, the length of 80 obtained from a surface of constant Born-time, is a coordinate-dependent quantity. The spacelike curve being measured isn't even locally compatible with (infinitesimal) Einstein synchronisation, so even a tiny section of this curve has a spacetime-length that doesn't at all match the local space-length than a local observer (at rest in the train) would measure.

20. Dec 8, 2015

### Staff: Mentor

I'm not sure I agree. Again, this is the "adding up" issue. Also, there is the issue that, as soon as we move beyond infinitesimal distances in the rotating frame, there is no unique notion of "distance"; different methods of measurement give different answers.

There is no surface that is orthogonal to the congruence of worldlines of points in the train. Each individual worldline is orthogonal to a different surface.

No, it isn't. It is picked out by the symmetry of the scenario. In any other spacelike hypersurface, the curve occupied by the train will not be a circle; it will be an ellipse.

21. Dec 8, 2015

### JVNY

I developed the scenario while grappling with that faq and the note at the end. The basic idea is to avoid the thorny question of what happens when you take a circular object from rest into rotation (what shear forces might exist, etc.), while not giving up on identifying how the object came to be in rotation (which seems to me still to have some relevance, particularly when you can use straight line motion to start the analysis, because I find that easier to understand). Thus the set up using Born rigid acceleration first, followed by entry into the circular section of track.

Understood, and I have now found a paper making the same point, saying that the rotating object is not like the car/garage paradox for the same reason you mention. It is on page 5 of http://areeweb.polito.it/ricerca/relgrav/solciclos/nikolic_d.pdf.

I agree that one can use the radar method over infinitesimal distances, and others use that approach, such as Bini and Jantzen here on page 8, specifically saying that the result for the rotating observer is that the length of the total circumference is Lorentz expanded. http://areeweb.polito.it/ricerca/relgrav/solciclos/bini_jantzen_d.pdf

I also agree that the radar method only works over infinitesimal distances on a Born rigidly accelerating rod. However, I believe that it works over larger distances on a rotating rim. Say you mirror the inner surface of a rim and fix a clock unit on a point on the rim. The clock unit is also a light emitter, reflector, and absorber. The unit sends a light signal around the inner surface in one direction; when the signal strikes the unit after a complete circuit it reflects and returns in the opposite direction along the surface. When it returns again to the unit, the unit absorbs the signal and records the elapsed time for the around and back trip. Divide the elapsed time by 2, and you get the same Lorentz expanded result as from the prior citation. See pages 6-7 here: http://areeweb.polito.it/ricerca/relgrav/solciclos/tartaglia_d.pdf. However, note that the author (Tartaglia) thinks that the Lorentz expanded result is a function of forces that affect the mirror, not a reflection (no pun intended) of the length of the circumference.

Another approach, though, is to consider how a single observer on the train measures the length of the circle. I revert to a polygon in part because of the argument that if you use an n-gon, the observer will find each side to be length contracted by gamma (just like the astronaut who travels to Andromeda, or a muon that travels downward past a mountain), and therefore the entire circumference to be its ground length contracted by gamma. See http://abacus.bates.edu/~msemon/WortelMalinSemon.pdf [Broken], page 1125.

This results in a circumference for a train observer in our scenario of 64 rather than the Born coordinates' of 80, and rather than Bini and Jantzen's (and Einstein's) view that for the rotating observer the circumference is a Lorentz expanded 100. This may well be misleading (see more below).

In any event, I appreciate the discussion very much. It is helping me to understand that coordinates are not necessarily critical to understanding what happens to the train, and also to understand the concept that coordinates do not have to have an immediate metrical meaning.

Say the train is very brittle, just able to handle the bending that results from going around the circle, but no more deformation without shattering. If it is accelerated Born rigidly along the straight section of track by many rockets arrayed along its length then it does not shatter. If instead the rockets accelerate all of its parts at the same rate in the ground frame along the straight track, it will shatter (the rockets will pull it apart all along its length). Whether the train shatters or not is invariant in all frames.

Now assume that the brittle train is accelerated Born rigidly in the straight section of track, then enters the circle (say it even stays in the circle as A.T. postulates). I think that everyone would agree that the train does not shatter. It does not shatter during the Born rigid acceleration because the rockets accelerate all parts in a way that maintains their combined proper length. It does not shatter going into or remaining in the circle because . . . here I am less sure of how to describe the why. One way of putting it is that using the radar method, its length is still 100.

So coordinates are not necessary to understand what happens to the train: it does not matter that we cannot come up with coordinates that give the train a length of 100 while rotating, or that the parts of the train are not at rest in an inertial frame. Also, the coordinates we can identify "may be misleading": if I focus on the Born coordinate length of 80, or the conclusion that a single observer would measure the circumference to be 64, I might be led to conclude that the brittle train has compressed to 80 or 64 within the circular section of track and therefore shattered. But again, I don't think that anyone would conclude that the brittle train shatters.

Last edited by a moderator: May 7, 2017
22. Dec 8, 2015

### JVNY

Some people argue that if you cannot globally synchronize clocks then there cannot be a well defined length:

Only if this global synchronization were possible there would exist a well defined spatial length between different points on the boundary of the rotating disk. http://areeweb.polito.it/ricerca/relgrav/solciclos/pascual_sanchez_d.pdf, page 2.​

Others that the measurement of the circumference is

completely independent from the problem of synchronization. http://areeweb.polito.it/ricerca/relgrav/solciclos/tartaglia_d.pdf, page 6.​

And,

many relativists have come to consider Einstein synchronization merely a convention, or gauge, that affects no measurable quantities . . . http://areeweb.polito.it/ricerca/relgrav/solciclos/klauber_d.pdf, page 3.​

So whether synchronizability is relevant to being able to measure the circumference seems to be one of the intractable questions -- at least, it seems to me, if you insist on using some kind of coordinate system. If instead you just use a single clock unit as described above and the radar method, you get a well defined 100 length for the rotating train (without relying on synchronization, coordinates, etc.).

Last edited: Dec 8, 2015
23. Dec 9, 2015

### A.T.

Exactly.

I would rather say: Without globally synchronized clocks, you cannot use the most gereneral defintion of length, which is also applicable to moving objects. But if your object is static then you don't have to care about synchronized clocks. Also as you note, a length based on synchronized clocks depends on the synchronization convention chosen, so it's always ambiguous.

24. Dec 9, 2015

### pervect

Staff Emeritus
Ditto. I agree with the three conclusions at the end of Tartaglia's article, but I haven't located the section that JVNY is talking about.

The curve I would describe as the length most natrually lies in the quotient space, also called "the Relative Space", a paper using this approach is http://arxiv.org/abs/gr-qc/0309020 "The Relative Space: Space Measurements on a Rotating Platform".

However, one can also think of it as a curve in space-time, it is, however, not a closed curve. On the diagram below (borrowed from wiki):

The curve is a blue helix on the diagram. It starts where the blue helix intersects the red worldline of an observer with constant coordinates in the Born chart (referred to by several people in this thread as "at rest in the rotating frame", and ends where it intersects the same worldline.

The "Relative Space" approach defines an abstract quotient space (which however has an operational meaning as Ruggiero points out) by identifying all points on the worldline of an observer "at rest" in the rotating frame (i.e. an observer with constant Born coordinates) as being a single point in the abstract quotient space. The distance between points in the abstract quotient space is well defined and is equal to the length of the curve that is everywhere orthogonal to the worldlines of the "at rest" observers.

While I would agree that one is adding up a large number of spacelike segments (performing an integration, in other words), I would dispute the claim that it does not make "physical sense". We know how to measure distances on the rotating platform operationally, all we need to do is add them together. In short, given that we know how to find the distance between nearby points, we know how to find the distance of points that are further away by performing an integral.

Physically, if we attempt to put short measuring rods (which we've agreed we know how to determine the length of) around the circle, we can only fit so many before they start to overlap. They overlap when the spatial parts of their Born coordinates overlap. The number of rulers we can place this way multipled by the length of each of the rulers gives the circumference.

Last edited by a moderator: May 7, 2017
25. Dec 9, 2015

### A.T.

I would even suggest, that it's the most physical way to determine the length. Because it corresponds to physically laying out rulers at rest, and doesn't depend on some convention of clock synchronization.