Is the train in the circular track paradox similar to the barn/pole paradox?

[JVNY]

I'm not sure what you're trying to say or ask here.

I will restate. Start with a train of proper length 100 at rest in a circular track of ground circumference 100 in an inertial ground frame. The train occupies the entire circumference of the track. The front and rear of the train are touching but are not connected. Here the track is a solid line and the train a dashed line, with the front and rear marked by solid circles.

Next, the train accelerates to 0.6c going around the track using the same acceleration pattern that it would use to accelerate Born rigidly if it were on a straight line of track (that is, greater proper acceleration at the rear, and lesser proper acceleration at the front). The train contracts in the ground frame to length 80 in the same way that it would if it were accelerating Born rigidly on a straight line of track (from the rear forward, because the rear has a greater speed in the ground frame sooner than the front). Now the train simply goes around and around the track at 0.6c. It looks like this in the ground frame:

If an observer at the rear of the train radar measures the train's length (say the inner side of the train is mirrored, so the radar signal skims along the train from rear to front, where it reflects and returns skimming along the train to the rear), the radar length remains 100. So the radar measured own length of the train is 100, whereas the ground length of the train is length contracted by gamma = 1.25 to be only 80.

Do you agree?

 

[pervect]

I will restate. Start with a train of proper length 100 at rest in a circular track of ground circumference 100 in an inertial ground frame. The train occupies the entire circumference of the track. The front and rear of the train are touching but are not connected. Here the track is a solid line and the train a dashed line, with the front and rear marked by solid circles.

View attachment 93270

Next, the train accelerates to 0.6c going around the track using the same acceleration pattern that it would use to accelerate Born rigidly if it were on a straight line of track (that is, greater proper acceleration at the rear, and lesser proper acceleration at the front). The train contracts in the ground frame to length 80 in the same way that it would if it were accelerating Born rigidly on a straight line of track (from the rear forward, because the rear has a greater speed in the ground frame sooner than the front). Now the train simply goes around and around the track at 0.6c. It looks like this in the ground frame:

View attachment 93271

If an observer at the rear of the train radar measures the train's length (say the inner side of the train is mirrored, so the radar signal skims along the train from rear to front, where it reflects and returns skimming along the train to the rear), the radar length remains 100. So the radar measured own length of the train is 100, whereas the ground length of the train is length contracted by gamma = 1.25 to be only 80.

Do you agree?

I was going to say no, but then I thought about it some more, and now I believe this works in the limit for a very thin train. This may raise a few objections and eyebrows, so let me give the details even though they are rather technical.

This is based on considering the following transformation. Let $T, R, \Theta, Z$ be polar coordinates with the usual metric $-dT^2 + dR^2 + R^2d\Theta^2 + dZ^2$ and we consider the transform to new coordinates $t,r,\theta,z$ via:

$$T = \theta \sinh gt \quad R=r \quad \Theta = \theta \cosh gt \quad Z=z$$

which is basically just the standard Rindler transformation with with the linear coordinate replaced with an angular one.

The expansion and shear are zero at r=R=1, but not elsewhere. So the motion won't be Born rigid for a "thick" train that has a finite thickness dR, but for a thin train, where we can idealize the "cars" as points, the cars/points will stay a constant distance apart from each other as measured along the track.

The equation of motion for a point on the train in lab coordinates is that $\Theta = \sqrt{T^2 + \theta^2}$, where $\Theta$ is the angular position of a point on the train at lab-frame time T, and $\theta$ is the starting angle of that point. $\theta$ cannot be zero, but we can imagine it varying from $\pi$ to $3 \pi$ to make an initially closed circle which shrinks as the train accelerates.

 

[DrGreg]

The expansion and shear are zero at r=R=1, but not elsewhere. So the motion won't be Born rigid for a "thick" train that has a finite thickness dR, but for a thin train, where we can idealize the "cars" as points, the cars/points will stay a constant distance apart from each other as measured along the track.

I haven't confirmed the calculation for expansion and shear, but this is what I expected. A train of non-negligible thickness in the radial direction would "buckle" as it accelerated, due to the changing geometry, which is another way of saying the train would need to have some flexibility in the radial direction even if it was "rigid" along its length and height.

...the cars/points will stay a constant distance apart from each other as measured along the track.

The arc-length along the track will remain constant but the straight-line length (chord), as measured in the rotating metric, will not.

By my calculation, if I haven't made a silly mistake, the metric in these coordinates is$$
ds^2 = g^2 \theta^2 ( \cosh^2 gt - r^2 \sinh^2 gt) dt^2 - dr^2 - (r^2 \cosh^2 gt - \sinh^2 gt) d\theta^2 + 2g\theta (1 - r^2) \sinh gt \cosh gt \, dt \, d\theta -dz^2
$$which simplifies to$$
ds^2 = g^2 \theta^2 \, dt^2 - dr^2 - r^2 \, d\theta^2 -dz^2
$$when $r=1$, but elsewhere the presence of the $dt \, d\theta$ term indicates non-orthogonal coordinates, as they must be. Note also the metric components are time-dependent, except at $r=1$.

 

[JVNY]

Thanks all. Let's call what we just discussed the first problem

The second problem is what happens when the train leaves the straight track and enters the circular track. I'm not sure the motion for the second case has been discussed in great detail . . .

So let's see whether we can agree on what happens in the second problem, when the train leaves the straight track. The scenario, using numbers that keep the circular track circumference consistent, is as follows.

Start with a very thin train of proper length 125 at rest on a straight track in an inertial ground frame (step 1 below). Next, the train accelerates Born rigidly on the straight track to 0.6c -- its ground length contracts to 100 (step 2 below). Next, a shunt directs it into the circular section of track (step 3 below).

Finally, the entire train is in the circular section of track and stays there, traveling around and around. The train has ground length 100, and it is on a circular track of ground circumference 100, so the train occupies the entire track. The train's front and rear are touching.

The two questions are:

(a) does the train's length deform in the process of entering the circular section of the track?

(b) is the train's radar measured proper length while it is in the track still 125? Assume again that the inner side of the train is mirrored, so an observer at the rear can send a radar signal toward the front; the signal skims around the surface of the train to the front, where it reflects and then skims back around to the rear.

 

[pervect]

Thanks all. Let's call what we just discussed the first problem
So let's see whether we can agree on what happens in the second problem, when the train leaves the straight track. The scenario, using numbers that keep the circular track circumference consistent, is as follows.

Start with a very thin train of proper length 125 at rest on a straight track in an inertial ground frame (step 1 below). Next, the train accelerates Born rigidly on the straight track to 0.6c -- its ground length contracts to 100 (step 2 below). Next, a shunt directs it into the circular section of track (step 3 below).

View attachment 93304

Finally, the entire train is in the circular section of track and stays there, traveling around and around. The train has ground length 100, and it is on a circular track of ground circumference 100, so the train occupies the entire track. The train's front and rear are touching.

View attachment 93305

The two questions are:

(a) does the train's length deform in the process of entering the circular section of the track?

The question is a bit ambiguous, but I believe that both the proper length of the train and the length of train in the lab frame remain constant as I answered the last go-around. From context, since you ask about the proper length of the train, I assume that when you say "length" you actually mean "length as measured in the lab frame". You have to specify the frame in which you measure non-proper length in order to be non-ambiguous, because the non-proper concept of length is frame dependent.

(b) is the train's radar measured proper length while it is in the track still 125? Assume again that the inner side of the train is mirrored, so an observer at the rear can send a radar signal toward the front; the signal skims around the surface of the train to the front, where it reflects and then skims back around to the rear.

Yes.

Additionally, as we also mentioned last time, in the lab frame the clocks synchronized in the train frame are not synchronized in the lab frame, and the offset in the lab frame is the same before, during, and after the train goes around the loop.

 

[JVNY]

Thanks.

(a) does the train's length deform in the process of entering the circular section of the track?

The question is a bit ambiguous, but I believe that both the proper length of the train and the length of train in the lab frame remain constant . . .

By "deform" I mean for example if the train is brittle then it cracks. Deforming (cracking, crushing, etc.) is frame independent. But if I have to choose between proper length and ground length, then in question (a) I am referring to the train's proper length.

So observers on the trains in both problems measure the length of the circumference that they are traveling around to be 125, whereas a ground observer measures it to be 100. As DrGreg writes:

I think the rotating observer and the inertial observer disagree on what the curvature of the same circular track is: they . . . disagree on its circumference

Circular motion is like the barn and pole for ground observers -- they observe the moving train to be length contracted, so the train of proper length 125 fits entirely in a circular track of ground length 100. But for an observer on the train, it is the opposite of the barn and pole. The train observers do not observe the ground length 100 circular track to be contracted -- instead, they observe it to be expanded to 125, because the train of proper length 125 fits entirely into the circular track.

So I think that the only other remaining issue is the curvature of the train in the first problem (train of proper length 100 accelerates to 0.6c within the circular track of ground length 100). The train started with proper length 100 at rest in and curved around in a circular track of circumference 100. It ends with the same proper length 100 but traveling around a circle of circumference that train observers measure to be 125.

it changes its shape in the radial direction . . . the curvature has decreased.

Do you agree that the train observers measure the curvature of the train to be decreased after the acceleration?

 

[pervect]

Thanks.

By "deform" I mean for example if the train is brittle then it cracks. Deforming (cracking, crushing, etc.) is frame independent. But if I have to choose between proper length and ground length, then in question (a) I am referring to the train's proper length.

The best way of understanding the deformation issue is to go back to the rotating disk, not this more compicated example.

As the train goes around the circular track, three basic things happens. It moves along the track. It's centripetally accelerated by the track. And it rotates - the car starts out horiziontal, tilts to the vertical, back to horizontal, but upside down, etc.

Only the last (rotation) is incompatible with Born rigidity. Note that in reality, we don't make the cars too long and couple them via couplings, so the train as a whole isn't rigid - it can conform to the track as the track goes up hills, etc. The suspension allows the wheels to tilt a bit (how much depends on the design, I gather, I haven't found anything really detailed, from what I did read some carriage designs had more wheel-swivel than others.

In any event, all we really have to ask what happens to a car-sized bit of the train when it rotates. It'd be hard to get relativistic effects unless the car was very long, or the rotation rate was very high. Maybe in your example the rotation rate is high. However, if you imagine a train with a centripetal acceleration of 1g, and it's moving at relativistic speeds, I'm thinking that the radius of curvature of the track will be in the neighborhood of a light year (because 1g = 1 light year/year^2, and I believe the centripetal acceleration is gamma v^2/r). And if we imagine a radius of curvature of the track of a light year, the train won't be rotating very fast at all for any reasonable gamma.

So I think that the only other remaining issue is the curvature of the train in the first problem (train of proper length 100 accelerates to 0.6c within the circular track of ground length 100). The train started with proper length 100 at rest in and curved around in a circular track of circumference 100. It ends with the same proper length 100 but traveling around a circle of circumference that train observers measure to be 125.
Do you agree that the train observers measure the curvature of the train to be decreased after the acceleration?

I haven't actually tried to calculate anything along this line.

 

[PeterDonis]

it rotates - the car starts out horiziontal, tilts to the vertical, back to horizontal, but upside down, etc.

Are you referring to the case of a train going at a constant angular velocity around the circle, or of the train accelerating while in a circle?

In the former case (constant angular velocity), the train car would not rotate as you describe. In the latter case, I'm not sure whether it would or not.

 

[A.T.]

Do you agree that the train observers measure the curvature of the train to be decreased after the acceleration?

What about the curvature of the tracks? Does it have to be locally the same as the curvature of the train?

I haven't actually tried to calculate anything along this line.

How would one do this, conceptually? Can you define a geodesic based on the spatial part of the rotating metric, which is tangential to the track circle? Would that define a local straight line of zero curvature, relative to which the local train curvature can be measured.

 

[JVNY]

I haven't actually tried to calculate anything along this line.

How would one do this, conceptually?

And if the curvature does reduce, is it possible to determine whether it would do so symmetrically (with both the front and the rear moving equally radially outward) or asymmetrically (e.g., the front does not move radially, but every part behind it moves radially outward, with the rear moving radially outward the most)? If the train's curvature does bend, then it should be possible to determine where it bends, because there will be physical effects at particular locations on the train (tearing apart if the train actually has some width), and these physical effects will not be frame dependent.

 

[pervect]

Oh, as far as deformation goes, I should mention that that's what the shear tensor I mentioned compute - how things deform. But if you're not familiar with it, I'll probably have to draw some diagrams to (hopefully) explain it. It might fit best in a different thread.

 

[pervect]

Are you referring to the case of a train going at a constant angular velocity around the circle, or of the train accelerating while in a circle?

In the former case (constant angular velocity), the train car would not rotate as you describe. In the latter case, I'm not sure whether it would or not.

It's not an exotic relativity effect like Thomas precession. It's just what normally happens when a train goes around a loop. See the diagram below.

[add]
If you imagine a gyroscope mounted on the train, the train body rotates relative to the gyroscope, that's why it's rotating. If you imagine a round train sliding around the loop without rotating, then there would be no rotation.

The physics point I'm trying to make is just this. It's possible to have a Born-rigid non-rotating object, and it's possible to have Born-rigid rotating object, but you can't Born-rigidly make a non-rotating object into a rotating one, or vice-versa.

 

[PeterDonis]

It's just what normally happens when a train goes around a loop.

In that case, the rotation is entirely in the plane of the loop. It looked to me like what you were describing was rotation in a plane perpendicular to the plane of the loop.

It's possible to have a Born-rigid non-rotating object, and it's possible to have Born-rigid rotating object, but you can't Born-rigidly make a non-rotating object into a rotating one, or vice-versa.

Agreed.

 

[pervect]

As far as the curvature of the track goes - if you consider the proper frame of the train, i.e a comoving inertial frame, the track is elliptical due to Lorentz contraction, as in the sketch below. So at the location where the train is, the track is more curved - since the track isn't circular, the curvature isn't constant.

 

[JVNY]

Oh, as far as deformation goes, I should mention that that's what the shear tensor I mentioned compute - how things deform. But if you're not familiar with it, I'll probably have to draw some diagrams to (hopefully) explain it. It might fit best in a different thread.

I am very unlikely to understand the math, but I suspect the other posters would. Diagrams would be great.

 

[A.T.]

As far as the curvature of the track goes - if you consider the proper frame of the train, i.e a comoving inertial frame, the track is elliptical due to Lorentz contraction, as in the sketch below. So at the location where the train is, the track is more curved - since the track isn't circular, the curvature isn't constant.

Is this the instantaneous inertial frame of a train car? Doesn't the proper centripetal acceleration of the car affect the geometry in the actual non-inertial rest frame of the car? Is there even clear way to determine that, or does it again depend on the simultaneity conventions in non-inertial frames?

 

[pervect]

Is this the instantaneous inertial frame of a train car? Doesn't the proper centripetal acceleration of the car affect the geometry in the actual non-inertial rest frame of the car? Is there even clear way to determine that, or does it again depend on the simultaneity conventions in non-inertial frames?

Yes, this is the instantaneous inertial frame of the train. And I assume the simultaneity convention of that co-moving observer. The issue I was concerned with was how to transport the basis vectors to create what is called in my textbook (MTW) "a proper frame" for the observer, when I realized that it was irrelevant to what the shape of the track was "now" at a point co-located with the train. Note that I didn't think in detail about other possible simultaneity conventions, though my current thinking is that the Einstein convention is expected.

As far as the shape of the track goes, basically, since we know what the basis vectors are for the train observer, we know the spatial geometry from the point of view of that co-moving observer - and it's just the Lorentz transform.