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

[A.T.]

Yes, but that's not just because the system is not changing with time; it's because the system is hypersurface orthogonal--the individual segment lengths, even if I measure them at different times, can all be time translated into a single spacelike hypersurface that is closed and orthogonal to the worldlines of all the segments.That can't be done for the rotating train.

So what if the fence is build on top of the rotating train? Can you not use the usual method to determine how much fence there is, if for example it needs to be replaced?

Once again, I think we agree on all the results of actual measurements. We just disagree on "interpretation"--on the ordinary language words that can be used to describe what's going on. "Interpretation" is not physics; the physics is the actual measurements.

I think we disagree which length measurements are physically relevant. How much fence would you order to replace the old one on top of the entire rotating train? 80 or 100?

 

[PeterDonis]

How much fence would you order to replace the old one on top of the entire rotating train? 80 or 100?

Hm, I see your point. It would have to be 100, since that's how much it would take if the train were at rest in an inertial frame, and I think we established earlier that the process of having the train go from straight line motion to circular motion with the same velocity would not stretch the train (at least in the idealized case where the discontinuity of acceleration has negligible effect).

 

[JVNY]

Here is one last follow up about the train. Say the train of 100 proper length starts at rest on a circular track of 100 circumference in the inertial ground frame. The front and rear of the train are touching, but not attached to each other. Is it possible to Born rigidly accelerate the train within the circular track? I think that it should be. The train can use the same acceleration program that it used on the straight section of track because proper acceleration is an invariant. So the train should accelerate Born rigidly (retain its 100 length) while length contracting in the ground frame (even though none of the parts of the train share a simultaneity surface because the train is traveling in a circle). In this case the train would not occupy the entire circular track because it would length contract to less than 100 length in the ground frame.

 

[PeterDonis]

Is it possible to Born rigidly accelerate the train within the circular track?

No. Born rigid acceleration is only possible in a straight line, not in a circle.

The train can use the same acceleration program that it used on the straight section of track because proper acceleration is an invariant.

"Invariant" just means the proper acceleration of a given point on the train has the same value in all frames. It does not mean that the proper acceleration of different points on the train will be the same, much less that the proper acceleration if it is in a straight line will be the same as if it is in a circle.

 

[JVNY]

It does not mean that the proper acceleration of different points on the train will be the same

Agreed. However, I am proposing that the proper acceleration at each point on the train is different, with the greatest proper acceleration at the rear, and lowest at the front. Assume that the train is driven by a large number of small rockets arrayed along its length, with the thrust of each governed by an accelerometer. The thrust programs are set when the train is in a straight stretch of track, and the train does several trials and confirms that the thrust programs generate the correct thrust for each part of the train to Born rigidly accelerate it in a straight line. Next the train moves into the circular section of track and comes to rest there. It then activates the same thrust programs -- so the rear gets the most thrust and proper acceleration, and the remaining parts of the train ahead of it lower proper acceleration (with the front getting the lowest proper acceleration).

In the inertial ground frame, the train length contracts exactly the same as it does in straight line acceleration (the rear has greater ground speed than the front). Are you stating that the train compresses along its length (so that it shatters if it is the brittle type of train that I describe above)? Or that the train stretches out along its length (so that it shatters)?

Say you are going to ride on the train and you take along some fencing to erect along its length after it accelerates and settles into constant speed of 0.6c. Per A.T.:

How much fence would you order

I think that you would order 100 of fence.

 

[DrGreg]

In the inertial ground frame, the train length contracts exactly the same as it does in straight line acceleration (the rear has greater ground speed than the front). Are you stating that the train compresses along its length (so that it shatters if it is the brittle type of train that I describe above)? Or that the train stretches out along its length (so that it shatters)?

Neither of those; it changes its shape in the radial direction, orthogonal to the length of the train; i.e the curvature of the train changes, a non-rigid movement. The train initially subtends an angle of $2\pi$ at the centre of the circle, but later subtends a smaller angle, yet its arc-length (in the rotating geometry) is still the same, so the curvature has decreased.

 

[PeterDonis]

I am proposing that the proper acceleration at each point on the train is different, with the greatest proper acceleration at the rear, and lowest at the front.

This is Born rigid acceleration if it is linear. But there is no way to do this in a Born rigid manner for circular motion.

 

[JVNY]

Neither of those; it changes its shape in the radial direction, . . . so the curvature has decreased.

But there is no way to do this in a Born rigid manner for circular motion.

Yes, agreed -- I now understand what you mean. Sorry about using that definition. I meant to refer only to the rigidity of the length. Would you both agree that the length of the train remains the same, as everyone agreed that it does when it goes into the circular section of track at a constant speed? In both cases the train bends, but I think that in both cases the train's length remains the same as measured by observers on the train. In both cases, one would order 100 of fence to install along the train.

 

[PeterDonis]

Would you both agree that the length of the train remains the same, as everyone agreed that it does when it goes into the circular section of track at a constant speed?

No. The case where it enters the circular track when already traveling at a constant speed, and maintains that constant speed around the circle, is different from the case where the train must be accelerated in a circular path. Any change in speed that also involves a change in direction, i.e., any acceleration involving a change in speed in anything other than a straight line, cannot be done in a Born rigid manner.

I think that in both cases the train's length remains the same as measured by observers on the train.

No. In the second case (change in speed around a circular path), the train's length as measured by observers on the train must change. More precisely, the length must change if we constrain the train to remain on the same circular shunt, as seen from the ground frame, as it accelerates--which is what I understand you to be proposing. The general result that underlies this is called the Herglotz-Noether theorem.

 

[JVNY]

That result certainly is the case if you try to set a disk or ring into rotation. But in the case I described, the front and rear are not connected (the train is not a ring), so the rear can travel faster in the ground frame than the front, and the train ends up length contracted to 80 in the ground frame, traveling at 0.6c in the ground frame. As such it is just like the train traveling around in a circle at 0.6c earlier in the thread, and as to its length:

The length will still be 100, regardless of coordinates. . . the method of calculating the length may not be obvious. . . One of these techniques . . . is . . . called the "radar method" for obvious reasons.

The radar method will work exactly the same for the train in this second scenario after it has accelerated to 0.6c in the ground frame and continues at that constant rate. A signal from the rear to the front will take ground time 80 / 0.4 = 200 to reach the front, then after reflecting will take ground time 80 / 1.6 = 50 to return to the rear, for a total round trip time of 250. Half of that is 125, and dilated by gamma of 1.25 means that the radar measured length is unchanged at 100.

How can the radar length be any different when the train has 80 ground length traveling at 0.6c in a 100 ground circumference track (the second scenario) than it does when it has 80 ground length traveling at 0.6c in an 80 ground circumference track (the first scenario)?

 

[PeterDonis]

in the case I described, the front and rear are not connected (the train is not a ring), so the rear can travel faster in the ground frame than the front, and the train ends up length contracted to 80 in the ground frame, traveling at 0.6c in the ground frame.

Still not possible. Basically, the Herglotz-Noether theorem says that, if you accelerate the train in anything but a straight line, either its length or its curvature has to change. You are stipulating that the curvature does not change. Therefore the length must change.

The radar method will work exactly the same for the train in this second scenario after it has accelerated to 0.6c in the ground frame and continues at that constant rate.

We're not talking about whether it's possible to have a train moving at a constant speed of 0.6c around the circle with length 100. Of course it is. We're talking about whether it's possible to take a train with length 100 at rest in the ground frame, placed around the circle, and accelerate it to a speed of 0.6c while keeping it confined in the same circle, without changing its length as seen by observers moving with the train. It isn't.

 

[DrGreg]

You are stipulating that the curvature does not change. Therefore the length must change.

Are you sure it's not the other way round? We're stipulating that the length does not change, but the curvature does change (because of the change from Euclidean 3D space to the rotating non-Euclidean 3D quotient space). (Another way of saying what I said in post #36.)

 

[PeterDonis]

We're stipulating that the length does not change, but the curvature does change

It depends on who is doing the stipulating. As I understand it, JVNY is proposing a scenario where the train is sitting at rest (in the ground frame) in a circular track with a circumference of 100; then it accelerates to an angular velocity corresponding to a speed of 0.6c, all while remaining confined in that same circular track. So in his version, the curvature cannot change; it's constrained to be the same as the curvature of the fixed track.

In the scenario you were envisioning earlier, the curvature was not constrained--but that would correspond to the track being able to move radially to accommodate the change in curvature of the train as it accelerates, in order to keep the proper length of the train constant. So at the end of the acceleration in your version, the train would still have a proper length of 100, but the track, as seen from the ground frame, would have gotten smaller--it would now have a circumference of only 80 (and a smaller radius to correspond). So your version has a different constraint from JVNY's version.

 

[DrGreg]

In the scenario you were envisioning earlier, the curvature was not constrained--but that would correspond to the track being able to move radially to accommodate the change in curvature of the train as it accelerates, in order to keep the proper length of the train constant.

But that wasn't what I meant. I think the rotating observer and the inertial observer disagree on what the curvature of the same circular track is: they agree on its radius but disagree on its circumference, so they disagree over the rate of change of angle with respect to arc length.

 

[PeterDonis]

I think the rotating observer and the inertial observer disagree on what the curvature of the same circular track is

That may be, but the fact remains that, if the proper length of the train is to remain the same during acceleration in a circle, the radius and circumference of the track as seen in the ground frame, and therefore the curvature as seen in the ground frame, must change. Whereas JVNY, as I understand it, was proposing acceleration in a circle that was constrained to stay in the same circle, with unchanging radius and circumference, as seen in the ground frame.

[Edit: On reading back, though, I see that JVNY is allowing for the possibility that, after the acceleration, the train does not occupy the complete circle, but only a portion of it. That would be yet a third scenario, which would still not be Born rigid--since no motion with changing angular velocity can be Born rigid--but it's unclear to me exactly how the failure of Born rigidity would show up in the motion.]

 

[A.T.]

if the proper length of the train is to remain the same during acceleration in a circle, the radius and circumference of the track as seen in the ground frame, and therefore the curvature as seen in the ground frame, must change.

In the inertial frame:
radius : const
arc length : decreases
angle : decreases
curvature : const

In the co-rotating frame:
radius : const
arc length : const
angle : decreases
curvature : decreases

 

[A.T.]

We're stipulating that the length does not change, but the curvature does change (because of the change from Euclidean 3D space to the rotating non-Euclidean 3D quotient space).

As an analogy:

If you draw a circle on a flat surface, the circumference has extrinsic curvature w.r.t to the surface. If you deform the surface to a sphere, such that your circle becomes a great circle, then the extrinsic curvature of the circumference w.r.t to the surface is reduced to zero.

 

[A.T.]

The train initially subtends an angle of 2π2\pi at the centre of the circle, but later subtends a smaller angle, yet its arc-length (in the rotating geometry) is still the same, so the curvature has decreased.

Thinking about this analogy:

If you draw a circle on a flat surface, the circumference has extrinsic curvature w.r.t to the surface. If you deform the surface to a sphere, such that your circle becomes a great circle, then the extrinsic curvature of the circumference w.r.t to the surface is reduced to zero.

I'm not so sure anymore if decreasing the angle, while keeping radius and arc length constant reduces the arc curvature. Bending the plane into a sphere (introducing positive curvature) will make an arc of constant length and radius subtend more angle, but the extrinsic curvature of the arc w.r.t to the surface will decrease.

However, in the rotating frame we are introducing negative curvature.

 

[JVNY]

It depends on who is doing the stipulating.

Neither of those; it changes its shape in the radial direction, . . . so the curvature has decreased.

Yes, agreed . . .

I have agreed with DrGreg that the curvature of the train decreases as measured by observers on the train. Here is a visual.

Say there are two parallel tracks very close to each other, each with a straight section leading into a circular section with ground circumference 100. A train of proper length 125 is on one of the straight tracks, and a train of proper length 100 is on the parallel straight track. Observers on the train paint meter marks along the trains, and verify that they line up between trains and that one has length 125 and the other 100.

Next, the 100 train moves into the circular section of its track and stops. It has the same length as the circle's circumference, so it occupies the entire track according to both its observers and ground observers.

Next, the 100 train accelerates as described before, using the same thrust programs that it would to accelerate Born rigidly if it were on a straight track to reach 0.6c, then remains at that speed. It contracts to 80 in the ground frame, but maintains its 100 length as measured by its own observers.

Finally, the 125 train accelerates Born rigidly on the straight section of its track to 0.6c. It length contracts to 100 in the ground frame but retains its 125 length as measured by its observers. It enters the circular section of its track. It has 100 ground length in a 100 ground circumference track, so it occupies the entire track in the ground frame; its front and rear ends touch.

The trains continue to move around the circular sections of the tracks. Here is how they look in the ground frame (I have separated the two for clarity; imagine that they are parallel and right next to each other):

The two trains are at rest with respect to each other. Observers on each train observe that neither has deformed; their meter marks still align; the one train still has proper length 100, and the other proper length 125. The 125 train is curved into a circle, with its front and rear touching, so observers on both trains agree that the circumference around which they are traveling has length 125. The ground circumference (100) is contracted compared to the trains' measured circumference (125) by gamma (1.25).

Finally, observers on the 100 train note that the curvature of the train is lower than it was at rest in the circular section of track. It started out curved with length 100 in a circle of circumference 100. Now it has length 100 traveling around a circle of circumference 125. In fact, if the 100 train accelerated to very nearly the speed of light, the length of the circumference that it would measure itself traveling on would increase to an arbitrarily large number, so observers on the train would measure their train to be nearly a straight line.

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

Exactly. This is the basis of Ehrenfest's Paradox. The length of the circumference is shorter as measured by ground observers (100) than it is as measured by train observers (125).

Basically, the Herglotz-Noether theorem says that, if you accelerate the train in anything but a straight line, either its length or its curvature has to change.

The descriptions of this theorem online are hard to understand, so I will take your summary to be correct. I have concluded that the 100 train's own measured length does not change (because of the measurement by the radar method). Thus I must conclude, as DrGreg pointed out, that it's own measured curvature changes. The train becomes less curved in its own measurement.

 

[pervect]

That result certainly is the case if you try to set a disk or ring into rotation. But in the case I described, the front and rear are not connected (the train is not a ring), so the rear can travel faster in the ground frame than the front, and the train ends up length contracted to 80 in the ground frame, traveling at 0.6c in the ground frame. As such it is just like the train traveling around in a circle at 0.6c earlier in the thread, and as to its length:

The radar method will work exactly the same for the train in this second scenario after it has accelerated to 0.6c in the ground frame and continues at that constant rate. A signal from the rear to the front will take ground time 80 / 0.4 = 200 to reach the front, then after reflecting will take ground time 80 / 1.6 = 50 to return to the rear, for a total round trip time of 250. Half of that is 125, and dilated by gamma of 1.25 means that the radar measured length is unchanged at 100.

How can the radar length be any different when the train has 80 ground length traveling at 0.6c in a 100 ground circumference track (the second scenario) than it does when it has 80 ground length traveling at 0.6c in an 80 ground circumference track (the first scenario)?

I'm not sure what you're trying to say or ask here. I'm assuming that "the case you described" is the train being accelerated Born rigidly. By definition, Born rigid acceleration doesn't change radar distance relationships between nearby elements, so the train has the same radar length after the Born rigid acceleration than it had before.

The coordinate length is a different animal. I think what you may be asking is why does the coordinate length change when you accelerate a train Born rigidly, while the proper length does not? Fundamentally, coordinate length is not the same as proper length, so you need to define which one you're talking about, you shouldn't expect them both to behave in identical manners. Yet you seem to be assuming that they do. Also, from your usage, I'm not sure which sort of length you're talking about (proper length or coordinate length) when you talk about "radar length".

So there are two problems here. The first is understanding what happens to both sorts of "length" when a train accelerates born-rigidly. 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, but since the confusion here seems to be about the first case, it should probably be sorted first, then the second case can perhaps be discussed in more detail.

 

[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.