How is SR applied to circular motion?

[lomidrevo]

Hi all,
I have a problem to fully understand how we can apply Special Relativity to a system where one observer is still in the center, and other one is moving in a circle around. For example, like a satellite orbiting Earth. In case of GPS, the clocks carried by satellite are running slower than clocks located on the ground - as a result of time dilation predicted by Special relativity due to their relative motion. (I know there is opposite and even more significant influence due to General relativity, but let's ignore this one).

Now to my question... My understanding is that SR is correct only for observes within inertial frames of reference, i.e. non-accelerating frames of reference. However, in case of satellite orbiting Earth, isn't there a permanent centripetal acceleration produced by gravitational force of the Earth (causing the curved/orbital path of the satellite)? Then how can we use SR to predict time dilation as described above?

I have two ideas in my mind, how can that be, but not sure whether any of them is correct to answer my question:

1) The orbital path is so large, so at any given infinitesimal time interval it can be considered as straight path. And consequently the relative motion can be seen as straight uniform motion.

2) Even there is the centripetal force acting on the satellite, it's frame of reference can be considered as inertial - it is in a free fall, and there won't be any net force acting on it.

Thanks in advance for your help!

 

[Ibix]

Special relativity is correct in flat spacetime - anywhere gravity is not important. It can perfectly well handle accelerating objects. It wouldn't be much of a theory of motion if it couldn't.

Einstein did originally construct special relativity in inertial reference frames, but it's perfectly possible to work in non-inertial frames. Just as it is in Newtonian physics, it's usually much easier to work in inertial frames but sometimes it's worth the pain. It's just a change of coordinates, after all.

 

[A.T.]

I have a problem to fully understand how we can apply Special Relativity to a system where one observer is still in the center, and other one is moving in a circle around. For example, like a satellite orbiting Earth.

That is a bad example, if you are interested in circular motion within Special Relativity, because it involves gravity and thus requires General Relativity. You cannot combine Special Relativity with Newtonian Gravity consistently.

 

[vanhees71]

No, circular motion in SR has nothing to do with gravity, and you can formulate special relativity also in rotating reference frames (although only for part of the spacetime, i.e., in the sense of a local reference frame). In Newtonian mechanics it's analogous: The inertial reference frames are special, and usually the theory takes the most simple form when described in coordinates relative to an inertial frame, but you can also describe Newtonian physics in rotating frames, which you sometimes do to, e.g., treat the Foucault pendulum.

It's of course true, if you use arbitrary generalized spacetime coordinates in SR, you are already pretty close in the mathematics to general relativity.

You find examples on the following problem set for our recent GR lecture (it's even in English :-)):

http://th.physik.uni-frankfurt.de/~hees/art-ws16/sheet02.pdf

The solutions are here:

http://th.physik.uni-frankfurt.de/~hees/art-ws16/lsg02.pdf

 

[lomidrevo]

Einstein did originally construct special relativity in inertial reference frames, but it's perfectly possible to work in non-inertial frames. Just as it is in Newtonian physics, it's usually much easier to work in inertial frames but sometimes it's worth the pain. It's just a change of coordinates, after all.

Thanks Ibix, could you pls provide me an example of such change of coordinates, so I can better understand what do you mean.

 

[Ibix]

Rindler coordinates describe the frame of reference of an observer eternally accelerating at a constant proper acceleration. Dolby and Gull wrote a paper on radar coordinates that is a more flexible system: https://arxiv.org/abs/gr-qc/0104077

 

[lomidrevo]

That is a bad example, if you are interested in circular motion within Special Relativity, because it involves gravity and thus requires General Relativity. You cannot combine Special Relativity with Newtonian Gravity consistently.

OK, what about this example: LHC collider, with a particle accelerated to a constant speed close to c (bearing it's virtual clocks). And a scientist sitting in the middle of LHC with it's own clocks. Did we get rid of gravity?

 

[lomidrevo]

Rindler coordinates describe the frame of reference of an observer eternally accelerating at a constant proper acceleration. Dolby and Gull wrote a paper on radar coordinates that is a more flexible system: https://arxiv.org/abs/gr-qc/0104077

I will need some time to go through it, but thanks :)

 

[A.T.]

OK, what about this example: LHC collider, with a particle accelerated to a constant speed close to c (bearing it's virtual clocks). And a scientist sitting in the middle of LHC with it's own clocks. Did we get rid of gravity?

Yes, here you can use SR to predict the time dilation of the particle clock, based on its speed in the inertial frame.

If you chose a rotating frame where the center and particle are both at rest, then you also introduce a centrifugal potential (similar to gravity). So in that frame the time dilation is explained by different positions in that potential.

 

[lomidrevo]

No, circular motion in SR has nothing to do with gravity, and you can formulate special relativity also in rotating reference frames (although only for part of the spacetime, i.e., in the sense of a local reference frame). In Newtonian mechanics it's analogous: The inertial reference frames are special, and usually the theory takes the most simple form when described in coordinates relative to an inertial frame, but you can also describe Newtonian physics in rotating frames, which you sometimes do to, e.g., treat the Foucault pendulum.

It's of course true, if you use arbitrary generalized spacetime coordinates in SR, you are already pretty close in the mathematics to general relativity.

You find examples on the following problem set for our recent GR lecture (it's even in English :-)):

http://th.physik.uni-frankfurt.de/~hees/art-ws16/sheet02.pdf

The solutions are here:

http://th.physik.uni-frankfurt.de/~hees/art-ws16/lsg02.pdf

Thank you, I will look at the examples

 

[PeterDonis]

in case of satellite orbiting Earth, isn't there a permanent centripetal acceleration produced by gravitational force of the Earth (causing the curved/orbital path of the satellite)?

No, because in relativity, "acceleration" doesn't mean the same thing it does in Newtonian mechanics. In relativity, "acceleration" is best viewed as proper acceleration, i.e., acceleration actually felt by a body (and measured with an accelerometer). A body moving solely under gravity, like a satellite orbiting the Earth, is in free fall, feeling zero acceleration. So, as others have pointed out, this is not a good case to use if you want to understand how SR treats circular motion; you need to have the circular motion produced by some non-gravitational force, like a centrifuge, or a ball being swung in a circle using a rope. (To put this another way, in relativity, gravity is not a force, and can't be treated the same way as non-gravitational forces, as is done in Newtonian mechanics.)

 

[lomidrevo]

Yes, here you can use SR to predict the time dilation of the particle clock, based on its speed in the inertial frame.

If you chose a rotating frame where the center and particle are both at rest, then you also introduce a centrifugal potential (similar to gravity). So in that frame the time dilation is explained by different positions in that potential.

Actually this is very interesting (to me), that we can find a frame in which both observers are in rest (no relative motion between them). And still their clocks will run at different rates. It somehow resembles to me the time dilation predicted by GR due to different position in gravitational field.

 

[PeterDonis]

we can find a frame in which both observers are in rest (no relative motion between them)

But this "frame" is not an inertial frame, so you can't reason about it as if it were.

 

[lomidrevo]

No, because in relativity, "acceleration" doesn't mean the same thing it does in Newtonian mechanics. In relativity, "acceleration" is best viewed as proper acceleration, i.e., acceleration actually felt by a body (and measured with an accelerometer). A body moving solely under gravity, like a satellite orbiting the Earth, is in free fall, feeling zero acceleration.

Actually this is what I meant by my idea Nr. 2 in the original post. Thanks for the explanation.

(To put this another way, in relativity, gravity is not a force, and can't be treated the same way as non-gravitational forces, as is done in Newtonian mechanics.)

Thanks for clarification, it make sense to me now

 

[sweet springs]

Hi. SR gives you more topics of interest, e.g.
- New ##\pi## is greater than 3.141592...
- Clocks still in the coordinate farther from the center, slower they tick.
- Center observes that right going light and left going light at distance have different speed.
Best.

 

[pervect]

Hi all,
I have a problem to fully understand how we can apply Special Relativity to a system where one observer is still in the center, and other one is moving in a circle around. For example, like a satellite orbiting Earth. In case of GPS, the clocks carried by satellite are running slower than clocks located on the ground - as a result of time dilation predicted by Special relativity due to their relative motion. (I know there is opposite and even more significant influence due to General relativity, but let's ignore this one).

Now to my question... My understanding is that SR is correct only for observes within inertial frames of reference, i.e. non-accelerating frames of reference.

When looked at in a specific way, that's reasonably close to being true. Some formulations of SR require one to do the analysis in an inertial frame of reference, just as some formulations of Newtonian mechanics are formulated to require that the problem be analyzed in an inertial frame of reference.

From these formulations in inertial frames of reference, one can find more general formulations of SR that aren't so restricted. The same comment can be said of Newtonian physics.

However, in case of satellite orbiting Earth, isn't there a permanent centripetal acceleration produced by gravitational force of the Earth (causing the curved/orbital path of the satellite)? Then how can we use SR to predict time dilation as described above?

Not really. You can certainly analyze an object forced to follow a circular orbit by a rocket, or a strong string in an inertial frame of reference, and when you do said analysis, you get the prediction of time dilation.

A point that you seem to be partially aware of is that this analysis won't apply to a satellite orbiting the Earth due to gravity, because gravity is not a force. To sucessfully incorporate gravity into special relativity requires general relativity. In order to proceed , we need to focus on the simplest problem that you can potentially analyze, and guide you to an analysis of this simplest possible problem. If you insist on running before you can walk, you won't get anywhere :(.

Now, I gather from your post that you don't understand what equations you need to analyze the simpler problem of a body forced to orbit by a string or rocket, so I'll present the equation you need to carry out this analysis. Here it is:

$$d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2$$

Here (t,x,y, and z) are coordinates in an inertial frame of reference, and ##\tau## is the proper time. The meaning of the term "proper time" can be understood to be just what a clock reads.

That's what you actually need to understand to work the problem. You seem to be going off in some directions in an attempt to solve the problem, but the directions you are heading in are probably not going to work. I can point you in a direction that will work, but that will only help if you actually follow where I'm pointing, if you, attracted by your own ideas, insist on following them, it'll be entirely up to your own efforts whether or not you succeed. From the fact that you are posting, you presumably want help, the help will come in the form of new ideas that may not occur to you naturally as to how to approach the problem.

I suppose you also have to understand that dx is a differential operator, which is part of calculus. I tend to take that for granted. I really don't know how to explain this without calculus, alas.

There are other problems that you probably want to understand, but they're more complicated than the above. One other problems that come to mind are the issues of how one goes about constructing an "accelerated frame of reference" in special relativity, and given such an accelerating frame, how does one go about modifying the solution process above?

The first part, how one defines an "accelerated frame of reference" is rather difficult, but I can address the second part somewhat. When you specify an accelerated frame of reference one of the things that comes out of the specification process is a metric.

Given a metric tensor, ##g_{ij}## we can re-write the previous solution as something like what appears below.

$$d\tau^2 = g_{00} dt^2 - g_{11} dx^2 - g_{22} dy^2 - g_{33} dz^2$$

Here all the ##g_{ii}## terms are the new addition, the metric tensor.

This isn't quite right though. At this point it likely becomes convenient to replace the x,y,z coordinates of the inertial frame with something more general. Cylindrical coordinates ##r, \theta, z## fit this particular problem very well, but we might as well make the leap to completely and totally general coordinates. We'll call these coordinates ##x^i##, and there will be four of them, one coordinate that replaces "t", and three coordinates that replace "x,y,z". We'll number these coordinates, starting at 0, so our general coordinates will be ##x^0, x^1, x^2, x^3##. Given these general coordinates, we can then write:

$$d\tau^2 = \sum_{i,j} \sum_{i,j} g_{ij} \, dx^i \, dx^j$$

and ##g_{ij}## is an array of 16 numbers - basically a matrix , though we call it a tensor. The matrix/tensor is symmetric, by and by the way, so that ##g_{ij}## turns out to be equal to ##g_{ji}##.

 

[lomidrevo]

thank you pervect for your extensive comment.

A point that you seem to be partially aware of is that this analysis won't apply to a satellite orbiting the Earth due to gravity, because gravity is not a force. To sucessfully incorporate gravity into special relativity requires general relativity.

Actually that's the initial bug in my head and reason for my original post. According to few sources (e.g. this one, sorry for the wikipedia), the overall time dilation in case of GPS satellite, consist of two components: "due to GR" and "due to SR". The component "due to SR" is calculated via Lorentz transformation similarly as it would be calculated for an observers in a uniform relative motion.
Thinking about that, maybe what is really confusing me, is that we can split the problem and calculate the two components separately, whereas the gravity plays role only in the component "due to GR".

In order to proceed , we need to focus on the simplest problem that you can potentially analyze, and guide you to an analysis of this simplest possible problem. If you insist on running before you can walk, you won't get anywhere :(.

Now, I gather from your post that you don't understand what equations you need to analyze the simpler problem of a body forced to orbit by a string or rocket, so I'll present the equation you need to carry out this analysis. Here it is:

$$d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2$$

Here (t,x,y, and z) are coordinates in an inertial frame of reference, and ##\tau## is the proper time. The meaning of the term "proper time" can be understood to be just what a clock reads.

That's what you actually need to understand to work the problem. You seem to be going off in some directions in an attempt to solve the problem, but the directions you are heading in are probably not going to work. I can point you in a direction that will work, but that will only help if you actually follow where I'm pointing, if you, attracted by your own ideas, insist on following them, it'll be entirely up to your own efforts whether or not you succeed. From the fact that you are posting, you presumably want help, the help will come in the form of new ideas that may not occur to you naturally as to how to approach the problem.

I suppose you also have to understand that dx is a differential operator, which is part of calculus. I tend to take that for granted. I really don't know how to explain this without calculus, alas.

I am definitely open to a new (verified) ideas, and I am not insisting on any of my previous ideas. That's why I am here on this forum - I'd like to understand how the things are really working :)
I know the calculus. And also I understand what is the proper time, and that it is an invariant in SR. If you wish to know, my current understanding of SR is within the scope of the Feynman Lectures, vol. I.

There are other problems that you probably want to understand, but they're more complicated than the above. One other problems that come to mind are the issues of how one goes about constructing an "accelerated frame of reference" in special relativity, and given such an accelerating frame, how does one go about modifying the solution process above?

The first part, how one defines an "accelerated frame of reference" is rather difficult, but I can address the second part somewhat. When you specify an accelerated frame of reference one of the things that comes out of the specification process is a metric.

Given a metric tensor, ##g_{ij}## we can re-write the previous solution as something like what appears below.

$$d\tau^2 = g_{00} dt^2 - g_{11} dx^2 - g_{22} dy^2 - g_{33} dz^2$$

Here all the ##g_{ii}## terms are the new addition, the metric tensor.

This isn't quite right though. At this point it likely becomes convenient to replace the x,y,z coordinates of the inertial frame with something more general. Cylindrical coordinates ##r, \theta, z## fit this particular problem very well, but we might as well make the leap to completely and totally general coordinates. We'll call these coordinates ##x^i##, and there will be four of them, one coordinate that replaces "t", and three coordinates that replace "x,y,z". We'll number these coordinates, starting at 0, so our general coordinates will be ##x^0, x^1, x^2, x^3##. Given these general coordinates, we can then write:

$$d\tau^2 = \sum_{i,j} \sum_{i,j} g_{ij} \, dx^i \, dx^j$$

and ##g_{ij}## is an array of 16 numbers - basically a matrix , though we call it a tensor. The matrix/tensor is symmetric, by and by the way, so that ##g_{ij}## turns out to be equal to ##g_{ji}##.

It means that for constructing an "accelerated frame of reference" (and getting the metric tensor) the GR is required, correct? I haven't studied the general relativity and metric tensors yet, but it is on my TODO list. Some day, I will start this long journey :)

 

[Ibix]

According to few sources (e.g. this one, sorry for the wikipedia), the overall time dilation in case of GPS satellite, consist of two components: "due to GR" and "due to SR".

That's not quite right, although the Wikipedia article is certainly open to that interpretation.

The time dilation between a ground station and a GPS satellite is completely described by general relativity. However, you can write it as a product of the time dilation between the ground station and a clock in a rocket hovering (not orbitting!) at the satellite altitude, and of the time dilation between that clock and a satellite passing its window. The former is purely gravitational. The latter is purely kinematic, and is exactly as predicted by SR.

So (in this case) you can decompose the time dilation into a component due to gravitational potential and a component due to kinematics. That doesn't really mean that "there's a component due to SR and a component due to GR". That makes them sound like two different theories, which they're not. It just means that once you've established the time dilation factor between the ground and the rocket, you only need the (much simpler) maths of special relativity to get the time dilation between the ground and the satellite.

 

[vanhees71]

GR is so much simpler, if you just take it seriously and don't try to split kinematical properties like time dilation into an SR and a GR part. GR, as the name "general" suggests, includes all effects in an ingenious way, making it to the most esthetical fundamental theory of physics ever.

 

[lomidrevo]

The conclusion seems to be clear: only GR can provide fully satisfactory answers (not only for this topic). I should start to study it - the sooner the better :)

Thank you all for your comments and feedback! I appreciate it. It gave me at least some clue how should I look at problems like this.

 

[vanhees71]

Well, for the study of physics in a rotating reference frame in Minkowski space, you don't need the full glory of GR. You just do the coordinate transformation in Minkowski space. See the following problem sheet + solutions (Problem 4):

http://th.physik.uni-frankfurt.de/~hees/art-ws16/sheet02.pdf
http://th.physik.uni-frankfurt.de/~hees/art-ws16/lsg02.pdf

 

[lomidrevo]

Well, for the study of physics in a rotating reference frame in Minkowski space, you don't need the full glory of GR. You just do the coordinate transformation in Minkowski space. See the following problem sheet + solutions (Problem 4):

http://th.physik.uni-frankfurt.de/~hees/art-ws16/sheet02.pdf
http://th.physik.uni-frankfurt.de/~hees/art-ws16/lsg02.pdf

Ah, good to know... I had a quick look at these papers when you post it for the first time in this thread, but apparently I was discouraged by the tittle: Tutorial “General Relativity”

What I deduced from the sheet with solutions:
- x^μ are coordinates in the inertial frame
- x'^ν are coordinates in the rotating frame
- in the equation (13), there is the infinitesimal interval ds² expressed in the coordinates of the rotating frame. I see now that this metric can be written in matrix format, equation (14), and that it corresponds to matrix (tensor) mentioned by pervect in post #16.
- for an observer sitting in the center (x=y=z=0, x^'=y^'=z^'=0): dτ² = ds² / c². I can imagine dτ as one tick on it's clocks
- the tick occurs in the center (not moving), therefore dx=dy=dz=0, dx^'=dy^'=dz^'=0 for any observer)
- then for an observer sitting at constant distance from center in the rotating frame, e.g. (x^'=R, y^'=z^'=0), using the equation (13) divided by c², I can get the "tick period" as seen by him dt^' by rearranging and square root of: dτ² = dt^'2 (1 - ω²R²/c²).

Is that correct?

However I don't understand the parts b) to d) of that problem, but if I got it right, it is not important for the topic of this thread.

 

[vanhees71]

This looks good.

Parts (b)-(d) are indeed to get used to general-relativity math, which of course applies also to the special case of arbitrary curvilinear coordinates in flat Minkowski spacetime. To learn it, is a very well invested time since of course it's also applicable in Euclidean 3D space, i.e., usual vector analysis. It let's you see the standard methods, using Cartesian coordinates and "div, grad, curl, and all that" from another more general perspective. A very nice introduction can be found in

H. Stephani, Relativity - An Introduction to Special and General Relativity, Cambridge University Press (2004) (3rd edition)

 

[lomidrevo]

Thank you for the hint, I will check that

 

[Demystifier]

I have a problem to fully understand how we can apply Special Relativity to a system where one observer is still in the center, and other one is moving in a circle around.

https://arxiv.org/abs/gr-qc/9904078

 

[sweet springs]

Hi. You may know Section 89 Rotation of the textbook,
https://archive.org/details/TheClassicalTheoryOfFields Makes best use of SR. Regards.

 

[PAllen]

Hi. You may know Section 89 Rotation of the textbook,
https://archive.org/details/TheClassicalTheoryOfFields Makes best use of SR. Regards.

Note that Demystifier's paper cites this, and expresses several disagreements with it. Having read Demystifier's paper, I am convinced that it is correct on these points of disagreement.

 

[pervect]

I'd like to say that I don't really agree with certain elements of Demistifyer's paper, however.

I agree with many of his statements, though.

I agree that is generally accepted that the space-time line-element is given by the Fermat metric, [3] in Demystifer's paper. Demystifier doesn't really explore "why" this line element is generally accepted though, and, if I'm reading the paper correctly, he winds up proposes a different line element. That's mostly the part I disagree with. Note that I don't disagree with the notion that it's fair to criticize "what's generally accepted" in a paper, what I'm a bit cautious about is giving this paper as the very first thing for a student to read on the topic.

I would say that the expression (5), ##2 \pi r / \sqrt{1-\omega^2\,r^2}## is not the proper circumference of the disk, because, as Demystifier points out, there isn't any single proper frame that represents the disk.

There is, however, a description of the rotating disk that's not a "proper frame", but something else, something called a quotient manifold. This approach is described by Ruggerio in "Relative space: space measurements on a rotating platform". <<link>> Ruggerio winds up calling this the "relative space" of the rotating disk, a term that has never caught on to the best of my knowledge.

Ruggerio makes the point that measurements in this quotient space, which he terms "relative space", have an operational definition in terms of the SI defintion of the meter. This is to my mind a compelling argument.

Ruggerio's claim, talking about "what he terms relative space, that it is "... recognized as the only space having an operational meaning in the study of the space geometry of a rotating disk." could perhaps use more scrutiny, it is such a strong statement. I believe it is reasonably clear that this relative space and it's associated Fermat line element does have an operational basis, and that this operatoinal basis is just the current SI definition of the meter. This leads one to ask: does the line element proposed by Demystifier have such an operational meaning, or is it a counterexample to Ruggerio's claim?

Stepping back a bit, though. Given the sheer mass of literature on the Ehrenfest paradox, and the number of different authors proposing resolutions which are similar though not all identical, the question arises as to "what is well-accepted" resolution so that we know what to present to students as a starting point for "the accepted approach". The only thought that comes to mind is to dig into the "impact factor" that the myriad papers on the topic have had, and sort out the top contenders in citations. Unfortunately, I don't have access to any such "impact database(s)".I

 

[sweet springs]

https://arxiv.org/abs/gr-qc/9904078

Hm... The author seems to believe Lorentz contraction is must. But sometimes a kind of "Lorentz extension" take place in rotation system. Say contraction for right tangent movement, extension for left tangent movement to circumpherence, i.e. less rotation up to staying still in an inertial system.

Space geometry
[tex]\gamma_{22}=\frac{r^2}{1-u^2}[/tex]
works even in limit ##r\rightarrow 0,\omega \rightarrow \infty## and their product u=const. acceleration does not matter. Am I wrong?

 

[sweet springs]

PS Rotation system share the clocks with the inertia frame of reference. These clocks seem not synchronized to rotation system in the above limit. This brings contraction/extension asymmetry of tangent direction.

 

[Demystifier]

The author seems to believe Lorentz contraction is must. But sometimes a kind of "Lorentz extension" take place in rotation system.

At the end of page 9, the paper says:
"Now, as in Section 2, assume that the rotating ring is a series of independent short rods, uniformly distributed along the gutter. Each rod is relativistically contracted, but the ring is not. This means that the distances between the neighboring ends of the neighboring rods are larger than those for a nonrotating ring, so the proper length of the ring is also larger than that of a nonrotating ring. This is concluded also in [2]. This situation mimics a more realistic ring made of elastic material, where atoms play the role of short rigid rods. Owing to the rotation the distances between neighboring atoms increase, so there are tensile stresses in the material."
This quote shows that "Lorentz extension" is included.

 

[sweet springs]

Thanks for your suggestion. I agree him on this point.

This situation mimics a more realistic ring made of elastic material, where atoms play the role of short rigid rods. Owing to the rotation the distances between neighboring atoms increase, so there are tensile stresses in the material."

More ideal case is ring made of incompressible fluid rotating with speed u through the gutter with no friction. Depth or width of fluid from the gutter r outward side would become shallower or narrower as the speed u increases. Or metal ring melted by heat should congeal in new rotating environment. No ##\phi##-stress inside the metal but get compressed when it stops. Best.

 

[PAllen]

At the end of page 9, the paper says:
"Now, as in Section 2, assume that the rotating ring is a series of independent short rods, uniformly distributed along the gutter. Each rod is relativistically contracted, but the ring is not. This means that the distances between the neighboring ends of the neighboring rods are larger than those for a nonrotating ring, so the proper length of the ring is also larger than that of a nonrotating ring. This is concluded also in [2]. This situation mimics a more realistic ring made of elastic material, where atoms play the role of short rigid rods. Owing to the rotation the distances between neighboring atoms increase, so there are tensile stresses in the material."
This quote shows that "Lorentz extension" is included.

You also noted in your paper that this is equivalent to linear cases, which I thought was a key observation. In my words, you have a rolled up version of Bell spaceship paradox. Because the circumference is constrained by the gutter to be the same as in the inertial frame, the local stretching increases without bound as rim speed approaches c. Just like a string constrained to maintain a fixed length in an inertial frame as it accelerates to c experiences unbounded local stretch.

 

[Demystifier]

You also noted in your paper that this is equivalent to linear cases, which I thought was a key observation. In my words, you have a rolled up version of Bell spaceship paradox. Because the circumference is constrained by the gutter to be the same as in the inertial frame, the local stretching increases without bound as rim speed approaches c. Just like a string constrained to maintain a fixed lenegth in an inertial frame as it accelerates to c experiences unbounded local stretch.

Yes, exactly!