Geometry in Rotating Reference Frames

Discussion in 'Physics & Math' started by Neddy Bate, Mar 28, 2010.

  1. Neddy Bate Valued Senior Member

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    Hello Everyone,

    I am trying to explore a rotating reference frame, in the absence of any gravitational bodies. I am interested in the geometry of such a frame, as well as the simultaneity, etc.

    I thought a good way to start would be to look at a wheel on a rolling tank. This is better than just a car or a train, because the wheels are connected by a continuous tread, and that helps make sure everything stays consistent.

    Note: If you have trouble seeing the slides, most browsers have a zoom feature that helps quite a bit. In Firefox and IE, you can hold down the Ctrl key, and then roll the wheel on your mouse. To get the zoom back to normal, Ctrl+0 works for those browsers, (that is Ctrl+zero).

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    SLIDE 0:
    Here is a slide showing the tank in its own reference frame:

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    As you can see, the tank is yellow, the tread is light blue, and the road is green. The tread is not physically stretched, because it was constructed while the tank was in motion. This was achieved by the tank wheels picking up measuring rods from the road, and connecting them end-to-end. This ensures that the measuring rods in all frames are the same length, because none are physically stretched.

    ---

    SLIDE 1:
    Now, from the reference frame of the road, things are a little different:

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    In the above slide, we start to explore the rotating reference frame. Notice the arrow pointing to "Bottom". You can think of this as an observer who is about to get picked up by the wheel. But I don't think I am really looking to explore his "observations", as in, "what he sees with his eyes." Since that is not the way relativity works in inertial frames, I don't think that is the idea here either. But anyway, Bottom is like an observer who will represent the rotating reference frame.

    As Bottom gets picked up by the wheel, he moves up the wheel ever so slightly. At this point he is feeling the acceleration, but it need not be very strong. The wheel might be really huge, so he might only be accelerating very slightly. Anyway, when he moves up the wheel just a little, the relative motion between himself, the top of the tread, and bottom of the tread is not changed by very much. So I think this means that all of the geometry in this frame is valid for Bottom, but I am not sure. Does Bottom really consider the wheel to be elliptical in its own frame? If not, what has to change?

    ---

    SLIDE 2:
    Now Bottom has completed 25% of the way around the wheel:

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    By this point, Bottom is in a position where the relative motion between himself, the top of the tread, and the bottom of the tread are equal, but in opposite directions. So I think the geometry is as shown. But the wheel itself is still elliptical, which is a bit troublesome.

    In the interest of keeping bandwidth down, and keeping this post short, I will provide the other three slides as hyperlinks.


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    SLIDE 3:
    Now Bottom has completed 50% of the way around the wheel:

    http://farm5.static.flickr.com/4013/4469938907_f61da620aa.jpg


    ---

    SLIDE 4:
    Now Bottom has completed 75% of the way around the wheel:

    http://farm5.static.flickr.com/4051/4469938953_493d7353d5.jpg


    ---

    SLIDE 5:
    Now Bottom has completed 100% of the way around the wheel:

    http://farm3.static.flickr.com/2797/4470718710_c921b9daa6.jpg

    And the loop can repeat from there.

    One good test of simultaneity is to make use of the Barn-Pole paradox by instantaneously "trapping" the wheel between two walls in the road frame. If the walls appear simultaneously, the wheel must be able to fit between them. If the walls appear one after the other, then the wheel does not necessarily have to fit between them.

    Does anyone know if I am on the right track? I don't think the wheel should be elliptical in its own frame, but I am not sure how else to make all the geometry work. Thanks in advance for all of your input!
     
    Last edited: Mar 28, 2010
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  3. Billy T Use Sugar Cane Alcohol car Fuel Valued Senior Member

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    Hi Needy. Good to see you posting, but I confessed I did not read much. I just wanted to let you know, in case you do not, that there was a thread about tank moving at relativistic speed with tread at rest on the earth. It was too complex for me to try to keep up. That is part of why I skipped most of yours.
     
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  5. CptBork Valued Senior Member

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    Hi Neddy. Wow, it's pretty rare that someone goes into detail and posts nice explicit demonstrative slides like yours. Did you code them in LaTex or something? Anyhow, I can't see anything wrong with your reasoning at first glance, but I have to caution that in Relativity the notion of a rigid body doesn't really hold. Circular motion due to a central force in one reference frame doesn't necessarily have to be circular motion in another frame, as far as I see it. I also believe central forces in one frame wouldn't necessarily have to be central forces in other frames, but I haven't checked the math on this one.

    P.S. I didn't even know FireFox had a zoom function... sweet, thanks for the tips!
     
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  7. Jack_ Banned Banned

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    How are you "trapping" the wheel between two walls in the road frame?
     
  8. Pete It's not rocket surgery Registered Senior Member

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    The wheel doesn't have an inertial rest frame - there is no inertial rest frame in which the whole wheel is at rest.
    The only inertial rest frame in which the wheel is circular is the rest frame of the axel.

    The wheel is always elliptical in the instantaneous inertial rest frame of Bottom, for the same reason that it is elliptical in the rest frame of the road.
    Describing the wheel according to Bottom is difficult, since his rest frame is not inertial, and simultaneity might be hard to define.

    Are you familiar with The barn and the pole? Neddy is thinking of something similar, with a wheel instead of the pole.
     
  9. Neddy Bate Valued Senior Member

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    Hi Billy T. Yes there was a thread a long time ago with a relativistic tank, but the rotating reference frame of the wheel was never even considered. In this thread, the rotating frame is the main subject. The road and tank frames are only included here for reference.
     
  10. Neddy Bate Valued Senior Member

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    Thanks, I'm glad you like the slides. I drew them in a CAD program so I could make sure everything was scaled properly. And yes, I am very happy there is a zoom feature in my browser!

    I'm not sure if the non-rigid body concept helps solve this one. The wheel has a lot of strange things going on, but I tried to avoid most of them by building the wheel from measuring rods picked up from the road. This way, nothing should be physically stressed or stretched.

    As far as circular motion not being circular motion in other frames: It is true that the wheel is not rotating in the frame of the wheel itself. In that frame, the road and the tank are rotating around the wheel. In all other frames though, I think the wheel is always rotating.
     
  11. Neddy Bate Valued Senior Member

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    2,548
    The idea was this: Add two walls in the road frame. Let them exist instantaneously and simultaneously in that frame. Have them spaced just far enough apart that they "trap" the wheel between them.

    Now, if the wheel is elliptical in its own frame, then it is narrow enough to fit between the walls. That suggests the simultaneity of the walls can also be simultaneous in the wheel frame. That might tell us something about simultaneity on the wheel.

    On the other hand, if the wheel is circular in its own frame, the wheel would not fit between the walls. This would suggest that the walls are not simultaneous in the wheel frame.

    But right now, I cannot figure out how to make the wheel circular in its own frame, even though it seems like it should be.
     
  12. Neddy Bate Valued Senior Member

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    2,548
    Hi Pete!


    Yes, but those are all inertial frames, at least instantaneously. What about the rotating frame? It's not inertial, but there is no relative motion between any points on the wheel in that frame. Shouldn't the wheel be circular because of symmetry concerns?


    Does the definition of simultaneity change the resulting shape of the wheel?

    In slide 1, when Bottom moves up the wheel ever so slightly, the relative motion of the other parts of the tread hardly change. This is strong support for the idea that all of the geometry is still very similar to what is shown there as the road frame. This suggests the wheel is elliptical in its own frame, but it doesn't seem like it should be, does it?
     
  13. Jack_ Banned Banned

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    But right now, I cannot figure out how to make the wheel circular in its own frame, even though it seems like it should be.

    It simply is as a rest radius.

    You drawings are good though.

    What exactly are you trying to prove?
     
  14. Pete It's not rocket surgery Registered Senior Member

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    Defining simultaneity is important, because the shape of the wheel at a given instant is determined by the simultaneous locations of each part of the wheel at that instant.
    But as you say, symmetry dictates that the wheel should be circular in its rest frame.

    No, it's only elliptical in that frame because it's an inertial frame - most parts of the wheel are moving in that frame. In the wheel's rest frame, it is not just Bottom that is stationary, but all parts of the wheel.

    The puzzle of the wheel's rest frame has been touched on before in this forum. See rpenner's initial response to this thread, and the links attached: [post=2432376]A Mystery in Topological Geometry[/post]

    Central to the puzzle is that in the wheel's rest frame, its circumference seems to be more than pi times its diameter.
     
  15. Jack_ Banned Banned

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    I read this but RPenner's interpretation contradicts GPS.

    The rings of Saturn remain in sync relative to the origin of Saturn otherwise GPS does not work.
     
  16. Pete It's not rocket surgery Registered Senior Member

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    No contradiction there. Firstly, the rings of Saturn were used as an analogy regarding the conflict between reality and inadequate models.
    Secondly, there's no synchronisation problem in reality - the puzzle here is an artificial one, that of finding a good model that adequately describes the wheel with constant location coordinates. In reality, the problem can be overcome by brute force by artificially synchronising rotating clocks to an arbitrary standard (this is what is done in the GPS, for example).

    But that's getting off-topic. If you want to discuss the GPS, you should open a new thread or find an old one to reply to. (It has been address in several threads (eg [thread=57834]Determining Absolute Motion Between Inertial Frames of Reference[/thread], and [thread=46583]Physics & Math forum - policy changes?[/thread], but I don't know if there's a thread specifically devoted to it.)
     
  17. Neddy Bate Valued Senior Member

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    I run into problems when I try to make the wheel a circle in its own frame. For example, in Slide 1, the top part of the tread and the bottom part of the tread are contracted to different degrees. If Bottom considers the wheel to be a circle as soon as he gets picked up by the wheel, then it seems like the top and bottom parts of the tread become equally contracted, as they are in Slide 0.

    Thanks. I'm not really trying to prove anything. I think I have provided a pretty good foundation for the geometry of a rotating reference frame, but it doesn't seem complete as long as the wheel is not a circlular shape in its own frame.
     
  18. Neddy Bate Valued Senior Member

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    2,548
    If we just assume there is no true simultaneity on the wheel, does that mean its shape is meaningless? If so, then perhaps the elliptical shape might be acceptable after all.

    That thread deals with the problem of the wheel/torus needing to be non-rigid in order to become length contracted as it is spun up to higher speeds.

    My wheel is always going the same speed, and it is made of measuring rods picked up from the road, so it is not deformed in any way. It could in fact be perfectly rigid, if there were such a thing.

    I have tried to include that fact in the geometry I have so far:

    In slide 0, you can see that it is possible for four measuring rods of length (1/gamma) to fit around the cyan wheel. Only two are shown though, because I only show measuring rods on the tread, not the wheel itself. In the wheel frame, these rods would not be contracted though.

    So, in slide 1, you can see that the cyan rod along the bottom part of the wheel is nearly un-contracted to its full length of (1), but only truly so at the very bottom. The rest of the rod is still contracted toward the top, so it doesn't quite equal (1) overall.

    And, by slide 2, I had to use zig-zag lines to represent that the two measuring rods on the bottom half of the wheel are actually full size. You have to really zoom in a lot to see that though, so just look for the darker shaded rods in that slide, and imagine that those rods are full size in the wheel frame. I cannot draw non-euclidean geometry on the screen.

    But none of this prevents the wheel from coming out elliptical instead of circular.
     
    Last edited: Mar 30, 2010
  19. Neddy Bate Valued Senior Member

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    This is interesting. I think it could mean that my trying determining the shape of the wheel is an artificial quest.

    But so far, the elliptical shape stands unrefuted. No one has explicitly shown how the circular shape can work. Perhaps all of the other geometry has to change as well, but no one has tried yet to explain how it should be changed.
     
  20. Jack_ Banned Banned

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    What does slide 0 show?
     
  21. Neddy Bate Valued Senior Member

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    Slide 0 shows that the rotating wheel is a circular shape in the reference frame of the tank. This is the same reference frame as the axle about which the wheel rotates. It makes perfect sense that the wheel should be circular in this reference frame, because all parts of the wheel are moving at the same speed.

    By that reasoning, however, the wheel should also be a circular shape in its own reference frame. That particular reference frame is not an inertial frame though, so it is difficult to establish the wheel's shape in that frame. The geometry shown in slides 1-5 seem to suggest that the wheel would be elliptical in that rotating frame. That doesn't seem right to me. Does it seem right to you?
     
  22. Jack_ Banned Banned

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    OK, Now I see what you are driving at.



    I do not think rotatiion in the tanks frame has any impact other than sagnac and then the tank's frame then it appears inertial but rotatiing.

    I will reread all your stuff now.

    I see your bottom line point. Nice idea.

    [Edit to add] Perhaps the wheel is constricted in the tank frame. I am thinking on the fly.

    It has to be under SR.I cannot see a change in the circular shape but perhaps someone here has equations. I have not seen this problem mathematically. I can see the integral in my mind causing constriction but not a shape change because of the circle and the uniform changing of the x-component.
     
    Last edited: Mar 31, 2010
  23. CptBork Valued Senior Member

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    I'm not referring to rotating frames though, I should have specified inertial. What I meant is that, say you had an electron orbiting a proton like the classical model of the hydrogen atom and the only force you consider is the classical electromagnetic force, treating both particles as charged points. Assume the proton mass is much greater than the electron mass, so the proton doesn't substantially accelerate in any direction, and assume the electron orbit is circular.

    If you boost to another frame using a velocity transformation, the new boosted observer sees a proton moving with constant velocity, but if they traced the path of the orbiting electron and measured its distance to the proton at any given time, I think the orbit would probably look elliptical instead of spherical. I'd have to play with the equations a bit to make certain. Assuming the orbit were elliptical in some non-stationary reference frame (non-stationary relative to the proton, that is), I don't think it would be that strange. In the rest frame you would argue that the electron only feels an electric force due to the static electric field of the stationary proton, always pulling in the direction of the proton. In another reference frame, the observer will argue that the electron experiences a changing electric field and a changing magnetic field coming from the moving proton, and the resulting forces need not point towards the "current" position of the proton (as measured by the observer in this boosted frame).

    I think the last point is kind of crucial if you want to make sense of rotating objects. Simple classical models don't always transition well to the relativistic regime. You have to account for what kind of forces are present in the model and see if those force laws transform properly between reference frames, otherwise you can expect to get some fairly strange and counterintuitive results at times.
     

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