Geometry in Rotating Reference Frames

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

  1. Pete It's not rocket surgery Registered Senior Member

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    You could construct a reference frame to make it any shape you like, but it would be very artificial. I think that the simplest ways of defining or enforcing simultaneity will always give a symmetrical result.

    [thread=98071]That thread[/thread] also deals with the state of the wheel when it is in constant motion, and related reference frames. rpenner's post, for example, explains that in the instantaneous inertial reference frame of any part of the wheel (such as Bottom, for example), the wheel is neither rigid nor moving simply. Further down in [post=2510505]posts 15 and 16[/post], I explicitly attempted to explore how the wheel's non-inertial rest frame could be defined.

    The links in rpenner's first post to that thread are also very relevant, if difficult to follow through (I get lost fairly quickly):
    Born coordinates define a coordinate system in which the wheel is stationary, which I think is derived from the Langevin observers like Bottom. There are some interesting issues with this system that I don't fully grasp. For example, see the discontinuity in a Langevin-observer's simultaneous slice of space-time, and the various distinct notions of distance.
    The Ehrenfest paradox is about both the problem of spinning up the wheel from rest, and the problem that the circumference of the wheel is more than pi times its diameter. Examination of the paradox seems to converge onto the difficulty of synchronizing clocks in a rotating reference frame.

    The equivalence principle of general relativity might provide some clues. I think that the wheel's rest frame has similarities to an inverted gravity field, a field that repels objects from the wheel's axel. This tells us that in our rotating reference frame, the axel is a special place, and that any natural coordinate system must be based on the axel, and symmetrical about the axel.

    For example, if all clocks on the wheel are artificially synchronized to a signal from the axel (similarly to GPS synchronization), and one spatial coordinate is the distance from the axel, then any system in which these definitions are uniform will describe a uniform circular wheel.


    Resources:
    Einstein's general theory of relativity: with modern applications in cosmology
    Section 5, Non-Inertial reference frames.

    Relativity in rotating frames (Guido Rizzi, Matteo Luca Ruggiero, 2004. Preview on Google Books).
    This looks like an excellent, readable, and complete coverage on exactly this topic. It includes in full the key historical and contemporary papers, and a transcript of a round table discussion between experts as they explore difficulties and disagreements over things like defining simultaneity and length in rotating frames.
    I'd like to find the time to get it from a library and read it properly.
     
    Last edited: Mar 31, 2010
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  3. Neddy Bate Valued Senior Member

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    Yes, absolutely, that makes perfectly good sense to me. I misunderstood your earlier post, because I read "circular motion" to mean "rotation in general." I should have been more careful in my interpretation.

    In your non-stationary frame (relative to the proton), the electron is not simply rotating around the proton in a circle, of course that is true. The centripital forces involved may very well be changing form all the time, also. I have no problem with that, or the shape of the orbit being elliptical in the non-stationary (relative to the proton) frame.

    However, my concern is to get the electron to describe the shape of its own orbit. Or, getting back to my slides, to get a point on the edge of the wheel to regard the shape of the wheel.

    I would think it must be circular, but it is very difficult to get all of the surrounding geometry to jibe with that concept. I am trying to get the tank tread, the road, and the tank itself to fit together with the circular wheel in its own frame, but it doesn't seem to fit very well.
     
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  5. Uno Hoo Registered Senior Member

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    In response to your post #1, I have a comment, or, perhaps, you could see it as a question.

    You say that you wish to examine the geometry of a rotating reference frame. Well, in terms of Special Relativity, we have to make an assumption germane to The Theory. I presume that you are writing in a favorable light to The Theory, and are not an Anti-Relativity heretic?

    According to The Special Theory Of Relativity, any examiner, or, in Einstein's quaint linguistic, any observer, must consider himself as being the reference-body of his own personal inertial reference frame. So, are you posing the query as being what the inertial observer would "see", or, if not, what is your reference-body and reference frame for the sense of your thread?
     
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  7. Pete It's not rocket surgery Registered Senior Member

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    I think that the tank, the road, and the tread are likely to be curved in most sensible description of wheel's rest frame. They might also be stretched, and there might even be some kind of singularity (where the circumferential speed in the reference frame exceeds c) that makes the tank and track of infinite length. Maybe.
     
  8. Neddy Bate Valued Senior Member

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    2,365

    Can you construct a reference frame for the wheel in which it is circular in its own frame, and yet is also generally consistent with my other slides? If you can, I'd be very interested to hear the explanation. I have to warn that I think there are some strong implications about the geometry in those slides.



    I like posts 15 and 16, and I see that you have already been thinking about this problem. Your idea of a singularity surrounding the wheel at some radial distance is very interesting.


    I still think rigidity is a completely different problem. Don't you agree that the rods on my wheel can all be perfectly rigid?



    I agree that the above represents a circular wheel. You have forced the simultaneity of the tank onto the wheel, which is perfectly reasonable. But this does not seem to be compatible with Slide 1. Bottom is about to be picked up by the wheel, and the top of the tread is very different from the bottom of the tread. Do you want these two parts of the tread to suddenly shift to be equal, albeit in opposite directions? In my geometry, this is not acheived until Slide 2. This is part of the strength of wheel being elliptical in its own frame, there are no sudden shifts for any contracted measuring rods.



    Excellent references. I hope they can provide a solution.
     
    Last edited: Mar 31, 2010
  9. Neddy Bate Valued Senior Member

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    2,365

    This is probably the most pertinent question. What the heck measurement method can the examiner rely upon? Maybe that's the whole problem.

    All I know is that the geometry I have presented here seems perfectly consistent with relativity. Yet it seems to run me into a position where the wheel is elliptical in its own frame. I would think that all of the slides should fit together seemlessly. It just seems like something is wrong...
     
  10. Pete It's not rocket surgery Registered Senior Member

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    I don't think so, no. In your slides (which are very impressive), there is always one part of the wheel at rest, while the rest of the wheel is in motion.

    Constructing a frame so that all parts of the wheel are at rest will make the tank, tread, and road look pretty ugly, I think.

    Yes, rigidity is a different problem.
    The things I mentioned address both this problem and the rigidity problem, so they're relevant and worth reading.

    There's no sudden shift involved, I think - it's about considering the whole situation from start to finish from a different perspective. Like starting from slide zero, and instead of transforming to slide 1 where Bottom is at rest, you instead transform to a slide in which all parts of the wheel are at rest.
     
    Last edited: Mar 31, 2010
  11. Pete It's not rocket surgery Registered Senior Member

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    I've had a lightbulb moment. We should expect symmetry if we consider all parts of the wheel to be equal... :idea: But! We don't have to! And in this case we are not. We're treating Bottom as special.

    For Bottom, the axel is certainly a special place, but it is not the most special. The most special place for Bottom is Bottom's own location. I think that's why we're getting an elliptical wheel.

    Bottom's world is not symmetrical around Bottom (because that special place, the axel, is off to one side). So when we construct a reference frame around Bottom (ie Bottom's location is a defining feature of the frame), I think it's natrual that we get some assymetry.

    So, I think:
    • that your slides do represent a meaningful and valid non-inertial reference frame,
    • that the elliptical representation of the wheel is correct in that reference frame,
    • that the surprising asymmetry is because of the artifical choice of Bottom as our reference point, and
    • that a precise and rigorous description of the frame including issues of simultaneity is just an interesting sidetrack for further study.
     
    Last edited: Mar 31, 2010
  12. Neddy Bate Valued Senior Member

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    2,365
    Sorry, I don't think I answered this properly yesterday. I answered right before sleep, and I was quite tired.

    In slides 1-5, The observer/examiner is always located on the edge of the wheel. His frame is not inertial. The reference body which is at rest for him is the wheel itself. But other, moving reference bodies also exist, such as the tank tread, the road, and the tank itself. These bodies are useful to help make sure no contradictions arise. I am questioning the geometry in Slides 1-5.

    Slide 0 was only provided for reference, and I am not really questioning that one.
     
    Last edited: Mar 31, 2010
  13. Neddy Bate Valued Senior Member

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    2,365
    How does curving the road, tank, and tread help resolve the geometry? If it could work, I'd be very interested in trying it. Are you trying to get the top part of the wheel in Slide 1 to have normal size rods? If you could explain what you are trying to do, I might be able to draw it from your descriptions.

    As for the singularity, I think that only applies to masses on the wheel. They cannot exceed a certain radius because they cannot travel at c. However, masses not on the wheel can be any distance away from the axle. The light from them can eventually reach the wheel. These are side issues, albeit interesting ones.
     
  14. Neddy Bate Valued Senior Member

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    2,365

    Thanks. If you think of Slides 1-5 as inertial frames, then you can say that there is always one part of the wheel at rest, and the rest of the wheel in motion. However, I was trying for the geometry of the rotating frame. If you look at Slides 1-5 that way, then all parts of the wheel should be at rest relative to each other.

    Wait, that might be part of the answer. Consider Slide 1 first as the inertial frame of the road. The top of the tread is in motion relative to Bottom, but the bottom part of the tread is not. This is evident from the relative sizes of their measuring rods. As soon as Bottom gets picked up onto the wheel, the drawing should change a little. All parts of the wheel should have the same size measuring rods, and the wheel should be circular. Yet the top part of the tread should still be contracted and the bottom tread un-contracted, somehow, because the relative motion of the tread has not changed much after Bottom gets picked up. Hmmm, I'll have to try this.


    That's alright, I still want to know it, if possible.


    But Slide 1 is useful, because it shows the instant where Bottom changes from the road frame to the wheel frame. There should not be a sudden shift, but i do think something should change, as explained above.
     
    Last edited: Mar 31, 2010
  15. Neddy Bate Valued Senior Member

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    2,365
    I like your idea, and it may end up being the best explanation we can find. But as I explained in my last couple of posts, I think I should try one more time to make the wheel circular.

    I think:
    • Bottom considers the rods on the wheel to be at rest,
    • Bottom considers the rods on the wheel to all be full sized,
    • Bottom considers the wheel to be circular,
    • Bottom considers the tread, tank and road to be pretty much as they are in Slides 1-5.
     
    Last edited: Mar 31, 2010
  16. Neddy Bate Valued Senior Member

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    2,365
    I was able to do all the things in my above list, except I still have not yet been able to make the wheel circular in its own frame.

    Slide 1A:
    Bottom is represented by a small white circle at the bottom of the left wheel.

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    In this slide, Bottom is still in the same reference frame as the road. The measuring rod nearest him on the wheel is not contracted, in his immediate vicinity. It does become contracted toward the top of the wheel, however. This is represented by the 115 degree arc along the edge of the wheel. For comparison, you can refer to a full size rod along the road.

    The measuring rod near the top of the wheel is contracted to a large degree. This is indicated by the 65 degree arc along the edge of the wheel. Notice how the length contracted rods on the wheel are reasonably consistent with the length contracted rods on the top and bottom parts of the tread.

    ---

    Slide 1B:
    Bottom is still represented by a small white circle at the bottom of the left wheel.

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    Now Bottom is in the rotating reference frame of the wheel. The measuring rods on the wheel are no longer contracted, yet each one of them fits into a 90 degree arc on the wheel, because of non-Euclidean geometry. This is represented by the zig-zag lines in each quadrant, which if unfolded would make each quadrant as long as one measuring rod on the road.

    Bottom does not consider the wheel rod near him to have become smaller, even though it went from 115 degrees to 90 degrees. On the contrary, the rod has become longer, and so have all of the other rods on the wheel.

    One odd thing is that the top rod on the wheel is much longer than the adjacent contracted rod on the top of the tread, even though these two rods are touching each other, end to end. This seems like a problem, but it starts to disappear in Slide 2, and is gone by Slide3. Note that Slide 3 should be updated to have zig-zag lines, very similar to an upside version of Slide 1B.

    It looks like some progress has been made, but the shape is still coming out elliptical instead of circular. Perhaps the unfolding of the zig-zag lines makes the wheel round again in its own frame?
     
  17. Pete It's not rocket surgery Registered Senior Member

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    10,167
    To do that, I think that you will have to start from slide zero. From slide 1, Bottom is given preferential treatment, which introduces the asymmetry. Start from slide zero again, and don't single out any part of the wheel for special treatment except the axle.
     
  18. Pete It's not rocket surgery Registered Senior Member

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    I think that this post was just wrong.
     
  19. Pete It's not rocket surgery Registered Senior Member

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    You're right. There is no horizon or singularity at the critical radius.
    Beyond that radius is like an ergosphere around a rotating black hole - in our artificial reference frame, everything beyond that radius has to be rotating (ie the wheel can't reach or exceed the critical radius, as you said).
     
  20. James R Just this guy, you know? Staff Member

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    Moderator note: posts regarding GPS have been deleted as off-topic for the current thread.
     
  21. Neddy Bate Valued Senior Member

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    2,365
    I think this would be a lot like Slide#0, except the whole slide should be rotating around the axle. Furthermore, instead of contracted rods at the edge of the wheel, there should be full-length rods. So I'll have to put zig-zag lines to represent the non-Euclidean geometry, because a full-length rod somehow fits in one quadrant on the edge of the wheel.

    This does create a circular wheel shape, and it also suggests that the simultaneity of the rotating frame is the same as that of the tank itself. Of course clocks on the edge of the wheel are always slower than a clock at the axle, but other than this rate difference, simultaneity is essentially the same.
     
  22. Neddy Bate Valued Senior Member

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    2,365
    Here is my newest concept of the rotating frame's geometry, in animated GIF format. Finally I have the wheel circular in its own frame!

    IMPORTANT: Direct your attention to the little white circle, centered at the bottom of the animation. There is a reason that circle is always in the same place in the animation:

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    The little white circle represents a point that is making its way all the way around the the tank tread. Sometimes the point is on a wheel, and other times the point is not on a wheel. Thus, the geometry is sometimes from an inertial frame, and other times the geometry is from a rotating frame.

    Next, take note of when the wheel shape is circular, and when it is elliptical. It is circular in the rotating frames, and elliptical in the inertial frames. Also, whenever the wheel is elliptical, the length of the tank is also contracted, as represented by the four yellow measuring rods which span from one axle to the other.

    Finally, the really strange thing is the non-Euclidean geometry. Whenever a measuring rod is represented by a zig-zag line, the measuring rod is actually as long as the zig-zag would be if it were unfolded like an accordion. The odd thing about the rotating frame geometry is that basically everything is non-Euclidean except the yellow measuring rods between the axles. This means the wheel circumference, road, and tread are all full length in the rotating frame, even though none of this would normally fit either on the wheels, or in between the wheels.

    If anyone has any objections to this geometry, please let me know. I don't see any inconsistencies, but it is a little strange.
     

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