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Natural Science Forum / Physics / Relativity / November 2004Interpreting the rotating disk metric
I ask for help in interpreting the space-time diagram of a rotating disk. Refer to this treatise, http://www.smcm.edu/nsm/physics/SMP03S/KeatingB.doc.pdf and go down to page 17, to the diagram labelled "Figure 4" (but it's actually figure 3). It shows a light flash emitting from point Y2 and travelling to points Y1 and Y3 (okay, maybe the character is gamma, or upsilon; whatever). It clearly shows the light flash reaching Y1 at an earlier time than the light flash reaching Y3. My question is this: it appears that the entire diagram is a depiction from the FoR/viewpoint of a non-rotating onlooker. Can anyone verify this? the text is not abundantly clear about that issue. And yes, this is at least the third time that I've broached this particular perplexing question. The text of the aforementioned treatise makes one thing perfectly clear: any attempt to define a "locus of simultaneity" around the perimeter of the *native* metric of a rotating disk, would be a fool's undertaking (to paraphrase). I might be using the term "metric" uncoolly, 'not sure. Renewing my opinion, it seems rather obvious to me that in the (screwy) native coordinate system of a rotating platform, light's behavior should be equanimitous over relatively short open spans about the perimeter (Sagnac notwithstanding) -- just as it is equanimitous in a roughly comoving inertial frame. I think that perhaps the Sagnac generality applies as it does only when there is at least one mirror reflection employed. -Ben Bean > My question is this: it appears that the entire diagram is a depiction from > the FoR/viewpoint of a non-rotating onlooker. Can anyone verify this? the > text is not abundantly clear about that issue. Yes, I'm pretty sure it's the non-rotating observer. > And yes, this is at least the third time that I've broached this particular > perplexing question. The text of the aforementioned treatise makes one thing > perfectly clear: any attempt to define a "locus of simultaneity" around the > perimeter of the *native* metric of a rotating disk, would be a fool's > undertaking (to paraphrase). The "native" metric is the metric of the quotient space (in the topological and geometrical sense) of spacetime modulo the worldlines of the material points of the disc. As such it has no well-defined time coordinate. Topologically this quotient is still a 2D disc but its quotient geometry has negative curvature. This explains the seeming inconsistency of having a negatively curved disc apparently embedded into the flat Minkowski spacetime *without* it being bent into a hyperboloid: it is not embedded at all, it is a quotient space, not a subspace. This quotient geometry is what Moller and many other textbooks calculate. Another way of describing this geometry can be seen from following the behaviour of objects sitting on the disc and rotating with it (ropes, measuring sticks, etc.): it is composed of infinitesimal patches of spacelike disc slices that are Lorentz-orthogonal to every helical material point worldline. The author of this paper is incorrect, BTW, when he says just before his Conclusion that "We therefore see that the curvature obtained by the methods suggested in Moller are a result of artificially 'forcing' the endpoints A and B in figure 4 together". Instead, the calculations in Moller - and scores of other books - compute the curvature of the quotient I mentioned above, that's why A and B are identified. There is nothing artificial about this. > Renewing my opinion, it seems rather obvious to me that in the (screwy) > native coordinate system of a rotating platform, light's behavior should be > equanimitous over relatively short open spans about the perimeter (Sagnac > notwithstanding) -- just as it is equanimitous in a roughly comoving > inertial frame. I think that perhaps the Sagnac generality applies as it > does only when there is at least one mirror reflection employed. Light's deviation from the usual rules it obeys in inertial systems typically increses with distance, mirror or not. Probably the best and very general article on the subject in by J. Anandan "Sagnac effect in relativistic and nonrelativistic physics" (1981) (10.1103/PhysRevD.24.338). It pretty much contains everything one needs to resolve the "paradox". This paper, and the Rizzi/Tartaglia paper, do not mention Anandan. Perhaps they hadn't read it. <sarcasm>Perhaps if they had, they wouldn't have written these papers.</sarcasm> Jan Bielawski Many many thanks to you all, especially Jan Bielawski! -BB > I ask for help in interpreting the space-time diagram of a rotating disk. > Refer to this treatise, [quoted text clipped - 15 lines] > undertaking (to paraphrase). I might be using the term "metric" uncoolly, > 'not sure. What is for you perplexing about it? > Renewing my opinion, it seems rather obvious to me that in the (screwy) > native coordinate system of a rotating platform, light's behavior should be > equanimitous over relatively short open spans about the perimeter (Sagnac > notwithstanding) -- just as it is equanimitous in a roughly comoving > inertial frame. If you calibrate your clocks immediately before your measurement to the current position, and they are for example along the perimeter, yes - because then you *made* an inertial reference system. > I think that perhaps the Sagnac generality applies as it > does only when there is at least one mirror reflection employed. It has nothing to do with mirrors, and everything with inertial systems. Harald |
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