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Geodesics in a rotating coordinate system

  1. Jun 22, 2012 #1
    In a uniformly rotating coordinate system the trajectories of freely moving objects are influenced by an apparent centrifugal and Coriolis force. Is there a coordinate system or metric (or both) in which these trajectories are geodesics instead?
  2. jcsd
  3. Jun 22, 2012 #2


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    The property of being a geodesic doesn't depend on the coordinate system. The trajectories of freely moving objects are always geodesics, whatever coordinate system you use.

    In non-rotating Cartesian coordinates [itex]x = vt[/itex] is a geodesic, which in rotating Cartesian coordinates might become [itex]X \cos \omega T + Y \sin \omega T = VT[/itex]. In these coordinates [itex]X = VT[/itex] would not be a geodesic.

    I'm not sure if that answers your question.
  4. Jun 22, 2012 #3


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    The trajectories of freely moving objects will be geodesics in any coordinate system. The condition for a path being a geodesic is that there are no "real" forces influencing the path. In an inertial frame, there are no real and no apparent forces on a geodesic trajectory. In a non-inertial frame, such as your rotating frame, there may be apparent forces on a geodesic trajectory, but there are still no "real" forces.
  5. Jun 23, 2012 #4
    Hmm.. Thank you both. Must think about this some more.
  6. Jun 23, 2012 #5


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    I think the metric you are looking for is given in the last paragraph of chapter 2 in this:
    http://www.projects.science.uu.nl/igg/dieks/rotation.pdf [Broken]
    Last edited by a moderator: May 6, 2017
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