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Momentarily Comoving Reference Frame

  1. Nov 4, 2011 #1
    - http://www.physicsinsights.org/glossary.html

    I'm often unsure whether people are using "frame" to mean a chart, or a basis field of tangent vectors, or a basis for an individual tangent space. Have I got this (more or less) right?

    (1) A MCRF of a pointlike object is a function that assigns to each event p in the object's worldline an orthonormal basis for the corresponding tangent space, TpM, such that the timelike basis tangent vector E0|p is equal to the proper velocity of the object at that event. At each event, the choice of spacelike tangent vectors is arbitrary, so there's no unique choice of MCRF for a given worldline.

    (2) A MCRF of a fluid is defined similarly but over a 4-dimensional region, with E0|p equal to the limit as V --> 0 of any function which outputs the average proper velocities of fluid particles in a suitably well-defined (to allow the necessary integration) set of events containing p and whose input is some measure V of the volume of that set.

    (3) A MCRF of a chart x is a function which assigns an orthonormal basis to the tangent space at each event p in the chart's domain such that

    [tex]E_0 |_p = \frac{\partial_0}{\sqrt{|g(\partial_0,\partial_0)|}},[/tex]

    supposing the chart induces one and only one timelike coordinate basis tangent vector at each point; otherwise I suppose there is no such family of MCRFs associated with the chart.
     
  2. jcsd
  3. Nov 4, 2011 #2

    Dale

    Staff: Mentor

    My understanding is that technically a frame is a field of orthonormal vectors at each point in the manifold:
    http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

    So the correct usage is essentially your second one. However, (and I am particularly guilty of this), it is pretty common to mis-use "frame" as a synonym for "coordinate system", particularly in SR where the distinction is less important. So your first usage is incorrect but very common mis-usage. I have never seen someone use frame as a basis for a tangent space at a single point in the manifold.
     
  4. Nov 4, 2011 #3
    Thank you, DaleSpam. I've been parsing "frame field" like "vector field"; that's what led me to imagine people sometimes meant a basis for the tangent space at a point. And, of course, the word "vector" is often used as short-hand for "vector field". But reading the Wikipedia article now, it seems like they really are using "frame" and "frame field" as synonyms.
     
  5. Nov 4, 2011 #4
    Also relevant: http://en.wikipedia.org/wiki/Proper_frame

    In Semi-Riemannian Geometry, O'Neill defines a "frame field" as an orthonormal basis field of tangent vectors (he also sometimes calls it simply a "frame"). He defines a Lorentz (=inertial) coordinate system as a time preserving isometry from Minkowsi space (conceived as an affine space) to the corresponding vector space R41. A frame (field) determines a family of such charts by the exponential map; they're the normal charts each specified by the orthonormal basis at a point. I wonder if these are the "frames of reference" which Wikipedia: Proper frame mentions, as it explicitly defines "frame of reference" as a coordinate system, or perhaps that article is just conflating the concepts of chart and frame.
     
  6. Nov 5, 2011 #5
    Ah, and here's more from O'Neill. It seems "field of reference" can also be used for a certain kind of vector field, rather than a chart:

    "An observer field on an arbitrary spacetime M is a timelike, future-pointing, unit vector field U. Each integral curve of U is indeed an observer paramatrized by proper time. Thus U describes a family of U observers filling M. [...] (Observer fields are called reference frames in [SW], [...]"

    ([SW] = Sachs & Wu: General Relativity for Mathematicians.)
     
    Last edited: Nov 5, 2011
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