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Angle between space-like and time-like vectors

  1. May 3, 2005 #1
    I would like to learn about the angle between space-like vectors and time-like vectors. Is there anyone who can help me repeatly, please?
  2. jcsd
  3. May 3, 2005 #2


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    There's no such thing.Angles and directions can be thoroughly defined only for spaces with positively defined metric (because they're defined using a scalar product).Minkowski space doesn't have this property,be it curved or flat.

  4. May 3, 2005 #3


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    It seems the best you can do is determine that "a nonzero spacelike vector is orthogonal to a nonzero timelike one" if their Minkowski-dot product is zero.

    Angle measure is usually defined as the ratio of arc-length on a "circle" to the radius. With two future-timelike vectors, one can use the intercepted arc-length of a unit hyperbola to define the angle between two future-timelike vectors. (This is called the rapidity.) However, there is no such hyperbola for a timelike and a spacelike vector. (One could also think of an angle as a parameter in the Lorentz Transformation to "rotate" one vector into another... However, Lorentz Transformations preserve the timelike (or, respectively, spacelike or null) nature of a vector.)

    A while back I thought about defining the angle you seek in terms of the angle between a timelike vector and a timelike-vector-orthogonal-to-the-spacelike-vector...of course, all three vectors in a common plane. However, it seems that one needs to play around with signs to get things to be consistent... however, this scheme looked rather unnatural to me... and didn't seem to have an immediate physical or geometrical interpretation.
  5. Dec 3, 2009 #4
    I can say at "Lorentz space" if v and w at the same timecone than,

    g(v,w)= -ııvıı.ııwıı.chQ, Q>=0

    if v and w are not at the same timecone than,

    ı g(v,w) ı = ııvıı.ııwıı.chQ

    if v and w spacelike vectors,

    g(v,w)= ııvıı.ııwıı.cosQ, 0<=Q<=pi

    Note that if v and w are together timelike or spacelike vectors than v are not

    orthogonal to w ( if v orthogonal to w than a once is timelike the other is must be spacelike)
  6. Dec 4, 2009 #5
    If I've understood this right, the rapidity is a hyperbolic angle, which is defined a little differently to a circular angle. The absolute value of a hyperbolic angle corresponds to the area under the unit hyperbola (on both sides of the axis), rather than arc length. Rapidity is labelled u on the diagram at the top of the first link:

    http://www.chartwellyorke.com/sketchpad/downloads/Minkowski_Overview.pdf [Broken]
    Last edited by a moderator: May 4, 2017
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