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Hyperbolic angle

  1. Jun 16, 2009 #1
    Is there a general way of writing the Lorentz transformation for (2+1) dimension or higher, in terms of its hyperbolic angle, sinh and cosh ?
     
  2. jcsd
  3. Jun 16, 2009 #2

    Mentz114

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    Gold Member

    This is a boost in the x direction with velocity [itex]\beta[/itex]
    [tex]\left[ \begin{array}{ccc}
    \cosh(\beta) & \sinh(\beta) & 0 \\\
    \sinh(\beta) & \cosh(\beta) & 0 \\\
    0 & 0 & 1 \end{array} \right]
    [/tex]
     
  4. Jun 16, 2009 #3
    Lut, isn't that a boost in the -x direction?
     
  5. Jun 16, 2009 #4
    What about in a 2D plane ? I don't think it's just a matrix multiplication between x-axis boost and y-axis boost, or is it ? At least, addition of velocity is not the case.
     
    Last edited: Jun 16, 2009
  6. Jun 16, 2009 #5

    jtbell

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    Staff: Mentor

    For three dimensions (but without hyperbolic angles) see Wikipedia. Scroll down to where it says "More generally for a boost in an arbitrary direction."
     
  7. Jun 16, 2009 #6
    Yes, unless I'm mistaken, a boost in the positive x direction (output frame moving in the positive x direction wrt the input frame) has minus signs in front of the sinhs. Also, I think the input for cosh and sinh should be the velocity parameter or "rapidity", sometimes written phi, which is artanh(beta) = artanh(v/c), rather than the velocity itself - artanh being the inverse hyperbolic tangent.

    http://en.wikipedia.org/wiki/Rapidity
     
  8. Jun 16, 2009 #7

    DrGreg

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    Under the convention that a 4-vector is written as

    [tex]\left[ \begin{array}{c}
    ct \\\
    \textbf{x}
    \end{array} \right]
    [/tex]​

    (where x is the spatial 3-vector) it is

    [tex]\left[ \begin{array}{ccc}
    \cosh \psi & & -\textbf{e}^T \sinh \psi \\\
    -\textbf{e} \sinh \psi & & \textbf{I} + (\cosh \psi - 1) \textbf{ee}^T}
    \end{array} \right]
    [/tex]​

    where [itex]\textbf{e} c \tanh \psi[/itex] is the 3-velocity vector of the boost (e being a unit 3-vector in the spatial direction of the velocity.)

    If you want an answer entirely in trig-angles and hyperbolic-angles, write e in spherical polar coordinates:

    [tex]\textbf{e} = \left[ \begin{array}{c}
    \cos \phi \sin \theta \\\
    \sin \phi \sin \theta \\\
    \cos\theta
    \end{array} \right]
    [/tex]​

    and expand the matrix!
     
  9. Jun 17, 2009 #8

    Mentz114

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    Gold Member

    Only for retardyons moving at sub-light speeds:wink:.

    You're right, sinh(beta) is negative for beta < 1.
     
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