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Hyperbolic Triangles

  1. May 16, 2014 #1
    Hello, I'm going through Landau and Lifshitz "The Classical Theory of Fields" this summer with a friend and in section 4 I've come to a bit of a math problem.

    Assume you have an inertial frame [itex]K'[/itex] moving at speed [itex]V[/itex] relative to an inertial frame [itex]K[/itex] in the [itex]x[/itex]-direction. In order for invariant intervals we require:

    (ct)^2 - x^2 = (ct')^2 - (x')^2

    For this to be true:

    x = x'\cosh{\Psi}+ct'\sinh{\Psi}

    Where [itex]\Psi[/itex] is the angle of rotation in the [itex]xt[/itex] plane. Which makes sense. Now if we just look at the origin of the [itex]K'[/itex] frame moving these can be reduced to:


    and dividing the equations yields:


    But the speed [itex]V[/itex] is given by [itex]V=\frac{x}{t}[/itex] so:


    The next part is what is a bit confusing. If this were regular trigonometry to find both [itex]\sin{\Psi}[/itex] and [itex]\cos{\Psi}[/itex] would just require the construction of a triangle. In this case however, they end up with very, very similar results except where the hypoteneuse would be they end up with something like this:


    If anyone can point me in the right direction of deriving these relationships either algebraically or using geometry that would be extremely helpful!
    Last edited: May 16, 2014
  2. jcsd
  3. May 16, 2014 #2


    Staff: Mentor

  4. May 16, 2014 #3


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  5. May 16, 2014 #4
    Okay hmm, assuming the Wikipedia section on "Comparison with circular functions" is correct then this is what I make of it all:


    Sorry for the mess, I've been working all day and am too tired to be perfect.
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