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Geodesic equation

  1. Apr 1, 2012 #1
    1. The problem statement, all variables and given/known data
    If a general parameter ##t=f(s)## is used to parameterize a straight line in Euclidean space, then the geodesic equation takes the form ##\frac{d^2u^i}{dt^2}+\Gamma^i_{jk}\frac{du^j}{dt}\frac{du^k}{dt}=h(s)\frac{du^i}{dt}##, where ##h(s)=-\frac{d^2t}{ds^2}{(\frac{dt}{ds})}^{-2}##. Show that this reduces to the simple form ##t=f(s)## is used to parameterize a straight line in Euclidean space, then the geodesic equation takes the form ##\frac{d^2u^i}{dt^2}+\Gamma^i_{jk}\frac{du^j}{dt}\frac{du^k}{dt}=0## if and only if ##t=As+B##, where##A, B## are constants (##A##≠##0##)

    3. The attempt at a solution
    I can not prove the inverse statment, i.e., if the geodesic equation is of the form ##\frac{d^2u^i}{dt^2}+\Gamma^i_{jk}\frac{du^j}{dt}\frac{du^k}{dt}=0##, then ##t=As+B##.
     
  2. jcsd
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