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What is the geodesic in this case?

  1. May 23, 2015 #1
    1. The problem statement, all variables and given/known data

    Using the geodesic equation, find the conditions on christoffel symbols for ##x^\mu(\tau)## geodesics where ##x^0 = c\tau, x^i = constant##.
    Show the metric is of the form ##ds^2 = -c^2 d\tau^2 + g_{ij}dx^i dx^j##.

    2. Relevant equations


    3. The attempt at a solution

    The geodesic equation is
    [tex]\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha \beta} \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0 [/tex]
    [tex] \Gamma^\mu_{\alpha \beta} = \frac{1}{2} g^{\mu \gamma} \left( \partial_\alpha g_{\gamma \beta} + \partial_\beta g_{\alpha \gamma} - \partial_\gamma g_{\alpha \beta} \right)[/tex]

    For ##x^0 = c\tau##, we have that ##\Gamma^0_{00} = 0##. This means that ##\partial_0 g_{\gamma 0} = \frac{1}{2} \partial_\gamma g_{00}##. How does this help??
     
  2. jcsd
  3. May 25, 2015 #2
  4. May 31, 2015 #3
    bumpp
     
  5. Jun 1, 2015 #4
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