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Solution help for the Geodesic Equation

  1. Mar 29, 2009 #1
    Let’s say an asteroid is about to enter earth’s atmosphere(it will burn up of course). The initially sitting is at:

    [tex]point = [x_{0},y_{0},z_{0},t_{0}] = [r_{0},\\theta_{0},\\phi_{0},t_{0}][/tex]

    With a 4-velocity:

    [tex]V = [v_{1},v_{2},v_{3},v_{4}][/tex]

    The momentum at 0+

    [tex]p_{0+} = mass*V = mass*[v_{1},v_{2},v_{3},v_{4}][/tex]

    The Riemann Geometry is Riemann Space with Torsion = 0. The metric is the Schwarzchild metric. Affinity is the Christoffel Symbol. Curvature is non-zero. And, the Ricci tensor is null.

    My question is how do you formulate [tex]x^{i}[/tex] to attempt solve for the path, "s" in the second-order geodesic differential equation from the given initial 4-velocity vector and starting point?
     
    Last edited: Mar 30, 2009
  2. jcsd
  3. Apr 2, 2009 #2
    I am looking for how to solve the geodesic path, I. Something like this except taking into account initial conditions:

    [tex]I = \int[(1-2m/r)^{-1} + r^{2}(d\theta/dr)^{2} + r^{2}(sin\theta)^{2}(d\phi/dr)^{2} - (1-2m/r)(dt/dr)^{2}]^{1/2} dr[/tex]
     
    Last edited: Apr 2, 2009
  4. Apr 8, 2009 #3
    The geodesic is a graph of many curves. You have to go piece-meal or one part at a time. Use the Geodesic equation to lesson the amount of unknowns:
     
    Last edited: Apr 8, 2009
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