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Geodesic equation

  1. Feb 20, 2008 #1
    1. The problem statement, all variables and given/known data
    I would like to manipulate the geodesic equation.

    2. Relevant equations
    The geodesic equation is usually written as
    [tex]k^{a}{}_{;b} k^{b}=\kappa k^{a}[/tex] (it is important for my purpose to keep it in non-affine form).
    It is clear that by contracting with the metric we may write alternatively
    [tex]k_{a ;b} k^{b} = \kappa k_{a}[/tex].
    What I would like to know is how to raise to a contravariant indices in the derivative on the left-hand side.

    3. The attempt at a solution
    If I had to guess, I would like to be able to write something like.
    [tex]k^{a ;b} k_{b}=\kappa k^{a}[/tex].
    Is this a valid form of the geodesic equation?
  2. jcsd
  3. Feb 20, 2008 #2


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    Sure. [tex]g^{a b} k_{;b}=k^{;a}[/tex].
  4. Feb 20, 2008 #3
    Thank you for your reply Dick.
    But how do you explain the lowering of the b index in the second factor on the left-hand side?
  5. Feb 20, 2008 #4


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    On the LHS you've got [tex]{k^a}_{;b}k^b=k^{a;c}g_{cb}k^b=k^{a;c}k_c=k^{a;b}k_b[/tex]
  6. Feb 20, 2008 #5
    Excellent! Thank you both very much.
  7. Feb 20, 2008 #6


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    The crucial point is that the covariant derivative transforms as a tensor, unlike say, the partial derivative.
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