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Problem with killing vectors

  1. Apr 12, 2006 #1
    Here is a piece http://www.photodump.com/direct/Bbking22/departurefromgeodesity.jpg from "The large scale structure of spacetime" and there is noway for me to reproduce the relation for departure from geodesity and the previous relation :cry: . Is there any idea about the right process for the calculations. Thanks for your time!
     
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  3. Apr 12, 2006 #2

    robphy

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    Did you use the fact that V is a unit vector?
     
  4. Apr 12, 2006 #3

    George Jones

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    Since [itex]K[/itex] is a Killing vector, it satisfies the antisymmetry property [itex]K_{a;b}=-K_{b;a}[/itex]. From [itex]f^{2}=-K^{c}K_{c}[/itex] and antisymmetry,

    [tex]
    ff_{;b}=-K_{c;b}K^{c}=K_{b;c}K^{c}
    [/tex]

    From [itex]V^{a}=f^{-1}K^{a}[/itex],

    [tex]
    \begin{align*}
    V^{a}{}_{;b}V^{b} & =-f^{-3}f_{;b}K^{a}K^{b}+f^{-2}K^{a}{}_{;b}K^{b}\\
    & =-f^{-4}K_{b;c}K^{b}K^{c}K^{a}+f^{-2}K_{c;b}K^{b}g^{ca}.
    \end{align*}
    [/tex]

    The first term on the left vanishes because of the combination of antisymmetry and symmetry in [itex]K_{b;c}K^{b}K^{c}[/itex]. The second term on the left, when used with the [itex]ff_{;b}[/itex] equation, gives the desired result.

    Regards,
    George
     
    Last edited: Apr 12, 2006
  5. Apr 13, 2006 #4
  6. Apr 19, 2006 #5
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