How Do Killing Vectors Affect Geodesic Deviation?

[Nikos]

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 . Is there any idea about the right process for the calculations. Thanks for your time!

 

[robphy]

Did you use the fact that V is a unit vector?

 

[George Jones]

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

$$ff_{;b}=-K_{c;b}K^{c}=K_{b;c}K^{c}$$

From $V^{a}=f^{-1}K^{a}$,

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

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

Regards,
George

 

[Nikos]

Thank you guys for your time . George thank you very much for the help and the nice derivation of the relation! Here is my answear for the previous relation http://www.photodump.com/direct/Bbking22/hyperrelation.jpg . I would like you to check it out and tell me if you agree.

 

[Nikos]

Can someone give some further help about the http://www.photodump.com/direct/Bbking22/hyperrelation.jpg , and tell me if that is the right way for the http://www.photodump.com/direct/Bbking22/departurefromgeodesity.jpg