# Problem with killing vectors

1. Apr 12, 2006

### 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!

2. Apr 12, 2006

### robphy

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

3. Apr 12, 2006

### George Jones

Staff Emeritus
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

Last edited: Apr 12, 2006
4. Apr 13, 2006

### Nikos

5. Apr 19, 2006