# Covariant derivative, timelike vector field (Kerr- Schild Class)

1. Apr 12, 2010

### Lurian

Hi!
It seems I have some major problems in understanding covariant derivatives in concrete calculations. I discovered in an article dealing with Kerr- Schild classes the following:
In calculating the Riemann Tensor from the Christoffel- symbols one gets the following expression:

$$C^{a}_{ac}$$=$$1/2$$$$\left\{$$$$\partial_{a}\left(k^{a}k_{c}f\right)+\partial_{c}\left(k^{a}k_{a}f\right)-\partial^{a}\left(k_{c}k_{a}f\right)+fk^{a}\left(k\partial\right)\left(k_{c}k_{a}\right)\right\}$$

where since $$k^{a}$$ is lightlike $$k^{a}k_{a}=0$$ holds,and since we are dealing with Kerr- Schild class the indices are raised and lowered by $$\eta^{ab}$$ the Lorentz metric and $$\left(k^{a}\nabla_a\right)k^{c}=\left(k^{a}\partial_a\right)k^{c}=0$$ geodetic and f is a scalar function.
This expression should give zero which I cannot reproduce. I used the product rule on every term except the second one which is obviously zero. I get the following result

$$C^{a}_{ac}=k_{c}k^{a}\partial_{a}f+k_{c}f\partial_{a}k^{a}+fk^{a}\partial_{a}k_{c}-f k_{a}\partial^{a}k_{c}-fk_{c}\partial^{a}k_{a}-k_{c}k_{a}\partial^{a}f+f^{2}k^{a}k_{c}\left(k\partial\right)k_{a}$$

where all terms should cancel to zero. Can someone explain this to me, would be very grateful. I think one need not know much about Kerr- Schild classes to explain this.
Thank you

Last edited by a moderator: Apr 12, 2010