Covariant derivative of Killing vector and Riemann Tensor

[dwellexity]

I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. $\xi$ is a Killing vector.
I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$
I can't figure out a way to get the required form from my solution. Please suggest a way to get the required form from my solution.

 

[PeterDonis]

What happens when you raise the $\alpha$ index on each side?

 

[dwellexity]

What happens when you raise the $\alpha$ index on each side?

Just raising the $\alpha$ index doesn't solve the problem. There is still the minus sign and the order of $\mu$ and $\nu$ indices is wrong.

 

[PeterDonis]

There is still the minus sign and the order of $\mu$ and $\nu$ indices is wrong.

What happens when you swap those indices on the Riemann tensor?

 

[dwellexity]

What happens when you swap those indices on the Riemann tensor?

I do not know of any symmetry involving swapping two middle indices of the Riemann tensor. The symmetries I know involve first two or last two or pair of first two and last two.

 

[dwellexity]

@PeterDonis Did you delete the last post? I can't see it anymore. Anyway, I tried what you suggested and it didn't get me anywhere. Maybe I am doing something wrong. Could you please show me a few steps?

 

[JorisL]

The relation you have found so far is correct (see e.g. Carroll's book, eq 3.176).
I'm not sure if it can be rewritten using the Bianchi identity $R_{\mu[\nu\rho\sigma]}=0$ you can check out this question+the answers for inspiration.
http://math.stackexchange.com/quest...t-killing-vector-and-riemann-curvature-tensor

Edit:
As Peter remarks below I was unclear, I verified that your result is a valid expression.
However I'm not sure whether your desired expression is valid or not and give some pointers (which are likely irrelevant since you already used them to reach the result so far).

 

[PeterDonis]

Did you delete the last post?

Yes, because I realized I was misreading the index order in the second equation in your OP. Where did you obtain the first equation in your OP, the one you are trying to prove?

 

[JorisL]

Added an edit, somehow it seems I forgot to clarify what I was actually saying.

 

[dwellexity]

Yes, because I realized I was misreading the index order in the second equation in your OP. Where did you obtain the first equation in your OP, the one you are trying to prove?

It's a question in an assignment I have to do.

 

[vanhees71]

You should also be aware that there are many different sign conventions in different textbooks (or even different editions of the same textbook!) in the definition of the curvature tensor. This is very unfortunate, but you can't help it. There's a table of conventions in the famous book by Misner, Thorne, and Wheeler.

 

[PeterDonis]

It's a question in an assignment I have to do.

Then this whole thread should be in the homework forum. And you should have filled out the homework template. Please do not post assignment questions in the regular forums in future; that is against PF rules.

Edit: I have moved the thread to Advanced Physics Homework. dwellexity, please post the actual question from your assignment.