A problem Sean Carroll's book about Killing vectors

[shichao116]

Homework Statement

(This is problem 12 of Chapter 3 in Sean Carroll's book: Spacetime and Geometry: An Introduction to General Relativity.)
Show that any Killing vector K^\mu satisfies the following relations:
\nabla_\mu\nabla_\sigma K^\rho = R^\rho_{\sigma\mu\nu}K^\nu
K^\lambda\nabla_\lambda R = 0

Where R is Riemann tensor.

Homework Equations

The Attempt at a Solution

I can prove the first one by using the definition of Riemann tensor, i.e. the commutator of two covariant derivatives, Killing equations associated with Killing vector, and Bianchi identity.

But for the second one, in the book it is said that we can prove it by contracting the first equation, i.e.
\nabla_\mu\nabla_\sigma K^\mu = R_{\sigma\nu}K^\nu
and the contracted Bianchi identity
\nabla_\mu(R^{\mu\nu}-1/2g^{\mu\nu}R)=0

What I do is multiplying Killing vector to the contracted Bianchi identity and then I get to where I stuck:
1/2K^\mu\nabla_\mu R = K_\nu\nabla_\mu R^{\mu\nu}

obviously the left hand side is what we need to prove to be zero. But I failed to show the right hand side to be zero after tried many ways.

Can anyone give me some clue how to do that ?

Thanks a lot

 

[fzero]

What I do is multiplying Killing vector to the contracted Bianchi identity and then I get to where I stuck:
1/2K^\mu\nabla_\mu R = K_\nu\nabla_\mu R^{\mu\nu}

Use the product rule

K_\nu\nabla_\mu R^{\mu\nu} =\nabla_\mu( K_\nu R^{\mu\nu}) - R^{\mu\nu}\nabla_\mu K_\nu.

Then you should be able to use the first identity and the Killing equation to see that both terms vanish.