I'm now stuck in the second part of problem 12 in Chapter 3. The problem is " Show that any Killing vector [itex]K^\mu[/itex] satisfies the following relations:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\nabla_\mu\nabla_\sigma K^\rho = R^\rho_{\sigma\mu\nu}K^\nu[/tex]

[tex]K^\lambda\nabla_\lambda R = 0[/tex]

Where R is Riemann tensor.

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.

[tex]\nabla_\mu\nabla_\sigma K^\mu = R_{\sigma\nu}K^\nu[/tex]

and the contracted Bianchi identity

[tex]\nabla_\mu(R^{\mu\nu}-1/2g^{\mu\nu}R)=0[/tex]

What I do is multiplying Killing vector to the contracted Bianchi identity and then I get to where I stuck:

[tex]1/2K^\mu\nabla_\mu R = K_\nu\nabla_\mu R^{\mu\nu} [/tex]

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

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# A problem from Sean Carroll's about Killing vectors

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