(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm currently self-studying Carroll's GR book and get stuck by proving

the following identity:

[tex] K^\lambda \nabla _\lambda R = 0 [/tex]

where K is Killing vector and R is the Ricci Scalar

2. Relevant equations

Mr.Carroll said that it is suffice to show this by knowing:

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

Bianchi identity [tex] \nabla ^ \mu R_{\rho \mu} = \frac{1}{2} \nabla _\rho R [/tex]

and Killing equation [tex] \nabla _\mu K_\nu + \nabla _\nu K_\mu = 0 [/tex]

3. The attempt at a solution

The work I done so far :

[tex] K^\lambda \nabla _\lambda R = 2 K^\lambda \nabla ^\mu R_{\mu \lambda} = 2 \left( \nabla ^\mu R_{\mu \lambda} K^\lambda -R_{\mu \lambda} \nabla ^\mu K^\lambda \right) = 2 \nabla ^\mu \nabla _\sigma \nabla _\mu K^\sigma [/tex]

Note that [tex]R_{\mu \lambda} \nabla ^\mu K^\lambda =0 [/tex], since

[tex]R_{\mu \lambda} \nabla ^\mu K^\lambda = - R_{\mu \lambda} \nabla^\lambda K^\mu = -R_{\lambda\mu} \nabla^\lambda K^\mu = -R_{\mu \lambda} \nabla ^\mu K^\lambda[/tex]

And I can't get any further

Could someone help?? Thanks in advace

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# Homework Help: Killing Vector and Ricci curvature scalar

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