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In Sean Carroll's Spacetime and Geometry, a formula is given as ${\nabla _\mu }{\nabla _\sigma }{K^\rho } = {R^\rho }_{\sigma \mu \nu }{K^\nu },$

where $K^\mu$ is a Killing vector satisfying Killing's equation ${\nabla _\mu }{K_\nu } +{\nabla _\nu }{K_\mu }=0$ and the convention of Riemann curvature tensor is

$\left[\nabla_{\mu},\nabla_{\nu}\right]V^{\rho}={R^\rho}_{\sigma\mu\nu}V^{\sigma}.$

So how to prove the this formula (the connection is Levi-Civita)?

2 Answers 2

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Permit me the use of Latin indices instead of Greek indices and the convention $\nabla_a K_b=K_{b;a} $. So we wish to prove $\newcommand{\Tud}[3]{{#1}^{#2}_{\phantom{#2}{#3}}}$ $\Tud{K}{a}{;b c} = \Tud{R}{a}{b c d} K^d$ where $\Tud{V}{a}{;b c} - \Tud{V}{a}{;c b} = \Tud{R}{a}{d c b} V^d$ and $K_{a ; b} + K_{b ; a} = 0$

Differentiating the last equation, we get $K_{a ; b c} + K_{b ; a c} = 0$ so, relabelling and summing, $K_{a ; b c} + K_{b ; a c} - K_{b ; c a} - K_{c ; b a} + K_{c ; a b} + K_{a ; c b} = 0$ hence, $K_{a; b c} + K_{a ; c b} = R_{b d a c} K^d + R_{c d a b} K^d$ By the interchange symmetry $R_{a b c d} = R_{c d a b}$, and raising indices, we get $\Tud{K}{a}{;b c} - \Tud{R}{a}{b c d} K^d = -(\Tud{K}{a}{;c b} - \Tud{R}{a}{c b d} K^d)$

On the other hand, by the first Bianchi identity and antisymmetry, we have $\Tud{R}{a}{d c b} = \Tud{R}{a}{b c d} + \Tud{R}{a}{c d b}$ Hence we get $\Tud{K}{a}{;b c} = \Tud{K}{a}{; c b} + \Tud{R}{a}{d c b} K^d = \Tud{K}{a}{;c b} + \Tud{R}{a}{b c d} K^d + \Tud{R}{a}{c d b} K^d$ and therefore $\Tud{K}{a}{;b c} - \Tud{R}{a}{b c d} K^d = \Tud{K}{a}{;c b} - \Tud{R}{a}{c b d} K^d$ The conclusion follows.

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    It follows from the last equation and the equation fourth from the bottom together.2011-12-31
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@C.R. This is my 'simpler proof'; I'm pretty sure it's correct, and simpler than Zhen's one as well.

From the first Bianchi identity [Carroll, (3.132)] $R_{\mu\nu\rho\sigma}+R_{\mu\rho\sigma\nu}+R_{\mu\sigma\nu\rho}=0$ we have that, for every vector $V^\rho$, $\nabla_{[\mu}\nabla_\nu V_{\rho]}=\tfrac{1}{6}\big(R_{\rho\alpha\mu\nu}+R_{\mu\alpha\nu\rho}+R_{\rho\mu\nu\alpha}\big)V^\alpha=0,$ where in the last equation I used the symmetry properties of the indices in the Riemann tensor to reduce it to the Bianchi identity. This is a very useful formula! I'm quite sure about the index placement thanks to the metric compatibility, $\nabla_\mu g_{\nu\rho}=0$ [Carroll, (1.32)], which implies that the metric $g_{\nu\rho}$ commutes with the covariant derivative $\nabla_\mu$. Now, we take the Killing vector $K^\mu$ and we expand the antisymmetrization in the previous equation: $0=6\nabla_{[\mu}\nabla_\nu K_{\rho]}=\nabla_\mu\nabla_\nu K_\rho + \nabla_\nu \nabla_\rho K_\mu +\nabla_\rho\nabla_\mu K_\nu- \nabla_\nu\nabla_\mu K_\rho-\nabla_\mu\nabla_\rho K_\nu-\nabla_\rho\nabla_\mu K_\nu.$ Now we use the Killing's equation [Carroll, (3.174)], $\nabla_{(\nu}K_{\rho)}=0$ or $\nabla_\nu K_\rho =-\nabla_\rho K_\nu$, to simplify the previous equation: $\nabla_\mu\,\nabla_\nu K_\rho=-\nabla_\mu\,\nabla_\rho K_\nu\quad \Rightarrow\quad 0=2\nabla_\mu\nabla_\nu K_\rho+(\nabla_\nu\nabla_\rho-\nabla_\rho\nabla_\nu)K_\mu+\nabla_\rho\nabla_\mu K_\nu-\nabla_\nu\nabla_\mu K_\rho.$ Again, using Killing's equations $\nabla_\rho\nabla_\mu K_\nu=-\nabla_\rho\nabla_\nu K_\mu$ and $\nabla_\nu\nabla_\mu K_\rho=-\nabla_\nu\nabla_\rho K_\mu$ we get: $0=2\big(\nabla_\mu\nabla_\nu K_\rho+(\nabla_\nu\nabla_\rho-\nabla_\rho\nabla_\nu)K_\mu\big)=2\big(\nabla_\mu\nabla_\nu K_\rho+R_{\mu\alpha\nu\rho}K^\alpha\big)$ $\nabla_\mu\nabla_\nu K_\rho=-R_{\mu\alpha\nu\rho}K^\alpha=R_{\rho\nu\mu\alpha}K^\alpha.$ Then, we can rise the index $\rho$ multiplying with the metric $g^{\rho\sigma}$ (and using the metric compatibility): $\nabla_\mu\nabla_\nu K^\sigma=R^\sigma_{\phantom{\sigma}\nu\mu\alpha}K^\alpha.$ That's all folks! I hope it would be helpful (and correct)!

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    I've left the field of math and physics and am no longer able to decipher these symbols. Sorry about your first answer. But someone else who has the same problem may one day find you answer helpful.2013-09-10