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I'm trying to calculate a commutator of two covariant derivatives, as it was done in Caroll, on page 122. The problem is, I don't get the terms he does :-/

If ##\nabla_{\mu}, \nabla_{\nu}## denote two covariant derivatives and ##V^{\rho}## is a vector field, i need to compute ##[\nabla_{\mu}, \nabla_{\nu}]V^{\rho}##. Covariant derivative is defined as

[tex]\nabla_{\mu} = \partial_{\mu} + \Gamma^{\nu}_{\mu\lambda}[/tex]

Putting it into the definition of the commutator, one can write

[tex]

\begin{align}

[\nabla_{\mu}, \nabla_{\nu}]V^{\rho} &= \nabla_{\mu} \nabla_{\nu} V^{\rho} - \nabla_{\nu}\nabla_{\mu} V^{\rho} \nonumber \\ &=\partial_{\mu} (\nabla_{\nu} V^{\rho})+\Gamma^{\rho}_{\mu\sigma}(\nabla_{\nu}V^{\sigma})-\Gamma^{\lambda}_{\mu\nu}\nabla_{\lambda}V^{\rho}+(\mu \leftrightarrow \nu) \nonumber \\ &=\ldots \nonumber

\end{align}

[/tex]

What gives me problems is the 3rd term in the 2nd row. I don't know where this third term comes from. The expansion is even more problematic, Caroll expands these three terms into 7:

[tex]\ldots\partial_{\mu}\partial_{\nu}V^{\rho}+(\partial_{\mu}\Gamma^{\rho}_{\nu\sigma})V^{\sigma}+\Gamma^{\rho}{\gamma\sigma}\partial_{\mu}V^{\rho}-\Gamma^{\lambda}_{\mu\nu}\partial_{\lambda}V^{\rho}-\Gamma^{\lambda}_{\mu\nu}\Gamma^{\rho}_{\lambda\sigma}V^{\sigma}+\Gamma^{\rho}_{\mu\sigma}\partial_{\nu}V^{\sigma}+\Gamma^{\rho}_{\mu\sigma}V^{\lambda}-(\mu \leftrightarrow \nu)\ldots[/tex]

Any ideas would be very much appreciative.

:-)

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# Commutator of two covariant derivatives

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