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I have a quick question, in the derivation of the Raychaudhuri equation in Wald: General Relativity on page 218, equation 9.2.10, an identity for $\xi^a \nabla_a B_{ab}$ is derived, where $B_{ab} = \nabla_b \xi_a$ and $\xi^a$ is a timeline geodesic. From the second to the third line, the expression $\nabla_b(\xi^c \nabla_c \xi_a)$ is assumed to vanish. Why is that? Is that just the geodesic equation? Does this hold even though $\xi_a$ is a covariant vector?

Any help will be appreciated.

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That term vanishes for the reason that you suspect. It is simply the geodesic equation. A geodesic $\xi^a$ is a curve defined in such a way that $\nabla_\xi\xi^c=\xi^a\nabla_a\xi^c=0.$ If your concern is that the object inside the covariant derivative is $\xi_a$ rather than $\xi^a$ then you only need to use one more fact which is that the metric is preserved by the covariant derivative, i.e. $\nabla_a g_{bc} = 0$. Therefore: $\xi^a\nabla_a\xi_c = \xi^a\nabla_a g_{bc}\xi^b = g_{bc}\xi^a\nabla_a\xi^b = 0.$

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  • $\begingroup$ Thank you! One additional question, the expansion is defined in Wald as $\Theta = B^{ab} h_{ab}$, where $h_{ab} = g_{ab}+\xi_a \xi_b$. Is this equivalent to the definition $\Theta = \nabla_a \xi^a$? $\endgroup$
    – eb1612
    May 8, 2018 at 18:34
  • $\begingroup$ I don't have Wald with me, and I don't recall these variables. It might be best to open another question rather than asking in comments. $\endgroup$
    – enumaris
    May 8, 2018 at 18:36

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