Problem in deriving Killing equation

Question

I am studying derivation of Killing equation by Wald (also reading some other literature) but having some problem in understanding the math.

Let $\chi ^a$ is killing vector on the horizon $$\chi _{[a \nabla_{b}} \chi_{c}]=0$$ We will get 6 terms on the left hand side on expanding the above bracket (which I understand) and then by applying killing equation $\nabla_{a} \chi_{b}+\nabla_{b} \chi_{a}=0$ we will get $$\chi_{c} \nabla_{a} \chi_{b}=\chi_{b} \nabla_{a} \chi_{c}-\chi_{a} \nabla_{b} \chi_{c}$$ I understand this entire working. Now, How do we get the following equation? $$\chi_{c} \nabla_{a} \chi_{b}=-2 \chi_{[a \nabla_{b}]}\chi_{c}.$$

Second question: I came across another equation while deriving it $$\chi_{c}(\nabla^a \chi^b)(\chi_{a} \chi_{b})=-(\nabla^a \chi^b)(\chi_{a}\nabla_{b} \chi_{c})-(\nabla^b \chi^a)(\chi^b \nabla_{a} \chi_{c})$$ How do we get the following from the above equation? $$=-2(\chi_{a} \nabla^a \chi^b)(\nabla_{b} \chi_{c})$$

Another question: If $\chi^{a} \chi_{a}=0$ (which is indeed on the horizon) does it mean $\chi_{a}\nabla^{b} \chi^{a}=0$ as well?

Should I ask this question in mathematics stack exchange ??

Answer 1

Question one: this is just notation for the antisymmetric part of a tensor \begin{align} \chi_{[a}\nabla_{b]}\chi_c\equiv\frac{1}{2}(\chi_{a}\nabla_{b}\chi_c-\chi_{b}\nabla_{a}\chi_c) \end{align} See definition in Wald equation (2.4.4)

Then \begin{align} -2\chi_{[a}\nabla_{b]}\chi_c &\equiv-2\frac{1}{2}(\chi_{a}\nabla_{b}\chi_c-\chi_{b}\nabla_{a}\chi_c)\\ &=-\chi_{a}\nabla_{b}\chi_c+\chi_{b}\nabla_{a}\chi_c\\ &=\chi_{b}\nabla_{a}\chi_c-\chi_{a}\nabla_{b}\chi_c \end{align}