How to obtain reduced Einstein equations?

Question

It is well known that in wave coordinates, vacuum Einstein field equations are equivalent to the following so called reduced Einstein equations:

$$R_{\mu\nu}+g_{\sigma(\mu}\Gamma_{,\nu)}^{\sigma}=0.$$

But I don't know how to show this fact. The following are my questions and I'd appreciated it if someone could please help.

1. What does the notation $g_{\sigma(\mu}\Gamma_{,\nu)}^{\sigma}$ mean?

2. How to show this well known fact? If it is not easy to type, a reference showing how to use wave coordinates to obtain reduced Einstein equations is also appreciated.

Answer 1

My previous answer is wrong because it lowered the index $\sigma$ which breaks the summation convention. This shows that only the two lower indices $\mu$ and $\nu$ that are between the parentheses get symmetrized:

$$ g_{\sigma(\mu}{\Gamma^{\sigma}}_{,\nu)}=\frac{1}{2}( g_{\sigma\mu}{\Gamma^{\color{red}{\sigma}}}_{,\nu}+ g_{\sigma\nu}{\Gamma^{\color{red}{\sigma}}}_{,\mu} ). $$

Previous Answer

There are three indices $\mu,\sigma,\nu$ between the symmetrization parentheses. According to the link I gave $$ g_{\sigma(\mu}{\Gamma^{\sigma}}_{,\nu)}=\frac{1}{3!}( g_{\sigma\mu}{\Gamma^{\color{red}{\sigma}}}_{,\color{lightgreen}{\nu}}+ g_{\sigma\mu}{\Gamma^{\color{lightgreen}{\nu}}}_{,\color{red}{\sigma}}+ g_{\sigma\color{lightgreen}{\nu}}{\Gamma^{\color{red}{\sigma}}}_{,\mu}+ g_{\sigma\color{lightgreen}{\nu}}{\Gamma^{\mu}}_{,\color{red}{\sigma}}+ g_{\sigma\color{red}{\sigma}}{\Gamma^{\mu}}_{,\color{lightgreen}{\nu}}+ g_{\sigma\color{red}{\sigma}}{\Gamma^{\color{lightgreen}{\nu}}}_{,\mu})\,. $$ Just use all permutations of $\mu,\sigma,\nu\,.$