Explicit Einstein-tensor coefficient convention

Displayed formulas in this passage

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  $$\begin{array}{rl}{R}_{2,\mu \nu }=& {\partial }_{\rho }{{\Gamma }_2}^{\rho }{}_{\mu \nu }-{\partial }_{\nu }{{\Gamma }_2}^{\rho }{}_{\mu \rho }\\ & +{{\Gamma }_1}^{\rho }{}_{\rho \lambda }{{\Gamma }_1}^{\lambda }{}_{\mu \nu }-{{\Gamma }_1}^{\rho }{}_{\nu \lambda }{{\Gamma }_1}^{\lambda }{}_{\mu \rho }.\end{array}$$

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  $$\begin{array}{rl}{G}^{(2)}[h,h{]}_{\mu \nu }=& R_{2,\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }\left({\eta }^{ab}{R}_{2,ab}-h^{ab}{R}_{1,ab}\right)\\ & -\frac{1}{2}{h}_{\mu \nu }{\eta }^{ab}{R}_{1,ab}.\end{array}$$

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  $$H_{\mu \nu \sigma }={\partial }_{\mu }{h}_{\nu \sigma }+{\partial }_{\nu }{h}_{\mu \sigma }-{\partial }_{\sigma }{h}_{\mu \nu },$$

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  $${{\Gamma }_1}^{\rho }{}_{\mu \nu }=\frac{1}{2}{\eta }^{\rho \sigma }{H}_{\mu \nu \sigma },{{\Gamma }_2}^{\rho }{}_{\mu \nu }=-\frac{1}{2}{h}^{\rho \sigma }{H}_{\mu \nu \sigma }.$$

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  $$R_{1,\mu \nu }={\partial }_{\rho }{{\Gamma }_1}^{\rho }{}_{\mu \nu }-{\partial }_{\nu }{{\Gamma }_1}^{\rho }{}_{\mu \rho },$$

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  $$G^{(1)}[h{]}_{\mu \nu }=R_{1,\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }{\eta }^{ab}{R}_{1,ab},$$

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  $$G^{(1)}[2\sigma \eta {]}_{\mu \nu }=-2{\partial }_{\mu }{\partial }_{\nu }\sigma +2{\eta }_{\mu \nu }{\square }_{\eta }\sigma .$$

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  $${\partial }^{\mu }{T^{(0)}}_{\mu \nu }[{\varphi }_0]=(P_m{\varphi }_0){\partial }_{\nu }{\varphi }_0=0.$$