Explicit Einstein-tensor coefficients

Displayed formulas in this passage

• VII-A@1#eq:RicciConvention · line 2189

  $$R_{\mu \nu }={\partial }_{\rho }{{\Gamma }^{\rho }}_{\mu \nu }-{\partial }_{\nu }{{\Gamma }^{\rho }}_{\mu \rho }+{{\Gamma }^{\rho }}_{\rho \lambda }{{\Gamma }^{\lambda }}_{\mu \nu }-{{\Gamma }^{\rho }}_{\nu \lambda }{{\Gamma }^{\lambda }}_{\mu \rho }.$$

• VII-A@1#eq:Gamma1 · line 2200

  $$\begin{array}{rl}{{\Gamma }^{(1)\rho }}_{\mu \nu }[h]& =\frac{1}{2}{\eta }^{\rho \sigma }\left({\partial }_{\mu }{h}_{\nu \sigma }+{\partial }_{\nu }{h}_{\mu \sigma }-{\partial }_{\sigma }{h}_{\mu \nu }\right),\\ {{\Gamma }^{(2)\rho }}_{\mu \nu }[h,h]& =-\frac{1}{2}{h}^{\rho \sigma }\left({\partial }_{\mu }{h}_{\nu \sigma }+{\partial }_{\nu }{h}_{\mu \sigma }-{\partial }_{\sigma }{h}_{\mu \nu }\right).\end{array}$$

• VII-A@1#eq:Gamma2 · line 2200

  $$\begin{array}{rl}{{\Gamma }^{(1)\rho }}_{\mu \nu }[h]& =\frac{1}{2}{\eta }^{\rho \sigma }\left({\partial }_{\mu }{h}_{\nu \sigma }+{\partial }_{\nu }{h}_{\mu \sigma }-{\partial }_{\sigma }{h}_{\mu \nu }\right),\\ {{\Gamma }^{(2)\rho }}_{\mu \nu }[h,h]& =-\frac{1}{2}{h}^{\rho \sigma }\left({\partial }_{\mu }{h}_{\nu \sigma }+{\partial }_{\nu }{h}_{\mu \sigma }-{\partial }_{\sigma }{h}_{\mu \nu }\right).\end{array}$$

• VII-A@1#eq:R1 · line 2230

  $${R_{\mu \nu }}^{(1)}[h]={\partial }_{\rho }{{\Gamma }^{(1)\rho }}_{\mu \nu }[h]-{\partial }_{\nu }{{\Gamma }^{(1)\rho }}_{\mu \rho }[h],$$

• VII-A@1#eq:R2 · line 2238

  $$\begin{array}{rl}{R_{\mu \nu }}^{(2)}[h,h]& ={\partial }_{\rho }{{\Gamma }^{(2)\rho }}_{\mu \nu }[h,h]-{\partial }_{\nu }{{\Gamma }^{(2)\rho }}_{\mu \rho }[h,h]\\ & +{{\Gamma }^{(1)\rho }}_{\rho \lambda }[h]{{\Gamma }^{(1)\lambda }}_{\mu \nu }[h]-{{\Gamma }^{(1)\rho }}_{\nu \lambda }[h]{{\Gamma }^{(1)\lambda }}_{\mu \rho }[h].\end{array}$$

• VII-A@1#eq:r1 · line 2253

  $$\begin{array}{rl}{r}^{(1)}[h]& ={\eta }^{\mu \nu }{R_{\mu \nu }}^{(1)}[h],\\ r^{(2)}[h,h]& ={\eta }^{\mu \nu }{R_{\mu \nu }}^{(2)}[h,h]-h^{\mu \nu }{R_{\mu \nu }}^{(1)}[h].\end{array}$$

• VII-A@1#eq:r2 · line 2253

  $$\begin{array}{rl}{r}^{(1)}[h]& ={\eta }^{\mu \nu }{R_{\mu \nu }}^{(1)}[h],\\ r^{(2)}[h,h]& ={\eta }^{\mu \nu }{R_{\mu \nu }}^{(2)}[h,h]-h^{\mu \nu }{R_{\mu \nu }}^{(1)}[h].\end{array}$$

• VII-A@1#eq:G1explicit · line 2265

  $$\begin{array}{rl}{G_{\mu \nu }}^{(1)}[h]& ={R_{\mu \nu }}^{(1)}[h]-\frac{1}{2}{\eta }_{\mu \nu }{r}^{(1)}[h],\\ {G_{\mu \nu }}^{(2)}[h,h]& ={R_{\mu \nu }}^{(2)}[h,h]-\frac{1}{2}{\eta }_{\mu \nu }{r}^{(2)}[h,h]-\frac{1}{2}{h}_{\mu \nu }{r}^{(1)}[h].\end{array}$$

• VII-A@1#eq:G2explicit · line 2265

  $$\begin{array}{rl}{G_{\mu \nu }}^{(1)}[h]& ={R_{\mu \nu }}^{(1)}[h]-\frac{1}{2}{\eta }_{\mu \nu }{r}^{(1)}[h],\\ {G_{\mu \nu }}^{(2)}[h,h]& ={R_{\mu \nu }}^{(2)}[h,h]-\frac{1}{2}{\eta }_{\mu \nu }{r}^{(2)}[h,h]-\frac{1}{2}{h}_{\mu \nu }{r}^{(1)}[h].\end{array}$$

• unlabeled · line 2221

  $$\Gamma [\eta +\kappa h+{\kappa }^{2}k]=\kappa {\Gamma }^{(1)}[h]+{\kappa }^2({\Gamma }^{(1)}[k]+{\Gamma }^{(2)}[h,h])+\mathcal{O}({\kappa }^3).$$

• unlabeled · line 2283

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

• unlabeled · line 2292

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