Inverse metric and raw lapse

Displayed formulas in this passage

• XX-D@1#eq:inverse · line 406

  $$\begin{array}{rlrl}{R}_0& ={\bar{g}}^{-1},& R_1& =-{\bar{g}}^{-1}h{\bar{g}}^{-1},\\ R_2& =2{\bar{g}}^{-1}h{\bar{g}}^{-1}h{\bar{g}}^{-1}-{\bar{g}}^{-1}k{\bar{g}}^{-1}.& & \end{array}$$

• XX-D@1#eq:S · line 421

  $$\begin{array}{rl}{S}_1& =-{R_1}^{\mu \nu }{\partial }_{\mu }\bar{\varphi }{\partial }_{\nu }\bar{\varphi }-2{R_0}^{\mu \nu }{\partial }_{\mu }\bar{\varphi }{\partial }_{\nu }\phi ,\\ S_2& =-{R_2}^{\mu \nu }{\partial }_{\mu }\bar{\varphi }{\partial }_{\nu }\bar{\varphi }-4{R_1}^{\mu \nu }{\partial }_{\mu }\bar{\varphi }{\partial }_{\nu }\phi \\ & -2{R_0}^{\mu \nu }{\partial }_{\mu }\phi {\partial }_{\nu }\phi -2{R_0}^{\mu \nu }{\partial }_{\mu }\bar{\varphi }{\partial }_{\nu }\psi .\end{array}$$

• XX-D@1#eq:N12 · line 442

  $$N_1=-\frac{S_1}{2v^3},N_2=\frac{3{S_1}^2}{4v^5}-\frac{S_2}{2v^3}.$$

• XX-D@1#eq:Sexplicit · line 449

  $$\begin{array}{rl}{S}_1& =v^2{h}_{00}+2v\dot{\phi },\\ S_2& =2v^2{h_{00}}^2-2v^2{a}^{-2}{h}_{0i}{h}_{0i}+v^2{k}_{00}+4v{h}_{00}\dot{\phi }-4v{a}^{-2}{h}_{0i}{\partial }_i\phi \\ & +2{\dot{\phi }}^2-2a^{-2}{\partial }_i\phi {\partial }_i\phi +2v\dot{\psi }.\end{array}$$

• unlabeled · line 402

  $${g_s}^{-1}=R_0+s{R}_1+\frac{s^2}{2}{R}_2.$$

• unlabeled · line 416

  $$-{g_s}^{-1}(\mathrm{d}{\varphi }_s,\mathrm{d}{\varphi }_s)=v^2+s{S}_1+\frac{s^2}{2}{S}_2,$$