The actual slice metric

Displayed formulas in this passage

• XX-A@1#eq:induced · line 429

  $${\gamma }_{s,ij}=(g_s{)}_{ij}({\theta }_s,\mathbf{x})+(g_s{)}_{0i}({\theta }_s,\mathbf{x}){\partial }_j{\theta }_s+(g_s{)}_{0j}({\theta }_s,\mathbf{x}){\partial }_i{\theta }_s+(g_s{)}_{00}({\theta }_s,\mathbf{x}){\partial }_i{\theta }_s{\partial }_j{\theta }_s.$$

• XX-A@1#eq:gamma1 · line 443

  $$\begin{array}{rl}({\gamma }_1{)}_{ij}& =h_{ij}+2a^{2}H{u}_1{\delta }_{ij},\\ ({\gamma }_2{)}_{ij}& =k_{ij}+2u_1{\dot{h}}_{ij}+u_2{\partial }_t(a^2{\delta }_{ij})+{u_1}^2{{\partial }_t}^2(a^2{\delta }_{ij})\\ & +2h_{0i}{\partial }_j{u}_1+2h_{0j}{\partial }_i{u}_1-2{\partial }_i{u}_1{\partial }_j{u}_1.\end{array}$$

• XX-A@1#eq:gamma2 · line 443

  $$\begin{array}{rl}({\gamma }_1{)}_{ij}& =h_{ij}+2a^{2}H{u}_1{\delta }_{ij},\\ ({\gamma }_2{)}_{ij}& =k_{ij}+2u_1{\dot{h}}_{ij}+u_2{\partial }_t(a^2{\delta }_{ij})+{u_1}^2{{\partial }_t}^2(a^2{\delta }_{ij})\\ & +2h_{0i}{\partial }_j{u}_1+2h_{0j}{\partial }_i{u}_1-2{\partial }_i{u}_1{\partial }_j{u}_1.\end{array}$$

• unlabeled · line 424

  $${\iota }_s(\tau ,\mathbf{x})=({\theta }_s(\tau ,\mathbf{x}),\mathbf{x}).$$

• unlabeled · line 439

  $${\gamma }_s=a^2\delta +s{\gamma }_1+\frac{s^2}{2}{\gamma }_2.$$