Clock embedding and induced metric

Displayed formulas in this passage

• XX-D@1#eq:u · line 476

  $$\begin{array}{rl}{u}_1& =-\frac{\phi }{v}=-\frac{t}{A}\phi ,\\ u_2& =-\frac{\psi +2u_1\dot{\phi }+\dot{v}{u_1}^2}{v}\\ & =-\frac{t}{A}\psi +\frac{2t^2}{A^2}\phi \dot{\phi }+\frac{t}{A^2}{\phi }^2.\end{array}$$

• XX-D@1#eq:gamma1 · line 502

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

• XX-D@1#eq:gamma2 · line 502

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

• unlabeled · line 469

  $${\iota }_s(t,\mathbf{x})=\left(t+s{u}_1+\frac{s^2}{2}{u}_2,\mathbf{x}\right).$$

• unlabeled · line 488

  $$T_s=t+s\frac{t}{A}\phi +\frac{s^2}{2}\left(\frac{t}{A}\psi +\frac{t}{A^2}{\phi }^2\right),$$

• unlabeled · line 497

  $${\gamma }_s={\iota }_s^{\ast }{g}_s=a^{2}I+s{\gamma }_1+\frac{s^2}{2}{\gamma }_2.$$