Residual wave gauges

Displayed formulas in this passage

• III-B@1#eq:gauge-data · line 1179

  $$\begin{array}{rlrl}\delta A_{00}& =2b_0,& \delta B_{00}& =2\Delta a_0,\\ \delta A_{0i}& =b_i+{\partial }_i{a}_0,& \delta B_{0i}& =\Delta a_i+{\partial }_i{b}_0,\\ \delta A_{ij}& ={\partial }_i{a}_j+{\partial }_j{a}_i,& \delta B_{ij}& ={\partial }_i{b}_j+{\partial }_j{b}_i,\end{array}$$

• III-B@1#eq:scalar-data · line 1189

  $$\delta u=a^{\mu }({\partial }_{\mu }\Phi {)}_0,\delta v=b^{\mu }({\partial }_{\mu }\Phi {)}_0+a^{\mu }({\partial }_s{\partial }_{\mu }\Phi {)}_0.$$

• unlabeled · line 1160

  $${\mathcal{G}}_{r_{\ast }}={C^{\infty }}_{{\overline{B}}_{r_{\ast }}}({\mathbb{R}}^3;{\mathbb{R}}^4)\oplus {C^{\infty }}_{{\overline{B}}_{r_{\ast }}}({\mathbb{R}}^3;{\mathbb{R}}^4)$$

• unlabeled · line 1168

  $$\square {\xi }_{\mu }=0,{\xi }_{\mu }{|}_0=a_{\mu },{\partial }_s{\xi }_{\mu }{|}_0=b_{\mu }.$$

• unlabeled · line 1173

  $$(h,\phi )⟼(h+K\xi ,\phi +X_{\xi }\Phi ).$$