Harmonic expansion

Displayed formulas in this passage

• VII-B@1#eq:dcoeff · line 354

  $$\begin{array}{rlrl}{d}_1& =\frac{1}{2}h,& d_2& =\frac{1}{8}{h}^2-\frac{1}{4}{h}_{\rho \sigma }{h}^{\rho \sigma },\\ {A_1}^{\mu \nu }& =\frac{1}{2}h{\eta }^{\mu \nu }-h^{\mu \nu },& & \\ {A_2}^{\mu \nu }& =h^{\mu }{}_{\rho }{h}^{\rho \nu }-\frac{1}{2}h{h}^{\mu \nu }+d_2{\eta }^{\mu \nu }.& & \end{array}$$

• VII-B@1#eq:A1 · line 354

  $$\begin{array}{rlrl}{d}_1& =\frac{1}{2}h,& d_2& =\frac{1}{8}{h}^2-\frac{1}{4}{h}_{\rho \sigma }{h}^{\rho \sigma },\\ {A_1}^{\mu \nu }& =\frac{1}{2}h{\eta }^{\mu \nu }-h^{\mu \nu },& & \\ {A_2}^{\mu \nu }& =h^{\mu }{}_{\rho }{h}^{\rho \nu }-\frac{1}{2}h{h}^{\mu \nu }+d_2{\eta }^{\mu \nu }.& & \end{array}$$

• VII-B@1#eq:A2 · line 354

  $$\begin{array}{rlrl}{d}_1& =\frac{1}{2}h,& d_2& =\frac{1}{8}{h}^2-\frac{1}{4}{h}_{\rho \sigma }{h}^{\rho \sigma },\\ {A_1}^{\mu \nu }& =\frac{1}{2}h{\eta }^{\mu \nu }-h^{\mu \nu },& & \\ {A_2}^{\mu \nu }& =h^{\mu }{}_{\rho }{h}^{\rho \nu }-\frac{1}{2}h{h}^{\mu \nu }+d_2{\eta }^{\mu \nu }.& & \end{array}$$

• VII-B@1#eq:harmonic-density · line 370

  $${\partial }_{\nu }\left(A^{\nu \rho }(\kappa ){\partial }_{\rho }{{\mathsf{X}}^{\mu }}_{\mathbf{g}}\right)=0.$$

• VII-B@1#eq:v-definition · line 389

  $$v^{\mu }[h,h]=-{\mathsf{E}}_0\left\{{\partial }_{\nu }{A_2}^{\nu \mu }+{\partial }_{\nu }\left({A_1}^{\nu \rho }{\partial }_{\rho }{w}^{\mu }\right)\right\}.$$

• VII-B@1#eq:inverse · line 403

  $$({{\mathsf{X}}_{\mathbf{g}}}^{-1}{)}^{\mu }=x^{\mu }-\kappa w^{\mu }+{\kappa }^2\left(w^{\nu }{\partial }_{\nu }{w}^{\mu }-v^{\mu }\right)+O({\kappa }^3).$$

• VII-B@1#eq:B0 · line 412

  $$\begin{array}{rl}{{\mathcal{B}}_f}^{(0)}[h,\varphi ]& =\int f\varphi {\mathrm{d}}^{4}x,\\ {{\mathcal{B}}_f}^{(1)}[h,\varphi ]& =-\int f{w}^{\mu }{\partial }_{\mu }\varphi {\mathrm{d}}^{4}x,\\ {{\mathcal{B}}_f}^{(2)}[h,\varphi ]& =\int f\left[\left(w^{\nu }{\partial }_{\nu }{w}^{\mu }-v^{\mu }\right){\partial }_{\mu }\varphi +\frac{1}{2}{w}^{\mu }{w}^{\nu }{\partial }_{\mu }{\partial }_{\nu }\varphi \right]{\mathrm{d}}^{4}x.\end{array}$$

• VII-B@1#eq:B1 · line 412

  $$\begin{array}{rl}{{\mathcal{B}}_f}^{(0)}[h,\varphi ]& =\int f\varphi {\mathrm{d}}^{4}x,\\ {{\mathcal{B}}_f}^{(1)}[h,\varphi ]& =-\int f{w}^{\mu }{\partial }_{\mu }\varphi {\mathrm{d}}^{4}x,\\ {{\mathcal{B}}_f}^{(2)}[h,\varphi ]& =\int f\left[\left(w^{\nu }{\partial }_{\nu }{w}^{\mu }-v^{\mu }\right){\partial }_{\mu }\varphi +\frac{1}{2}{w}^{\mu }{w}^{\nu }{\partial }_{\mu }{\partial }_{\nu }\varphi \right]{\mathrm{d}}^{4}x.\end{array}$$

• VII-B@1#eq:B2 · line 412

  $$\begin{array}{rl}{{\mathcal{B}}_f}^{(0)}[h,\varphi ]& =\int f\varphi {\mathrm{d}}^{4}x,\\ {{\mathcal{B}}_f}^{(1)}[h,\varphi ]& =-\int f{w}^{\mu }{\partial }_{\mu }\varphi {\mathrm{d}}^{4}x,\\ {{\mathcal{B}}_f}^{(2)}[h,\varphi ]& =\int f\left[\left(w^{\nu }{\partial }_{\nu }{w}^{\mu }-v^{\mu }\right){\partial }_{\mu }\varphi +\frac{1}{2}{w}^{\mu }{w}^{\nu }{\partial }_{\mu }{\partial }_{\nu }\varphi \right]{\mathrm{d}}^{4}x.\end{array}$$

• unlabeled · line 345

  $$\mathbf{g}=\eta +\kappa h.$$

• unlabeled · line 349

  $$d(\kappa )=\sqrt{-\det (\eta +\kappa h)},A^{\mu \nu }(\kappa )=d(\kappa ){\mathbf{g}}^{\mu \nu }.$$

• unlabeled · line 377

  $${{\mathsf{X}}^{\mu }}_{\mathbf{g}}{|}_{{\Sigma }_0}=x^{\mu }{|}_{{\Sigma }_0},{\partial }_t{{\mathsf{X}}^{\mu }}_{\mathbf{g}}{|}_{{\Sigma }_0}={\partial }_t{x}^{\mu }{|}_{{\Sigma }_0}.$$

• unlabeled · line 384

  $${{\mathsf{X}}^{\mu }}_{\mathbf{g}}=x^{\mu }+\kappa w^{\mu }+{\kappa }^2{v}^{\mu }+O({\kappa }^3)$$