Differential operators

Displayed formulas in this passage

• V@1#eq:DK · line 470

  $$D(K\xi )=\square \xi .$$

• V@1#eq:LE · line 476

  $$\begin{array}{rl}(L_{\mathrm{E}}h{)}_{\mu \nu }=& -\frac{1}{2}\square h_{\mu \nu }-\frac{1}{2}{\partial }_{\mu }{\partial }_{\nu }{\mathrm{tr}}_{\eta }h+{\partial }^{\alpha }{\partial }_{(\mu }{h}_{\nu )\alpha }\\ & -\frac{1}{2}{\eta }_{\mu \nu }\left({\partial }^{\alpha }{\partial }^{\beta }{h}_{\alpha \beta }-\square {\mathrm{tr}}_{\eta }h\right).\end{array}$$

• V@1#eq:LEidentities · line 488

  $$L_{\mathrm{E}}K=0,{\partial }^{\mu }(L_{\mathrm{E}}h{)}_{\mu \nu }=0,2\overline{L_{\mathrm{E}}h}=-\square h+KD(h).$$

• V@1#eq:L · line 497

  $$L_h=-h^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }-D(h{)}^{\nu }{\partial }_{\nu }.$$

• V@1#eq:Duhamel · line 503

  $$P{S}_{t_0}(h)\varphi =-L_h\varphi ,S_{t_0}(h)\varphi (t)={{\int }_{t_0}}^t\frac{\sin ((t-s){\Omega }_m)}{{\Omega }_m}{L}_h\varphi (s)\mathrm{d}s.$$

• unlabeled · line 446

  $$\eta =\mathrm{diag}(-1,1,1,1),\square =-{{\partial }_t}^2+\Delta ,P=\square -m^2=-({{\partial }_t}^2+{{\Omega }_m}^2),{\Omega }_m=\sqrt{-\Delta +m^2}.$$

• unlabeled · line 457

  $$(K\xi {)}_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu },X_{\xi }={\xi }^{\mu }{\partial }_{\mu }.$$

• unlabeled · line 463

  $${\bar{h}}_{\mu \nu }=h_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }{\mathrm{tr}}_{\eta }h,D(h{)}^{\nu }={\partial }_{\mu }{\bar{h}}^{\mu \nu }.$$