Explicit coefficients

Displayed formulas in this passage

• X-A-conv@1#eq:bcoeff · line 632

  $$\begin{array}{rlrl}{b}_1(a)& =\frac{1}{2}\mathrm{tr}a,& b_2(a)& =\frac{1}{8}(\mathrm{tr}a{)}^2-\frac{1}{4}{a}_{\alpha \beta }{a}^{\alpha \beta },\\ a_1(a{)}^{\mu \nu }& =\frac{1}{2}(\mathrm{tr}a){\eta }^{\mu \nu }-a^{\mu \nu },& & \\ a_2(a{)}^{\mu \nu }& =a^{\mu }{}_{\alpha }{a}^{\alpha \nu }-\frac{1}{2}(\mathrm{tr}a)a^{\mu \nu }+b_2(a){\eta }^{\mu \nu }.& & \end{array}$$

• X-A-conv@1#eq:aone · line 632

  $$\begin{array}{rlrl}{b}_1(a)& =\frac{1}{2}\mathrm{tr}a,& b_2(a)& =\frac{1}{8}(\mathrm{tr}a{)}^2-\frac{1}{4}{a}_{\alpha \beta }{a}^{\alpha \beta },\\ a_1(a{)}^{\mu \nu }& =\frac{1}{2}(\mathrm{tr}a){\eta }^{\mu \nu }-a^{\mu \nu },& & \\ a_2(a{)}^{\mu \nu }& =a^{\mu }{}_{\alpha }{a}^{\alpha \nu }-\frac{1}{2}(\mathrm{tr}a)a^{\mu \nu }+b_2(a){\eta }^{\mu \nu }.& & \end{array}$$

• X-A-conv@1#eq:atwo · line 632

  $$\begin{array}{rlrl}{b}_1(a)& =\frac{1}{2}\mathrm{tr}a,& b_2(a)& =\frac{1}{8}(\mathrm{tr}a{)}^2-\frac{1}{4}{a}_{\alpha \beta }{a}^{\alpha \beta },\\ a_1(a{)}^{\mu \nu }& =\frac{1}{2}(\mathrm{tr}a){\eta }^{\mu \nu }-a^{\mu \nu },& & \\ a_2(a{)}^{\mu \nu }& =a^{\mu }{}_{\alpha }{a}^{\alpha \nu }-\frac{1}{2}(\mathrm{tr}a)a^{\mu \nu }+b_2(a){\eta }^{\mu \nu }.& & \end{array}$$

• X-A-conv@1#eq:gravzero · line 682

  $$\begin{array}{rl}{\ell }_{\mathrm{grav},0}& =-2{\eta }^{\mu \nu }{\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_1),\\ {\ell }_{\mathrm{grav},1}& =-2[a_1(h{)}^{\mu \nu }{\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_1)\\ & +{\eta }^{\mu \nu }({\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_2)+{\mathcal{Q}}_{\mu \nu }({\gamma }_2,{\gamma }_1))],\\ {\ell }_{\mathrm{grav},2}& =-2[a_2(h{)}^{\mu \nu }{\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_1)\\ & +a_1(h{)}^{\mu \nu }({\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_2)+{\mathcal{Q}}_{\mu \nu }({\gamma }_2,{\gamma }_1))\\ & +{\eta }^{\mu \nu }({\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_3)+{\mathcal{Q}}_{\mu \nu }({\gamma }_2,{\gamma }_2)+{\mathcal{Q}}_{\mu \nu }({\gamma }_3,{\gamma }_1))].\end{array}$$

• X-A-conv@1#eq:gravone · line 682

  $$\begin{array}{rl}{\ell }_{\mathrm{grav},0}& =-2{\eta }^{\mu \nu }{\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_1),\\ {\ell }_{\mathrm{grav},1}& =-2[a_1(h{)}^{\mu \nu }{\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_1)\\ & +{\eta }^{\mu \nu }({\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_2)+{\mathcal{Q}}_{\mu \nu }({\gamma }_2,{\gamma }_1))],\\ {\ell }_{\mathrm{grav},2}& =-2[a_2(h{)}^{\mu \nu }{\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_1)\\ & +a_1(h{)}^{\mu \nu }({\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_2)+{\mathcal{Q}}_{\mu \nu }({\gamma }_2,{\gamma }_1))\\ & +{\eta }^{\mu \nu }({\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_3)+{\mathcal{Q}}_{\mu \nu }({\gamma }_2,{\gamma }_2)+{\mathcal{Q}}_{\mu \nu }({\gamma }_3,{\gamma }_1))].\end{array}$$

• X-A-conv@1#eq:gravtwo · line 682

  $$\begin{array}{rl}{\ell }_{\mathrm{grav},0}& =-2{\eta }^{\mu \nu }{\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_1),\\ {\ell }_{\mathrm{grav},1}& =-2[a_1(h{)}^{\mu \nu }{\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_1)\\ & +{\eta }^{\mu \nu }({\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_2)+{\mathcal{Q}}_{\mu \nu }({\gamma }_2,{\gamma }_1))],\\ {\ell }_{\mathrm{grav},2}& =-2[a_2(h{)}^{\mu \nu }{\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_1)\\ & +a_1(h{)}^{\mu \nu }({\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_2)+{\mathcal{Q}}_{\mu \nu }({\gamma }_2,{\gamma }_1))\\ & +{\eta }^{\mu \nu }({\mathcal{Q}}_{\mu \nu }({\gamma }_1,{\gamma }_3)+{\mathcal{Q}}_{\mu \nu }({\gamma }_2,{\gamma }_2)+{\mathcal{Q}}_{\mu \nu }({\gamma }_3,{\gamma }_1))].\end{array}$$

• unlabeled · line 648

  $$d(\eta +\kappa h)=1+\kappa b_1(h)+{\kappa }^2{b}_2(h)+O({\kappa }^3),$$

• unlabeled · line 652

  $$A(\eta +\kappa h)=\eta +\kappa a_1(h)+{\kappa }^2{a}_2(h)+O({\kappa }^3).$$

• unlabeled · line 659

  $$T_{\lambda \mu \nu }(h)={\partial }_{\mu }{h}_{\lambda \nu }+{\partial }_{\nu }{h}_{\lambda \mu }-{\partial }_{\lambda }{h}_{\mu \nu },$$

• unlabeled · line 666

  $${{\gamma }_r}^{\rho }{}_{\mu \nu }=\frac{1}{2}(-1{)}^{r-1}(h^{r-1}{)}^{\rho }{}_{\sigma }{\eta }^{\sigma \lambda }{T}_{\lambda \mu \nu }(h),r=1,2,3,$$

• unlabeled · line 675

  $${\mathcal{Q}}_{\mu \nu }(P,Q)=P^{\rho }{}_{\mu \sigma }{Q}^{\sigma }{}_{\nu \rho }-P^{\rho }{}_{\mu \nu }{Q}^{\sigma }{}_{\rho \sigma }.$$

• unlabeled · line 718

  $${\ell }_{\mathrm{sc},0}=\frac{1}{2}{\eta }^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi ,{\ell }_{\mathrm{sc},j}=\frac{1}{2}{a}_j(h{)}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi ,j=1,2.$$