Euler operators and the unchanged matter defect

Displayed formulas in this passage

• XII-C@1#eq:scalar-euler · line 810

  $$E_{\varphi }=-{\partial }_{\mu }\left([(1-g){\eta }^{\mu \nu }+g\sqrt{-q}{q}^{\mu \nu }]{\partial }_{\nu }\varphi \right)-m^2[(1-g)+g\sqrt{-q}]\varphi .$$

• XII-C@1#eq:Z · line 821

  $${{Z_{\sigma }}_{\epsilon }}^{\mu \nu }=w{J}_3^{\mu \nu }+{{{\mathcal{D}}_{\sigma }}_{\epsilon }}^{\mu \nu }.$$

• XII-C@1#eq:eD · line 827

  $${e_{\mathrm{D}}}^{\mu \nu }=\frac{2}{\kappa }\sqrt{-q}{G}^{\mu \nu }-\frac{\kappa }{2}g\sqrt{-q}{T_q}^{\mu \nu }+\kappa {{Z_{\sigma }}_{\epsilon }}^{\mu \nu },$$

• XII-C@1#eq:Noperators · line 852

  $$\begin{array}{rl}{\mathcal{N}}_{\nu }(v)& =2{\partial }_{\mu }(v^{\mu \rho }{q}_{\rho \nu })-v^{\mu \rho }{\partial }_{\nu }{q}_{\mu \rho },\\ N_{0,\nu }(v)& =2{\eta }_{\rho \nu }{\partial }_{\mu }{v}^{\mu \rho },\\ Q_{\nu }(v)& =2{\partial }_{\mu }(v^{\mu \rho }{\gamma }_{\rho \nu })-v^{\mu \rho }{\partial }_{\nu }{\gamma }_{\mu \rho }.\end{array}$$

• XII-C@1#eq:matterdefect · line 866

  $${\mathcal{B}}_{g,\nu }=({\partial }_{\mu }g){T_0}^{\mu }{}_{\nu }-(1-g)(P_m\varphi ){\partial }_{\nu }\varphi +({\partial }_{\nu }g)({\mathfrak{L}}_{\eta }[\varphi ]-{\mathfrak{L}}_q[\varphi ]).$$

• unlabeled · line 803

  $${e_{\mathrm{D}}}^{\mu \nu }=\frac{\delta S_{\mathrm{D}}}{\delta {\gamma }_{\mu \nu }},E_{\varphi }=\frac{\delta S_{\mathrm{D}}}{\delta \varphi }.$$

• unlabeled · line 836

  $${T_q}^{\mu \nu }=q^{\mu \alpha }{q}^{\nu \beta }{\partial }_{\alpha }\varphi {\partial }_{\beta }\varphi -\frac{1}{2}{q}^{\mu \nu }(q^{\alpha \beta }{\partial }_{\alpha }\varphi {\partial }_{\beta }\varphi -m^2{\varphi }^2).$$

• unlabeled · line 844

  $$e_{\mathrm{D}}=A\gamma +\kappa \left(J_{3,\mathrm{D}}-\frac{1}{2}g{T}_0\right)+O({\kappa }^2).$$