A pure-trace layer check

Displayed formulas in this passage

• unlabeled · line 2190

  $$u_1=-\psi {\eta }^{-1},u_2={\psi }^2{\eta }^{-1},b_1=2\psi ,b_2={\psi }^2.$$

• unlabeled · line 2195

  $${\Gamma }_2=-\psi {\Gamma }_1,{{\Gamma }_1}^{\rho }{}_{\mu \nu }=\frac{1}{2}\left({{\delta }^{\rho }}_{\nu }{\partial }_{\mu }\psi +{{\delta }^{\rho }}_{\mu }{\partial }_{\nu }\psi -{\eta }_{\mu \nu }{\partial }^{\rho }\psi \right).$$

• unlabeled · line 2206

  $${{\mathcal{D}}_{\sigma }}^{\mu \nu }[\psi \eta ]=-\psi \left[{\eta }^{\mu \nu }({\partial }_{\rho }\sigma ){\partial }^{\rho }\psi +({\partial }^{\mu }\sigma ){\partial }^{\nu }\psi +({\partial }^{\nu }\sigma ){\partial }^{\mu }\psi \right],$$

• unlabeled · line 2216

  $${\eta }_{\mu \nu }{{\mathcal{D}}_{\sigma }}^{\mu \nu }[\psi \eta ]=-6({\partial }_{\rho }\sigma )\psi {\partial }^{\rho }\psi .$$

• unlabeled · line 2222

  $${\eta }_{\mu \nu }D{{\mathcal{D}}_{\sigma }}^{\mu \nu }[\psi \eta ](\zeta \eta )=-6({\partial }_{\rho }\sigma )\left(\zeta {\partial }^{\rho }\psi +\psi {\partial }^{\rho }\zeta \right).$$