Curvature and density representative

Displayed formulas in this passage

• X-A-conv@1#eq:ESaction · line 576

  $$S_{\mathrm{ES}}[q,\varphi ]=-\frac{2}{{\kappa }^2}\int d(q)R[q]{\mathrm{d}}^{4}x+\frac{1}{2}\int A(q{)}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi {\mathrm{d}}^{4}x.$$

• X-A-conv@1#eq:gravdensity · line 601

  $${\ell }_{\mathrm{grav}}(\kappa ,h)=-\frac{2}{{\kappa }^2}A(q{)}^{\mu \nu }\left({{\Gamma }^{\rho }}_{\mu \sigma }{{\Gamma }^{\sigma }}_{\nu \rho }-{{\Gamma }^{\rho }}_{\mu \nu }{{\Gamma }^{\sigma }}_{\rho \sigma }\right),$$

• X-A-conv@1#eq:ellseries · line 618

  $$\ell (\kappa ,h,\varphi )={\ell }_{\mathrm{grav}}+{\ell }_{\mathrm{sc}}={\ell }_0+\kappa {\ell }_1+{\kappa }^2{\ell }_2+O({\kappa }^3).$$

• unlabeled · line 562

  $$R_{\mu \nu }={\partial }_{\rho }{{\Gamma }^{\rho }}_{\mu \nu }-{\partial }_{\nu }{{\Gamma }^{\rho }}_{\mu \rho }+{{\Gamma }^{\rho }}_{\rho \lambda }{{\Gamma }^{\lambda }}_{\mu \nu }-{{\Gamma }^{\rho }}_{\nu \lambda }{{\Gamma }^{\lambda }}_{\mu \rho }.$$

• unlabeled · line 571

  $$d(q)=\sqrt{-\det q},A(q{)}^{\mu \nu }=d(q)q^{\mu \nu }.$$

• unlabeled · line 585

  $$T_{\mu \nu }={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -\frac{1}{2}{q}_{\mu \nu }{q}^{\alpha \beta }{\partial }_{\alpha }\varphi {\partial }_{\beta }\varphi ,$$

• unlabeled · line 593

  $$G_{\mu \nu }=\frac{{\kappa }^2}{4}{T}_{\mu \nu },{\square }_q\varphi =0.$$

• unlabeled · line 612

  $${\ell }_{\mathrm{sc}}(\kappa ,h,\varphi )=\frac{1}{2}A(q{)}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi .$$