One full action before expansion

Displayed formulas in this passage

• XI-A-ST@1#eq:curvature · line 524

  $$R_{\mu \nu }={\partial }_{\rho }{{\Gamma }^{\rho }}_{\mu \nu }-{\partial }_{\nu }{{\Gamma }^{\rho }}_{\mu \rho }+{{\Gamma }^{\rho }}_{\rho \lambda }{{\Gamma }^{\lambda }}_{\mu \nu }-{{\Gamma }^{\rho }}_{\nu \lambda }{{\Gamma }^{\lambda }}_{\mu \rho },G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}{q}_{\mu \nu }R.$$

• XI-A-ST@1#eq:EH · line 536

  $$\begin{array}{rl}{S}_{\mathrm{EH}}[q]& =-\frac{2}{{\kappa }^2}{\int }_{M_T}{\rho }_{q}R[q]{\mathrm{d}}^{4}x,\\ {\mathfrak{L}}_q[\varphi ]& =\frac{1}{2}{\rho }_q\left(q^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{\varphi }^2\right),\\ {\mathfrak{L}}_{\eta }[\varphi ]& =\frac{1}{2}\left({\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -m^2{\varphi }^2\right),\\ {S_g}^{\mathrm{sc}}[q,\varphi ]& ={\int }_{M_T}\left((1-g){\mathfrak{L}}_{\eta }[\varphi ]+g{\mathfrak{L}}_q[\varphi ]\right){\mathrm{d}}^{4}x,\\ S_{\mathrm{inv}}& =S_{\mathrm{EH}}+{S_g}^{\mathrm{sc}},S_{\mathrm{tot}}=S_{\mathrm{inv}}+S_{\mathrm{gf}}.\end{array}$$

• XI-A-ST@1#eq:Lq · line 536

  $$\begin{array}{rl}{S}_{\mathrm{EH}}[q]& =-\frac{2}{{\kappa }^2}{\int }_{M_T}{\rho }_{q}R[q]{\mathrm{d}}^{4}x,\\ {\mathfrak{L}}_q[\varphi ]& =\frac{1}{2}{\rho }_q\left(q^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{\varphi }^2\right),\\ {\mathfrak{L}}_{\eta }[\varphi ]& =\frac{1}{2}\left({\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -m^2{\varphi }^2\right),\\ {S_g}^{\mathrm{sc}}[q,\varphi ]& ={\int }_{M_T}\left((1-g){\mathfrak{L}}_{\eta }[\varphi ]+g{\mathfrak{L}}_q[\varphi ]\right){\mathrm{d}}^{4}x,\\ S_{\mathrm{inv}}& =S_{\mathrm{EH}}+{S_g}^{\mathrm{sc}},S_{\mathrm{tot}}=S_{\mathrm{inv}}+S_{\mathrm{gf}}.\end{array}$$

• XI-A-ST@1#eq:scalar-action · line 536

  $$\begin{array}{rl}{S}_{\mathrm{EH}}[q]& =-\frac{2}{{\kappa }^2}{\int }_{M_T}{\rho }_{q}R[q]{\mathrm{d}}^{4}x,\\ {\mathfrak{L}}_q[\varphi ]& =\frac{1}{2}{\rho }_q\left(q^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{\varphi }^2\right),\\ {\mathfrak{L}}_{\eta }[\varphi ]& =\frac{1}{2}\left({\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -m^2{\varphi }^2\right),\\ {S_g}^{\mathrm{sc}}[q,\varphi ]& ={\int }_{M_T}\left((1-g){\mathfrak{L}}_{\eta }[\varphi ]+g{\mathfrak{L}}_q[\varphi ]\right){\mathrm{d}}^{4}x,\\ S_{\mathrm{inv}}& =S_{\mathrm{EH}}+{S_g}^{\mathrm{sc}},S_{\mathrm{tot}}=S_{\mathrm{inv}}+S_{\mathrm{gf}}.\end{array}$$

• XI-A-ST@1#eq:Stot · line 536

  $$\begin{array}{rl}{S}_{\mathrm{EH}}[q]& =-\frac{2}{{\kappa }^2}{\int }_{M_T}{\rho }_{q}R[q]{\mathrm{d}}^{4}x,\\ {\mathfrak{L}}_q[\varphi ]& =\frac{1}{2}{\rho }_q\left(q^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -m^2{\varphi }^2\right),\\ {\mathfrak{L}}_{\eta }[\varphi ]& =\frac{1}{2}\left({\partial }_{\mu }\varphi {\partial }^{\mu }\varphi -m^2{\varphi }^2\right),\\ {S_g}^{\mathrm{sc}}[q,\varphi ]& ={\int }_{M_T}\left((1-g){\mathfrak{L}}_{\eta }[\varphi ]+g{\mathfrak{L}}_q[\varphi ]\right){\mathrm{d}}^{4}x,\\ S_{\mathrm{inv}}& =S_{\mathrm{EH}}+{S_g}^{\mathrm{sc}},S_{\mathrm{tot}}=S_{\mathrm{inv}}+S_{\mathrm{gf}}.\end{array}$$

• XI-A-ST@1#eq:gf · line 580

  $$S_{\mathrm{gf}}[\gamma ]={\int }_{M_T}{D}_{\nu }(\gamma )D^{\nu }(\gamma ){\mathrm{d}}^{4}x.$$

• unlabeled · line 573

  $$\tau ={\eta }^{\mu \nu }{\gamma }_{\mu \nu },{\bar{\gamma }}_{\mu \nu }={\gamma }_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }\tau ,D^{\nu }(\gamma )={\partial }_{\mu }{\bar{\gamma }}^{\mu \nu }.$$